Calculus II Exam Study Guidance

Q1. Which of the following is the Maclaurin series that represents the derivative of the function $f(x) = \cos(x^4)$?

Background

Topic: Power Series and Differentiation

This question tests your understanding of how to find the Maclaurin (Taylor at $x=0$) series for a function and its derivative, specifically for composite functions.

Key Terms and Formulas

• Maclaurin Series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$

• Derivative of a Series: Differentiate term-by-term.

• Chain Rule: $\frac{d}{dx} \cos(g(x)) = -\sin(g(x)) \cdot g'(x)$

• Maclaurin Series for $\cos(x)$: $\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$

Step-by-Step Guidance

1. Recall the Maclaurin series for $\cos(x)$ and substitute $x^4$ for $x$ to get the series for $\cos(x^4)$.

2. Differentiate the series term-by-term with respect to $x$ to find the series for $f'(x)$.

3. Apply the chain rule: for each term $x^{4n}$, the derivative is $4n x^{4n-1}$.

4. Write out the first few terms explicitly to compare with the given options.

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Q2. Identify the function represented by the following power series: $f(x) = \sum_{k=0}^{\infty} (-1)^k \frac{x^k}{10^k}$

Background

Topic: Power Series Representation of Functions

This question tests your ability to recognize a function from its power series, especially geometric series and their manipulations.

Key Terms and Formulas

• Geometric Series: $\sum_{k=0}^{\infty} ar^k = \frac{a}{1 - r}$ for $|r| < 1$

• Alternating Series: $(-1)^k$ introduces sign changes.

Step-by-Step Guidance

1. Identify the first term ($a$) and the common ratio ($r$) in the given series.

2. Express the series in the standard geometric form.

3. Write the sum as a closed-form function using the geometric series formula.

4. Compare your result to the given options to identify the correct function.

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Q3. Consider the line that passes through the points $(5, -1)$ and $(-3, 1)$. Which of the following represents this line oriented in the direction of decreasing $x$?

Background

Topic: Parametric Equations of Lines

This question tests your understanding of how to write parametric equations for a line through two points and how to orient the parameter to match a specified direction.

Key Terms and Formulas

• Parametric Equation: $x(t) = x_0 + at$, $y(t) = y_0 + bt$

• Direction Vector: $(a, b) = (x_2 - x_1, y_2 - y_1)$

• Orientation: Decreasing $x$ means $a < 0$

Step-by-Step Guidance

1. Find the direction vector from $(5, -1)$ to $(-3, 1)$.

2. Write the parametric equations using the direction vector and one of the points.

3. Check the sign of the $x$ component to ensure it represents decreasing $x$ as $t$ increases.

4. Compare your equations to the given options to select the correct one.

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Q4. What is the equation of the line tangent to the curve at the point corresponding to $t = \frac{\pi}{4}$ for $x(t) = \cos(t) - \sin(t)$, $y(t) = \sin(t) + \cos(t)$?

Background

Topic: Tangent Lines to Parametric Curves

This question tests your ability to find the equation of the tangent line to a parametric curve at a specific parameter value.

Key Terms and Formulas

• Tangent Line: $y - y_0 = m(x - x_0)$

• Slope: $m = \frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

• Evaluate $x(t)$ and $y(t)$ at $t = \frac{\pi}{4}$

Step-by-Step Guidance

1. Compute $x\left(\frac{\pi}{4}\right)$ and $y\left(\frac{\pi}{4}\right)$ to find the point of tangency.

2. Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$, then evaluate both at $t = \frac{\pi}{4}$.

3. Calculate the slope $m = \frac{dy/dt}{dx/dt}$ at $t = \frac{\pi}{4}$.

4. Write the equation of the tangent line in point-slope form using the point and slope found.

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Q5. What is the arc length $L$ of the parametric curve $x(t) = -4t + 7$, $y(t) = 3t + 5$ for $0 \leq t \leq 3$?

Background

Topic: Arc Length of Parametric Curves

This question tests your ability to compute the arc length of a curve given in parametric form over a specified interval.

Key Terms and Formulas

• Arc Length Formula: $L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

Step-by-Step Guidance

1. Compute $\frac{dx}{dt}$ and $\frac{dy}{dt}$ for the given parametric equations.

2. Substitute these derivatives into the arc length formula.

3. Simplify the expression under the square root.

4. Set up the definite integral with the given bounds $t = 0$ to $t = 3$.

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Q6. Express the polar coordinates $\left(4, -\frac{5\pi}{4}\right)$ in Cartesian coordinates.

Background

Topic: Polar to Cartesian Coordinate Conversion

This question tests your ability to convert from polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$.

Key Terms and Formulas

• $x = r \cos(\theta)$

• $y = r \sin(\theta)$

Step-by-Step Guidance

1. Identify $r = 4$ and $\theta = -\frac{5\pi}{4}$.

2. Compute $\cos\left(-\frac{5\pi}{4}\right)$ and $\sin\left(-\frac{5\pi}{4}\right)$.

3. Multiply $r$ by each trigonometric value to find $x$ and $y$.

4. Simplify the results to match one of the given options.

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Q7. Convert the polar equation $r = \frac{8}{2 \cos(\theta) + \sin(\theta)}$ to a Cartesian equation.

Background

Topic: Polar to Cartesian Equation Conversion

This question tests your ability to rewrite a polar equation in terms of $x$ and $y$ using the relationships between polar and Cartesian coordinates.

Key Terms and Formulas

• $x = r \cos(\theta)$

• $y = r \sin(\theta)$

• $r^2 = x^2 + y^2$

Step-by-Step Guidance

1. Multiply both sides of the equation by the denominator to clear the fraction.

2. Express $r \cos(\theta)$ and $r \sin(\theta)$ in terms of $x$ and $y$.

3. Rewrite the equation entirely in terms of $x$ and $y$.

4. Simplify the equation to match one of the given options.

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Q8. Determine the area $A$ enclosed within the polar region defined by $r = \sqrt{2 \cos(\theta)}$.

Background

Topic: Area in Polar Coordinates

This question tests your ability to compute the area enclosed by a polar curve using the appropriate integral formula.

Key Terms and Formulas

• Area Formula: $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta$

• Limits of integration are determined by the values of $\theta$ for which $r$ is real and non-negative.

Step-by-Step Guidance

1. Square $r$ to find $r^2 = 2 \cos(\theta)$.

2. Determine the interval for $\theta$ where $r$ is real (i.e., $\cos(\theta) \geq 0$).

3. Set up the area integral using the correct limits and the formula above.

4. Prepare to evaluate the integral, but stop before computing the final value.

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Q9. Which integral represents the area of the region inside the inner loop of the limaçon $r = 3 - 6 \cos(\theta)$?

Background

Topic: Area of Regions Bounded by Polar Curves

This question tests your ability to set up the correct integral for the area of a region defined by a polar curve, specifically the inner loop of a limaçon.

Key Terms and Formulas

• Area Formula: $A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 d\theta$

• Limits $\alpha$ and $\beta$ are the values of $\theta$ where the inner loop starts and ends (where $r = 0$).

Step-by-Step Guidance

1. Solve $3 - 6 \cos(\theta) = 0$ to find the bounds for $\theta$ corresponding to the inner loop.

2. Set up the area integral using these bounds and the formula above.

3. Compare your setup to the given options to select the correct integral.

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Q10. What is the arc length $L$ of the polar curve $r = 2 \cos(\theta)$?

Background

Topic: Arc Length of Polar Curves

This question tests your ability to compute the arc length of a curve given in polar coordinates over a full period.

Key Terms and Formulas

• Arc Length Formula (Polar): $L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$

• For $r = 2 \cos(\theta)$, determine $\frac{dr}{d\theta}$ and the appropriate bounds for $\theta$.

Step-by-Step Guidance

1. Compute $\frac{dr}{d\theta}$ for $r = 2 \cos(\theta)$.

2. Substitute $r$ and $\frac{dr}{d\theta}$ into the arc length formula.

3. Simplify the expression under the square root as much as possible.

4. Set up the definite integral with the correct bounds for $\theta$ (typically $0$to$\pi$ for a full circle in this case).

Try solving on your own before revealing the answer!