Business Calculus Final Exam Study Guide – Step-by-Step Guidance

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Find the limit."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Limits"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to evaluate limits, a foundational concept in calculus. Limits are used to describe the behavior of a function as the input approaches a certain value."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Limit:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a} f(x)"}},{"type":"text","text":" describes the value that "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" approaches as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Direct Substitution:"},{"type":"text","text":" Try plugging in the value directly, if possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Factoring/Simplifying:"},{"type":"text","text":" If direct substitution leads to an indeterminate form (like "},{"type":"inlineMath","attrs":{"latex":"0/0"}},{"type":"text","text":"), try to factor or simplify the expression."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the function and the value that "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" is approaching."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Attempt direct substitution: Plug the value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" into the function to see if you get a defined value."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If you get an indeterminate form, look for ways to factor, simplify, or use algebraic manipulation to resolve the indeterminate form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Once simplified, substitute the value again to see if the limit can be evaluated."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Find "},{"type":"inlineMath","attrs":{"latex":"\\frac{dy}{dx}"}},{"type":"text","marks":[{"type":"bold"}],"text":" for the indicated function "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Differentiation"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question is about finding the derivative of a function, which measures the rate at which "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" changes with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Derivative:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{dy}{dx}"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"y'"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Power Rule:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx} x^n = n x^{n-1}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Sum Rule:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write out the function "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" clearly."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify which differentiation rules apply (power, product, quotient, chain, etc.)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the appropriate rule to each term in the function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the resulting expression as much as possible."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Find the indicated derivatives:"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Differentiation (Product, Chain, and Power Rules)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These questions require you to find the derivative of given functions, possibly involving roots, products, or quotients."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Product Rule:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Chain Rule:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx} f(g(x)) = f'(g(x)) \\cdot g'(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Power Rule:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx} x^n = n x^{n-1}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite any roots or radicals as fractional exponents for easier differentiation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify if the function is a product, quotient, or composition, and select the appropriate rule."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the differentiation rule step by step to each part of the function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify your result as much as possible."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. The total profit from the sale of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" lawn mowers is "},{"type":"inlineMath","attrs":{"latex":"P(x) = 20x – 0.02x^2 – 320"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"0 \\leq x \\leq 1,000"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(A) Find the average profit per mower if 40 mowers are produced."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(B) Find the marginal average profit at a production level of 40 mowers and interpret the results."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Average and Marginal Analysis"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute average profit and marginal average profit, which are important in business calculus for understanding cost and profit behavior."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Average Profit:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\text{Average Profit} = \\frac{P(x)}{x}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Marginal Average Profit:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx}\\left(\\frac{P(x)}{x}\\right)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (A): Substitute "},{"type":"inlineMath","attrs":{"latex":"x = 40"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"P(x)"}},{"type":"text","text":" to find total profit, then divide by 40 to get average profit."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (B): Express average profit as "},{"type":"inlineMath","attrs":{"latex":"A(x) = \\frac{P(x)}{x}"}},{"type":"text","text":", then differentiate "},{"type":"inlineMath","attrs":{"latex":"A(x)"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the marginal average profit function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x = 40"}},{"type":"text","text":" into the marginal average profit function to find its value at that production level."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret what the marginal average profit means in this context (i.e., how average profit changes as production increases from 40 mowers)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. A family paid "},{"type":"inlineMath","attrs":{"latex":"99,000"}},{"type":"text","marks":[{"type":"bold"}],"text":" cash for a house. Fifteen years later, the house was sold for "},{"type":"inlineMath","attrs":{"latex":"195,000"}},{"type":"text","marks":[{"type":"bold"}],"text":". If interest is compounded continuously, what annual nominal rate of interest did the original $99,000$ investment earn?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Exponential Growth and Continuous Compounding"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the formula for continuous compounding to solve for the interest rate."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Continuous Compounding Formula:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"A = Pe^{rt}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"A"}},{"type":"text","text":" = final amount, "},{"type":"inlineMath","attrs":{"latex":"P"}},{"type":"text","text":" = principal, "},{"type":"inlineMath","attrs":{"latex":"r"}},{"type":"text","text":" = annual nominal rate, "},{"type":"inlineMath","attrs":{"latex":"t"}},{"type":"text","text":" = time in years"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the known values: "},{"type":"inlineMath","attrs":{"latex":"A = 195,000"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"P = 99,000"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"t = 15"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the formula: "},{"type":"inlineMath","attrs":{"latex":"A = Pe^{rt}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":"P"}},{"type":"text","text":" to isolate the exponential: "},{"type":"inlineMath","attrs":{"latex":"\\frac{A}{P} = e^{rt}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take the natural logarithm of both sides to solve for "},{"type":"inlineMath","attrs":{"latex":"r"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. The estimated salvage value "},{"type":"inlineMath","attrs":{"latex":"R"}},{"type":"text","marks":[{"type":"bold"}],"text":" (in dollars) of a company airplane after "},{"type":"inlineMath","attrs":{"latex":"t"}},{"type":"text","marks":[{"type":"bold"}],"text":" years is given by "},{"type":"inlineMath","attrs":{"latex":"R(t) = 20,000(0.86)^t"}},{"type":"text","marks":[{"type":"bold"}],"text":". What is the rate of depreciation (in dollars per year) after 1 year? 2 years?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Exponential Decay and Differentiation"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to find the rate at which the value of the airplane is decreasing at specific times, which involves finding the derivative of an exponential function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Derivative of Exponential Function:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dt} a^t = a^t \\ln a"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Rate of Depreciation:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"R'(t)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the function: "},{"type":"inlineMath","attrs":{"latex":"R(t) = 20,000(0.86)^t"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the derivative "},{"type":"inlineMath","attrs":{"latex":"R'(t)"}},{"type":"text","text":" using the formula for the derivative of an exponential function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"t = 1"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"t = 2"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"R'(t)"}},{"type":"text","text":" to find the rate of depreciation at those times."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. A research group developed the model "},{"type":"inlineMath","attrs":{"latex":"P(x) = 40 + 25 \\ln(x + 1)"}},{"type":"text","marks":[{"type":"bold"}],"text":" for systolic blood pressure, where "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" is age in years. What is the rate of change of pressure at the end of 10 years? At the end of 30 years?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Differentiation of Logarithmic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to find the rate of change (derivative) of a function involving a natural logarithm, and then evaluate it at specific values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Derivative of "},{"type":"inlineMath","attrs":{"latex":"\\ln(u)"}},{"type":"text","marks":[{"type":"bold"}],"text":":"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx} \\ln(u) = \\frac{1}{u} \\cdot u'"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the function: "},{"type":"inlineMath","attrs":{"latex":"P(x) = 40 + 25 \\ln(x + 1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the derivative "},{"type":"inlineMath","attrs":{"latex":"P'(x)"}},{"type":"text","text":" using the chain rule."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x = 10"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = 30"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"P'(x)"}},{"type":"text","text":" to find the rate of change at those ages."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. The total weekly cost of producing "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" tanks is "},{"type":"inlineMath","attrs":{"latex":"C(x) = 10,000 + 90x – 0.05x^2"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) Find the marginal cost function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) Find the marginal cost at a production level of 500 tanks per week."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c) Interpret the results of part (b)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Marginal Analysis"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the marginal cost (the derivative of the cost function) and interpret its meaning in a business context."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Marginal Cost:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"C'(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Derivative:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx} C(x)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Differentiate "},{"type":"inlineMath","attrs":{"latex":"C(x)"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"C'(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x = 500"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"C'(x)"}},{"type":"text","text":" to find the marginal cost at that production level."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret what the marginal cost means (i.e., the approximate cost of producing one more tank when 500 are already being produced)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Find "},{"type":"inlineMath","attrs":{"latex":"f'"}},{"type":"text","marks":[{"type":"bold"}],"text":" and simplify."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{1}{x^2}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"f(x) = (x^2 + 2)(x^2 - 3)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Differentiation (Power and Product Rules)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to find and simplify the derivative of given functions, using the power rule and product rule as needed."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Power Rule:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx} x^n = n x^{n-1}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Product Rule:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (a): Rewrite "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{x^2}"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"x^{-2}"}},{"type":"text","text":" to use the power rule."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (b): Apply the product rule to "},{"type":"inlineMath","attrs":{"latex":"(x^2 + 2)(x^2 - 3)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the resulting expressions as much as possible."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. Find the intervals on which "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" is increasing, decreasing, and the local extrema."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"f(x) = 3x^2 – 12x + 8"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"f(x) = x^3 + 5x + 5"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: First Derivative Test"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the first derivative to determine where a function is increasing or decreasing and to find local maxima and minima."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Critical Points:"},{"type":"text","text":" Where "},{"type":"inlineMath","attrs":{"latex":"f'(x) = 0"}},{"type":"text","text":" or is undefined."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"First Derivative Test:"},{"type":"text","text":" Analyze the sign of "},{"type":"inlineMath","attrs":{"latex":"f'(x)"}},{"type":"text","text":" to determine intervals of increase/decrease."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"f'(x)"}},{"type":"text","text":" for each function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve "},{"type":"inlineMath","attrs":{"latex":"f'(x) = 0"}},{"type":"text","text":" to find critical points."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Test intervals around the critical points to determine where "},{"type":"inlineMath","attrs":{"latex":"f'(x)"}},{"type":"text","text":" is positive (increasing) or negative (decreasing)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the results to identify local maxima and minima."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q11. Find the intervals on which the graph of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":" is concave upward, concave downward, and the inflection points."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"f(x) = x^3 - 3x^2 + 7x + 2"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Second Derivative Test and Concavity"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to use the second derivative to determine concavity and inflection points."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Second Derivative:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f''(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Concave Up:"},{"type":"text","text":" Where "},{"type":"inlineMath","attrs":{"latex":"f''(x) > 0"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Concave Down:"},{"type":"text","text":" Where "},{"type":"inlineMath","attrs":{"latex":"f''(x) < 0"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Inflection Point:"},{"type":"text","text":" Where "},{"type":"inlineMath","attrs":{"latex":"f''(x)"}},{"type":"text","text":" changes sign"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"f''(x)"}},{"type":"text","text":" by differentiating "},{"type":"inlineMath","attrs":{"latex":"f'(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve "},{"type":"inlineMath","attrs":{"latex":"f''(x) = 0"}},{"type":"text","text":" to find possible inflection points."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Test intervals around these points to determine where "},{"type":"inlineMath","attrs":{"latex":"f''(x)"}},{"type":"text","text":" is positive or negative."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the coordinates of the inflection points by substituting "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" values into "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. The monthly marginal profit is "},{"type":"inlineMath","attrs":{"latex":"P'(x) = 165 – 0.1x"}},{"type":"text","marks":[{"type":"bold"}],"text":". The company is currently manufacturing 1,500 HDTVs per month but is planning to increase production to 1,600 HDTVs. 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Find the absolute maximum and minimum of each function on the indicated intervals."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2x^2 – 4x"}},{"type":"text","text":" on [0, 2] and [2, 5]"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"f(x) = - x^2 + 8x"}},{"type":"text","text":" on [0, 5] and [5, 8]"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Extreme Value Theorem"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find absolute extrema on a closed interval by evaluating the function at critical points and endpoints."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Critical Points:"},{"type":"text","text":" Where "},{"type":"inlineMath","attrs":{"latex":"f'(x) = 0"}},{"type":"text","text":" or is undefined within the interval."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Absolute Maximum/Minimum:"},{"type":"text","text":" Largest/smallest value of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" on the interval."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"f'(x)"}},{"type":"text","text":" and solve "},{"type":"inlineMath","attrs":{"latex":"f'(x) = 0"}},{"type":"text","text":" to find critical points within the interval."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" at the critical points and at the endpoints of the interval."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare the values to determine the absolute maximum and minimum."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. Find the particular anti-derivative of "},{"type":"inlineMath","attrs":{"latex":"R'(x) = 500 – 0.4x"}},{"type":"text","marks":[{"type":"bold"}],"text":" that satisfies "},{"type":"inlineMath","attrs":{"latex":"R(0) = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Antiderivatives and Initial Conditions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to find an antiderivative (indefinite integral) and use an initial condition to solve for the constant of integration."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Antiderivative:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\int f(x) dx"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Initial Condition:"},{"type":"text","text":" Use "},{"type":"inlineMath","attrs":{"latex":"R(0) = 0"}},{"type":"text","text":" to solve for "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integrate "},{"type":"inlineMath","attrs":{"latex":"R'(x)"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"R(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the constant of integration "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"R(0) = 0"}},{"type":"text","text":" to solve for "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. Find the area bounded by the graphs of "},{"type":"inlineMath","attrs":{"latex":"y = x^2 – 4"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"y = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":", over "},{"type":"inlineMath","attrs":{"latex":"0 \\leq x \\leq 3"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Area Between Curves"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to set up and evaluate a definite integral to find the area between a curve and the x-axis."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Area:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\int_{a}^{b} |f(x)| dx"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the definite integral for the area: "},{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{3} [0 - (x^2 - 4)] dx"}},{"type":"text","text":" (if "},{"type":"inlineMath","attrs":{"latex":"x^2 - 4"}},{"type":"text","text":" is below the x-axis)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the integrand and integrate with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate the definite integral at the upper and lower limits."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. The rate of change of the income produced by a vending machine is given by "},{"type":"inlineMath","attrs":{"latex":"I'(t)"}},{"type":"text","marks":[{"type":"bold"}],"text":", where "},{"type":"inlineMath","attrs":{"latex":"t"}},{"type":"text","marks":[{"type":"bold"}],"text":" is time in years. Find the total income produced during the first five years."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Definite Integrals and Accumulated Change"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to find the total accumulated income by integrating the rate function over a given interval."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Total Income:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\int_{0}^{5} I'(t) dt"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the definite integral from "},{"type":"inlineMath","attrs":{"latex":"t = 0"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"t = 5"}},{"type":"text","text":" of "},{"type":"inlineMath","attrs":{"latex":"I'(t)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integrate "},{"type":"inlineMath","attrs":{"latex":"I'(t)"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"t"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate the result at the upper and lower limits and subtract."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Using integration by parts, find the indicated integrals."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Integration by Parts"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the integration by parts formula to evaluate integrals involving products of functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Integration by Parts:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\int u dv = uv - \\int v du"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify "},{"type":"inlineMath","attrs":{"latex":"u"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"dv"}},{"type":"text","text":" in the integrand."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"du"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"v"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the integration by parts formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the resulting expression and, if necessary, integrate again."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. The total cost of manufacturing "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" auto body frames is "},{"type":"inlineMath","attrs":{"latex":"C(x) = 60,000 + 300x"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(A) Find the average cost per unit if 500 frames are produced."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(B) Find the marginal average cost at a production level of 500 units and interpret the results."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(C) Use the results from (A) and (B) to estimate the average cost per frame if 501 frames are produced."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Average and Marginal Cost Analysis"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to compute average cost, marginal average cost, and use them to estimate the effect of producing one more unit."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Average Cost:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{C(x)}{x}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Marginal Average Cost:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{d}{dx}\\left(\\frac{C(x)}{x}\\right)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (A): Substitute "},{"type":"inlineMath","attrs":{"latex":"x = 500"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"C(x)"}},{"type":"text","text":" and divide by 500."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (B): Differentiate "},{"type":"inlineMath","attrs":{"latex":"A(x) = \\frac{C(x)}{x}"}},{"type":"text","text":" to find the marginal average cost function, then substitute "},{"type":"inlineMath","attrs":{"latex":"x = 500"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (C): Use the marginal average cost to estimate the average cost for 501 units."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. A company manufactures and sells "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" smartphones per week. The weekly price-demand and cost equations are "},{"type":"inlineMath","attrs":{"latex":"p=500 – 0.4x"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"C(x)=20,000 + 20x"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) What price should the company charge, and how many phones should be produced to maximize weekly revenue? What is the maximum weekly revenue?"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) What is the maximum weekly profit? How much should the company charge, and how many phones should be produced to realize the maximum weekly profit?"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Revenue and Profit Maximization"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and solve optimization problems involving revenue and profit functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Revenue:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"R(x) = p \\cdot x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Profit:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"P(x) = R(x) - C(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Maximization:"},{"type":"text","text":" Find "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"R'(x) = 0"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"P'(x) = 0"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (a): Express revenue as a function of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"R(x) = (500 - 0.4x)x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"R'(x)"}},{"type":"text","text":" and set it to zero to find the value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that maximizes revenue."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute this "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" back into the price equation to find the price to charge."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (b): Write the profit function "},{"type":"inlineMath","attrs":{"latex":"P(x) = R(x) - C(x)"}},{"type":"text","text":", find "},{"type":"inlineMath","attrs":{"latex":"P'(x)"}},{"type":"text","text":", and set it to zero to find the maximizing "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute this "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" into the price equation and profit function as needed."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q20. Find each indefinite integral and check the result by differentiating."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Basic Integration"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to find antiderivatives (indefinite integrals) and verify your results by differentiation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Power Rule for Integration:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\int x^n dx = \\frac{x^{n+1}}{n+1} + C"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"n \\neq -1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Check by Differentiation:"},{"type":"text","text":" Differentiate your answer to verify it matches the original integrand."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the integrand if necessary to use the power rule."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integrate each term separately."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the constant of integration "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Differentiate your result to check your work."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q21. The weekly marginal cost of producing "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" pairs of tennis shoes is "},{"type":"inlineMath","attrs":{"latex":"C' = 12 + \\frac{6}{\\sqrt{x}}"}},{"type":"text","marks":[{"type":"bold"}],"text":". If the fixed costs are "},{"type":"inlineMath","attrs":{"latex":"2,000"}},{"type":"text","marks":[{"type":"bold"}],"text":" per week, find the cost function. What is the average cost per pair if 1,000 pairs are produced each week?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Integration and Average Cost"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to find the total cost function by integrating the marginal cost and then compute average cost for a given production level."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Cost Function:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"C(x) = \\int C'(x) dx + C_0"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Average Cost:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{C(x)}{x}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Integrate "},{"type":"inlineMath","attrs":{"latex":"C'(x)"}},{"type":"text","text":" with respect to "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"C(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the fixed cost as the constant of integration."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x = 1,000"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"C(x)"}},{"type":"text","text":" and divide by 1,000 to find the average cost."