Q1. Evaluate

Background

Topic: Limits and Continuity

This question tests your understanding of how to evaluate limits, especially when direct substitution leads to an indeterminate form (like ).

Key Terms and Formulas:

• Limit: The value a function approaches as the input approaches a certain point.

• Indeterminate Form: An expression like that requires algebraic manipulation to resolve.

Step-by-Step Guidance

1. Substitute into the expression to check if it results in an indeterminate form.

2. If you get , factor the numerator to see if you can simplify the expression.

3. Cancel any common factors in the numerator and denominator.

4. Once simplified, substitute again to find the limit.

Try solving on your own before revealing the answer!

Q2. Is continuous at ? Justify your answer.

Background

Topic: Continuity of Piecewise Functions

This question tests your ability to determine continuity at a point for a piecewise function by checking the value of the function and the limit from both sides.

Key Terms and Formulas:

• Continuity: A function is continuous at if .

• Piecewise Function: A function defined by different expressions for different intervals.

Step-by-Step Guidance

1. Find using the definition for .

2. Calculate using the expression for .

3. Compare the limit and the function value at .

4. If they are equal, the function is continuous at ; otherwise, it is not.

Try solving on your own before revealing the answer!

Q3. Use the formula for the derivative of a function at to find the derivative of .

Background

Topic: Derivatives and Differentiation

This question tests your ability to apply the basic rules of differentiation to a polynomial function.

Key Terms and Formulas:

• Derivative: The rate at which a function changes with respect to its variable.

• Power Rule:

Step-by-Step Guidance

1. Identify the terms in : and .

2. Apply the power rule to each term separately.

3. Combine the results to write the derivative.

Try solving on your own before revealing the answer!

Q4. Find for .

Background

Topic: Differentiation of Products and Powers

This question tests your ability to differentiate products of power functions and constants.

Key Terms and Formulas:

• Product Rule:

• Power Rule:

Step-by-Step Guidance

1. Rewrite as .

2. Multiply and to simplify the expression.

3. Apply the power rule to the resulting term and to the constant .

Try solving on your own before revealing the answer!

Q5. Differentiate

Background

Topic: Basic Differentiation

This question tests your ability to differentiate a simple power function divided by a constant.

Key Terms and Formulas:

• Power Rule:

• Constant Multiple Rule:

Step-by-Step Guidance

1. Recognize that .

2. Apply the power rule to .

3. Multiply the result by .

Try solving on your own before revealing the answer!

Q6. Find the derivative of

Background

Topic: Product Rule for Differentiation

This question tests your ability to use the product rule to differentiate the product of two polynomial functions.

Key Terms and Formulas:

• Product Rule:

Step-by-Step Guidance

1. Let and .

2. Find and using the power rule.

3. Apply the product rule: .

4. Expand and combine like terms as needed.

Try solving on your own before revealing the answer!

Q7. Differentiate

Background

Topic: Chain Rule and Differentiation of Composite Functions

This question tests your ability to use the chain rule to differentiate a function raised to a negative power.

Key Terms and Formulas:

• Chain Rule:

• Power Rule:

• Derivative of :

Step-by-Step Guidance

1. Let , so .

2. Apply the power rule to , then multiply by (chain rule).

3. Find by differentiating .

4. Combine the results to write the derivative in terms of and .

Try solving on your own before revealing the answer!

Q8. Find an equation of the tangent line to the function at the point .

Background

Topic: Tangent Lines and Derivatives

This question tests your ability to find the equation of a tangent line using the derivative at a specific point.

Key Terms and Formulas:

• Tangent Line: A line that touches a curve at a single point and has the same slope as the curve at that point.

• Slope of Tangent: at the given point.

• Point-Slope Form:

Step-by-Step Guidance

1. Find for using the power rule.

2. Evaluate to get the slope at .

3. Use the point-slope form with the point and the slope found above.

Try solving on your own before revealing the answer!

Q9. A company finds that the cost and revenue, in dollars, of producing hundred items are given by and . Determine the rate at which the company's profit is changing when 900 items have been produced and sold.

Background

Topic: Marginal Analysis in Business Calculus

This question tests your ability to find the rate of change of profit (marginal profit) using derivatives of cost and revenue functions.

Key Terms and Formulas:

• Profit Function:

• Marginal Profit:

Step-by-Step Guidance

1. Write the profit function .

2. Find and using the power rule.

3. Calculate .

4. Substitute (since 900 items = 9 hundred) into to find the rate of change.

Try solving on your own before revealing the answer!

Q10. Differentiate

Background

Topic: Quotient Rule and Differentiation of Exponential and Logarithmic Functions

This question tests your ability to use the quotient rule to differentiate a function involving exponentials and logarithms.

Key Terms and Formulas:

• Quotient Rule:

• Derivative of :

• Derivative of :

• Derivative of :

Step-by-Step Guidance

1. Let and .

2. Find and using the rules for exponentials and logarithms.

3. Apply the quotient rule: .

4. Substitute the derivatives and simplify the expression.

Try solving on your own before revealing the answer!