Q1. Evaluate the derivative of the function:

Background

Topic: Derivatives of Inverse Trigonometric Functions

This question tests your ability to differentiate an inverse sine function with a composite argument.

Key Terms and Formulas

• : Inverse sine function

• Derivative formula:

Step-by-Step Guidance

1. Identify in the function .

2. Recall the derivative formula for : .

3. Compute for .

4. Substitute and into the formula.

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Q2. Consider two functions and on with given definite integrals. Evaluate the following integrals:

Background

Topic: Properties of Definite Integrals

This question tests your understanding of linearity, additivity, and manipulation of definite integrals.

Key Terms and Formulas

• Linearity:

• Additivity:

• Splitting intervals:

Step-by-Step Guidance

1. For each part, identify which property of integrals is being used (linearity, additivity, interval splitting).

2. Write the integral in terms of the given values, using the properties above.

3. For example, for , use linearity: .

4. For , use additivity: .

5. Continue for each part, expressing the integral in terms of the provided values.

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Q3. Find the derivative of

Background

Topic: Derivatives of Exponential Functions

This question tests your ability to differentiate an exponential function with a constant base.

Key Terms and Formulas

• Exponential function:

• Derivative:

Step-by-Step Guidance

1. Identify the base and the coefficient .

2. Recall the derivative formula: .

3. Multiply the derivative by the coefficient .

4. Write the derivative in terms of .

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Q4. Find the inverse function for , and graph both and

Background

Topic: Inverse Functions and Graph Symmetry

This question tests your ability to find the inverse of a function and understand the graphical relationship between a function and its inverse.

Key Terms and Formulas

• Inverse function:

• Graph symmetry: The graph of and are symmetric about the line .

Step-by-Step Guidance

1. Set and solve for in terms of .

2. Isolate .

3. Take the square root, considering the domain .

4. Express in terms of to get .

5. Sketch both graphs and check for symmetry about .

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Q5. Evaluate the derivative:

Background

Topic: Chain Rule for Derivatives

This question tests your ability to apply the chain rule to nested logarithmic functions.

Key Terms and Formulas

• Chain rule:

• Derivative of :

Step-by-Step Guidance

1. Let , so the function is .

2. Differentiate : .

3. Compute for .

4. Combine the results to get the derivative.

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Q6. Use a change of variables to evaluate the indefinite integral:

Background

Topic: Substitution in Integration

This question tests your ability to use substitution to simplify and evaluate an integral.

Key Terms and Formulas

• Substitution: Let

• Integral formula:

Step-by-Step Guidance

1. Let , then .

2. Express in terms of .

3. Rewrite the integral in terms of .

4. Apply the appropriate integral formula for .

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Q7. Evaluate the definite integral:

Background

Topic: Definite Integrals and Power Rule

This question tests your ability to apply the power rule to definite integrals.

Key Terms and Formulas

• Power rule: (for )

• Definite integral:

Step-by-Step Guidance

1. Apply the power rule to : .

2. Evaluate the antiderivative at the upper and lower limits: at and .

3. Subtract the values: .

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Q8. Find the derivative:

Background

Topic: Chain Rule for Derivatives

This question tests your ability to differentiate a logarithmic function with a quadratic argument.

Key Terms and Formulas

• Chain rule:

• Derivative of :

Step-by-Step Guidance

1. Let , so the function is .

2. Differentiate : .

3. Compute for .

4. Combine the results to get the derivative.

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Q9. Find the indefinite integral:

Background

Topic: Integration by Substitution

This question tests your ability to use substitution to integrate a function with a radical.

Key Terms and Formulas

• Substitution: Let

• Integral formula:

Step-by-Step Guidance

1. Let , then .

2. Express in terms of and rewrite the integral.

3. Rewrite the integrand in terms of and .

4. Apply the power rule for integration.

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Q10. Evaluate the definite integral: given areas of regions bounded by and the x-axis

Background

Topic: Area Under a Curve and Definite Integrals

This question tests your ability to interpret definite integrals as areas and use given values to compute net area.

Key Terms and Formulas

• Definite integral: represents the net area between and the x-axis from to .

• Net area: Positive for regions above the x-axis, negative for regions below.

Step-by-Step Guidance

1. Identify the regions and their areas as given in the problem.

2. Sum the areas, considering their signs (above or below the x-axis).

3. Express the integral as the sum of the given areas.

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Q11. Use substitution to evaluate the indefinite integral:

Background

Topic: Integration by Substitution

This question tests your ability to use substitution to integrate a function with a composite argument.

Key Terms and Formulas

• Substitution: Let

• Integral formula:

Step-by-Step Guidance

1. Let , then .

2. Express in terms of .

3. Rewrite the integral in terms of .

4. Apply the integral formula for .

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Q12. Suppose the slope of the curve at is . Find

Background

Topic: Derivative of Inverse Functions

This question tests your ability to use the formula for the derivative of an inverse function at a point.

Key Terms and Formulas

• Inverse function derivative: where

Step-by-Step Guidance

1. Identify the point where and .

2. Recall the formula: .

3. Plug in for and .

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Q13. Find the area of the region bounded by and the x-axis on

Background

Topic: Area Under a Curve and Definite Integrals

This question tests your ability to compute the area between a trigonometric function and the x-axis over a specified interval.

Key Terms and Formulas

• Area:

• Definite integral:

Step-by-Step Guidance

1. Set up the definite integral .

2. Find the antiderivative of , which is .

3. Evaluate at the upper and lower bounds.

4. Subtract the values to find the net area.

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Q14. Find the derivative of

Background

Topic: Product Rule for Derivatives

This question tests your ability to apply the product rule and power rule to differentiate a function.

Key Terms and Formulas

• Product rule:

• Power rule:

Step-by-Step Guidance

1. Let and .

2. Compute and .

3. Apply the product rule: .

4. Combine terms and factor where possible.

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Q15. Find the derivative of the inverse of , , at

Background

Topic: Derivative of Inverse Functions

This question tests your ability to use the formula for the derivative of an inverse function at a specific point.

Key Terms and Formulas

• Inverse function derivative: where

Step-by-Step Guidance

1. Identify the point , so .

2. Compute using the derivative of .

3. Apply the formula .

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Q16. Use the substitution to find the indefinite integral

Background

Topic: Integration by Substitution

This question tests your ability to use substitution to integrate a function with a radical and polynomial.

Key Terms and Formulas

• Substitution: ,

• Integral formula:

Step-by-Step Guidance

1. Let , so .

2. Rewrite the integral as .

3. Apply the power rule for integration.

4. Express the answer in terms of .

Try solving on your own before revealing the answer!