Q1. Evaluate

Background

Topic: Limits and Indeterminate Forms

This question tests your understanding of how to evaluate limits, especially when the expression initially appears to be indeterminate (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 numerator and denominator to check if the limit is indeterminate.

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 into the new expression to find the limit.

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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-defined function.

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 value to to determine continuity.

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

Background

Topic: Derivatives and Differentiation Rules

This question tests your ability to apply basic differentiation rules to polynomial functions.

Key Terms and Formulas:

• Derivative: represents the rate of change of with respect to .

• 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.

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Q4. Find for .

Background

Topic: Differentiation of Products and Power Functions

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. Identify and .

2. Find the derivatives and using the power rule.

3. Apply the product rule to and .

4. Differentiate the constant (which becomes $0$).

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Q5. Differentiate

Background

Topic: Quotient Rule for Differentiation

This question tests your ability to use the quotient rule to differentiate a function where the numerator and denominator are both functions of .

Key Terms and Formulas:

• Quotient Rule:

• Exponential Rule:

Step-by-Step Guidance

1. Let and .

2. Find and using the power and exponential rules.

3. Apply the quotient rule formula.

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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 formula.

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Q7. Differentiate

Background

Topic: Chain Rule for Differentiation

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 and .

2. Apply the power rule to .

3. Multiply by the derivative of using the chain rule.

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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 value.

• 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.

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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, which is the derivative of the profit function with respect to .

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 means $x = 9$ hundred) into .

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Q10. Differentiate

Background

Topic: Quotient Rule and Chain Rule for Differentiation

This question tests your ability to differentiate a quotient where the numerator involves both an exponential and a logarithmic function.

Key Terms and Formulas:

• Quotient Rule:

• Chain Rule:

• Derivative of :

• Derivative of :

Step-by-Step Guidance

1. Let and .

2. Find using the chain rule and logarithmic differentiation.

3. Find using the power rule.

4. Apply the quotient rule formula.

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