Step-by-Step Guidance for Calculus Exam Questions

Q1. Let $f(x) = x^2 + 2x + 5$. Find $f'(x)$.

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: $\frac{d}{dx} x^n = n x^{n-1}$

• Sum Rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$

Step-by-Step Guidance

1. Identify the function: $f(x) = x^2 + 2x + 5$.

2. Apply the power rule to $x^2$: $\frac{d}{dx} x^2 = 2x$.

3. Apply the constant multiple rule to $2x$: $\frac{d}{dx} 2x = 2$.

4. The derivative of a constant ($5$) is $0$.

Try solving on your own before revealing the answer!

Final Answer: $f'(x) = 2x + 2$

We differentiated each term separately and combined the results.

Q2. Compute $\int_0^1 (3x^2 + 2x + 1) dx$.

Background

Topic: Definite Integrals

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

Key Terms and Formulas:

• Definite Integral: $\int_a^b f(x) dx$ gives the area under $f(x)$ from $x=a$ to $x=b$.

• Power Rule for Integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

• Evaluate at bounds: $F(b) - F(a)$ where $F(x)$ is the antiderivative.

Step-by-Step Guidance

1. Find the antiderivative of $3x^2$: $\int 3x^2 dx = x^3$.

2. Find the antiderivative of $2x$: $\int 2x dx = x^2$.

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

4. Combine the antiderivatives: $F(x) = x^3 + x^2 + x$.

5. Set up the evaluation: $F(1) - F(0)$.

Try solving on your own before revealing the answer!

Final Answer: $3$

Plugging in the bounds gives $F(1) = 1 + 1 + 1 = 3$ and $F(0) = 0$.

Q3. The tangent line to the graph of $f(x)$ at $x=1$ is given by $y = 3x + 2$. Find $f(1)$ and $f'(1)$.

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 $x=a$ is $f'(a)$.

• The tangent line passes through $(a, f(a))$.

Step-by-Step Guidance

1. Identify the tangent line equation: $y = 3x + 2$.

2. At $x=1$, the tangent line passes through $(1, f(1))$.

3. Plug $x=1$ into the tangent line: $y = 3(1) + 2 = 5$.

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

Try solving on your own before revealing the answer!

Final Answer: $f(1) = 5$, $f'(1) = 3$

The tangent line gives both the value and the slope at $x=1$.

Q4. Find the area of the region bounded by $y = x^2$ and $y = x + 2$.

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: $\int_a^b [f(x) - g(x)] dx$ where $f(x)$ is the upper curve and $g(x)$ is the lower curve.

• Find intersection points to determine bounds $a$ and $b$.

Step-by-Step Guidance

1. Set $x^2 = x + 2$ to find intersection points.

2. Solve for $x$ to get the bounds $a$ and $b$.

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

4. Set up the integral: $\int_a^b [(x + 2) - x^2] dx$.

Try solving on your own before revealing the answer!

Final Answer: $\frac{9}{2}$

Evaluating the definite integral gives the area between the curves.

Q5. The region under the curve $y = x^2$ from $x = 0$ to $x = 1$ is rotated about the $x$-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 $x$-axis.

Key Terms and Formulas:

• Disk Method: $V = \pi \int_a^b [f(x)]^2 dx$

• Bounds: $a = 0$, $b = 1$

Step-by-Step Guidance

1. Set up the volume integral: $V = \pi \int_0^1 [x^2]^2 dx$.

2. Simplify the integrand: $[x^2]^2 = x^4$.

3. Set up the integral: $V = \pi \int_0^1 x^4 dx$.

4. Find the antiderivative: $\int x^4 dx = \frac{x^5}{5}$.

Try solving on your own before revealing the answer!

Final Answer: $\frac{\pi}{5}$

Plugging in the bounds gives the volume of the solid.

Q6. Let $f(x) = \sin x$. Find the equation of the tangent line at $x = \frac{\pi}{4}$ and verify that it passes through $\left(\frac{\pi}{4}, f\left(\frac{\pi}{4}\right)\right)$.

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 $\sin x$: $\frac{d}{dx} \sin x = \cos x$

• Tangent line at $x=a$: $y = f'(a)(x - a) + f(a)$

Step-by-Step Guidance

1. Find $f\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right)$.

2. Find $f'\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)$.

3. Write the tangent line equation: $y = f'\left(\frac{\pi}{4}\right)(x - \frac{\pi}{4}) + f\left(\frac{\pi}{4}\right)$.

4. Verify that the tangent line passes through $\left(\frac{\pi}{4}, f\left(\frac{\pi}{4}\right)\right)$ by plugging in $x = \frac{\pi}{4}$.

Try solving on your own before revealing the answer!

Final Answer: $y = \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) + \frac{\sqrt{2}}{2}$

Plugging in $x = \frac{\pi}{4}$ confirms the tangent line passes through the required point.