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Business Calculus Exam 1 Review – Step-by-Step Guidance

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Q1a. Write a linear demand function for a product where each $1 increase in price decreases quantity demanded by 5, and at $200, 50 items are sold.

Background

Topic: Linear Demand Functions

This question tests your ability to model demand as a linear function, using information about how price changes affect quantity demanded and a specific price-quantity pair.

Key Terms and Formulas

  • Linear function: , where is quantity, is price, is the slope, and is the intercept.

  • Slope (): Change in per unit change in .

Step-by-Step Guidance

  1. Recognize that the demand function is linear: .

  2. Use the information: For every pqmm = -5$.

  3. Plug in the known point: When , . Substitute these values into the equation to solve for .

  4. Set up the equation: .

  5. Now, solve for to complete the function.

Try solving on your own before revealing the answer!

Final Answer:

We found the slope is , and plugging in , gives . So the demand function is .

Q1b. Given supply and your demand function, find the equilibrium price and quantity.

Background

Topic: Market Equilibrium

This question tests your ability to find the price and quantity where supply equals demand.

Key Terms and Formulas

  • Equilibrium: The point where .

  • Set the demand and supply equations equal and solve for .

Step-by-Step Guidance

  1. Write the demand and supply equations: and .

  2. Set them equal: .

  3. Combine like terms to isolate on one side.

  4. Solve for .

  5. Once you have , substitute it back into either equation to find .

Try solving on your own before revealing the answer!

Final Answer: ,

Setting gives . Plugging $p = 10$ into either equation gives .

Q2a. $1200 annual interest. Compute the balance after $10$ years if compounded annually.

Background

Topic: Compound Interest (Annual Compounding)

This question tests your ability to use the compound interest formula for annual compounding.

Key Terms and Formulas

  • Compound interest formula:

  • = principal ($1200r), = number of times compounded per year ($1$), $t)

Step-by-Step Guidance

  1. Identify the values: , , , .

  2. Plug these into the formula: .

  3. Simplify inside the parentheses: .

  4. Raise the result to the $10$th power.

  5. Multiply by $1200$ to get the final amount.

Try solving on your own before revealing the answer!

Final Answer: dollars

After 10 years, the account balance is approximately .

Q2b. $1200 annual interest, compounded monthly. Find the balance after $10$ years.

Background

Topic: Compound Interest (Monthly Compounding)

This question tests your ability to use the compound interest formula with monthly compounding.

Key Terms and Formulas

  • Compound interest formula:

  • (monthly compounding)

Step-by-Step Guidance

  1. Identify the values: , , , .

  2. Plug into the formula: .

  3. Simplify and add to $1$.

  4. Calculate the exponent .

  5. Raise the base to the calculated exponent and multiply by $1200$.

Try solving on your own before revealing the answer!

Final Answer: dollars

Monthly compounding yields a slightly higher balance than annual compounding.

Q2c. $1200 annual interest, compounded continuously. Find the balance after $10$ years.

Background

Topic: Continuous Compounding

This question tests your ability to use the continuous compounding formula.

Key Terms and Formulas

  • Continuous compounding formula:

  • , ,

Step-by-Step Guidance

  1. Identify the values: , , .

  2. Plug into the formula: .

  3. Calculate the exponent .

  4. Compute raised to that exponent.

  5. Multiply by $1200$ to get the final amount.

Try solving on your own before revealing the answer!

Final Answer: dollars

Continuous compounding gives the highest balance among the three methods.

Q3. How long does it take for $1000 at annual interest compounded continuously?

Background

Topic: Solving for Time in Continuous Compounding

This question tests your ability to solve for time in the continuous compounding formula.

Key Terms and Formulas

  • Continuous compounding formula:

  • To solve for , use logarithms:

Step-by-Step Guidance

  1. Write the formula: .

  2. Divide both sides by $1000.

  3. Take the natural logarithm of both sides: .

  4. Solve for by dividing both sides by .

Try solving on your own before revealing the answer!

Final Answer: years

It takes about 32.19 years for the investment to grow from $1000 at compounded continuously.

Q4a. For and , identify the fixed cost, variable cost, and price per unit.

Background

Topic: Cost and Revenue Functions

This question tests your understanding of the components of cost and revenue functions.

Key Terms and Formulas

  • Fixed cost: The constant term in .

  • Variable cost: The term multiplied by in .

  • Price per unit: The coefficient of in .

Step-by-Step Guidance

  1. Identify the fixed cost as the constant in .

  2. Identify the variable cost as the coefficient of in .

  3. Identify the price per unit as the coefficient of in .

Try solving on your own before revealing the answer!

Final Answer: Fixed cost = $5000, Price per unit = $4$

The fixed cost is the constant in the cost function, the variable cost is the term, and the price per unit is the coefficient in the revenue function.

Q4b. Find the profit function and interpret .

Background

Topic: Profit Functions

This question tests your ability to construct and interpret a profit function from cost and revenue functions.

Key Terms and Formulas

  • Profit function:

Step-by-Step Guidance

  1. Write the profit function as .

  2. Substitute and into the formula.

  3. Simplify the expression to combine like terms.

  4. To interpret , substitute into your profit function and simplify.

Try solving on your own before revealing the answer!

Final Answer: ;

The profit function is . At , the profit is , meaning a loss of $4980$ when selling 10 units.

Q4c. Find the break-even point for the profit function.

Background

Topic: Break-Even Analysis

This question tests your ability to find the quantity where profit is zero.

Key Terms and Formulas

  • Break-even point: Set and solve for .

Step-by-Step Guidance

  1. Set the profit function equal to zero: .

  2. Solve for by isolating it on one side.

Try solving on your own before revealing the answer!

Final Answer:

The break-even point is at units, where profit is zero.

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