Q1. Let . Find .

Background

Topic: Differentiation

This question tests your ability to compute the derivative of a polynomial function using basic differentiation rules.

Key Terms and Formulas:

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

• Power Rule:

• Sum Rule:

Step-by-Step Guidance

1. Identify the function: .

2. Apply the power rule to : .

3. Apply the constant multiple rule to : .

4. The derivative of a constant ($5.

Try solving on your own before revealing the answer!

Q2. Compute .

Background

Topic: Definite Integrals

This question tests your ability to evaluate a definite integral of a polynomial function.

Key Terms and Formulas:

• Definite Integral: gives the area under from to .

• Power Rule for Integration:

• Evaluate at bounds: where is the antiderivative.

Step-by-Step Guidance

1. Find the antiderivative of : .

2. Find the antiderivative of : .

3. Find the antiderivative of $1\int 1 dx = x$.

4. Combine the antiderivatives: .

5. Set up the evaluation: .

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Q3. The tangent line to the graph of at is given by . Find and .

Background

Topic: Tangent Lines and Derivatives

This question tests your understanding of how the tangent line relates to the function and its derivative at a specific point.

Key Terms and Formulas:

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

• Slope of tangent line at is .

• The tangent line passes through .

Step-by-Step Guidance

1. Identify the tangent line equation: .

2. At , the tangent line passes through .

3. Plug into the tangent line: .

4. The slope of the tangent line is $3f'(1) = 3$.

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Q4. Find the area of the region bounded by and .

Background

Topic: Area Between Curves

This question tests your ability to set up and compute the area between two curves using definite integrals.

Key Terms and Formulas:

• Area between curves: where is the upper curve and is the lower curve.

• Find intersection points to determine bounds and .

Step-by-Step Guidance

1. Set to find intersection points.

2. Solve for to get the bounds and .

3. Determine which function is on top between the bounds.

4. Set up the integral: .

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Q5. The region under the curve from to is rotated about the -axis. Find the volume of the resulting solid.

Background

Topic: Volumes of Revolution

This question tests your ability to use the disk method to find the volume of a solid formed by rotating a curve about the -axis.

Key Terms and Formulas:

• Disk Method:

• Bounds: ,

Step-by-Step Guidance

1. Set up the volume integral: .

2. Simplify the integrand: .

3. Set up the integral: .

4. Find the antiderivative: .

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Q6. Let . Find the equation of the tangent line at and verify that it passes through .

Background

Topic: Tangent Lines to Trigonometric Functions

This question tests your ability to find the tangent line to a trigonometric function at a specific point and verify its properties.

Key Terms and Formulas:

• Derivative of :

• Tangent line at :

Step-by-Step Guidance

1. Find .

2. Find .

3. Write the tangent line equation: .

4. Verify that the tangent line passes through by plugging in .

Try solving on your own before revealing the answer!