Q: 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 $$x$$?

$$\begin{cases} x(t) = 5 - 4t \\ y(t) = -1 + t \end{cases},\ -6 \leq t \leq 5$$

Q: Suppose a curve is given in polar coordinates by $$r = 1 + \sin(\theta). $$What is the area enclosed by one loop of this curve?

$$\int_{0}^{2\pi} \frac{1}{2} (1 + \sin(\$$theta))^2$$ d\theta$$

Q: Which of the following parametric equations describes a line passing through $$(1,2)$$ with slope $$-3$$?

$$\begin{cases} x(t) = 1 + t \\ y(t) = 2 - 3t \end{cases}$$

Question 1

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

Accepted Answer

$$-4x^3$$ $$-4x^{11}$$ \over 2! + $$4x^{19}$$ \over 4! $$-4x^{27}$$ \over 6! + \cdots$$

Question 2

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

Accepted Answer

$$f(x) = \frac{10}{10 + x}$$

Question 3

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 $$x$$?

Accepted Answer

$$\begin{cases} x(t) = 5 - 4t \ y(t) = -1 + t \end{cases},\ -6 \leq t \leq 5$$

Question 4

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

Accepted Answer

$$y = \frac{\sqrt{2}}{2} x + \frac{\sqrt{2}}{2}$$

Question 5

What is the arc length $$L of $$the parametric curve $$\begin{cases} x(t) = -4t + 7 \ y(t) = 3t + 5 \end{cases}$$ for $$0 \leq t \leq 3$$?

Accepted Answer

$$L = 15$$

Question 6

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

Accepted Answer

$$(-2\sqrt{2}, 2\sqrt{2})$$

Question 7

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

Accepted Answer

$$y = -2x + 8$$

Question 8

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

Accepted Answer

$$A = 1$$

Question 9

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

Accepted Answer

$$\int_{\pi/3}^{5\pi/3} \frac{1}{2} (3 - 6\cos(\$$theta))^2$$ d	heta$$

Question 10

What is the arc length $$L of $$the polar curve given by $$r = 2\cos(	heta)$$?

Accepted Answer

$$2\pi$$

Question 11

A particle moves along the curve $$x(t) = t^2$, y(t$$) = $$t^3$$ for $$0 \leq t \leq 2. $$What is the total distance traveled by the particle?

Accepted Answer

$$\int_{0}^{2} \sqrt{(2t)^2$ + (3t^2$)^2$$} dt$

Question 12

Which of the following is the correct Maclaurin series for $$\sin(x^2$)$$?

Accepted Answer

$$x^2$$ - \frac{$$(x^2)^3$}{3!} + \frac{(x^2$)^5$$}{5!} - \cdots$$

Question 13

Suppose a curve is given in polar coordinates by $$r = 1 + \sin(	heta). $$What is the area enclosed by one loop of this curve?

Accepted Answer

$$\int_{0}^{2\pi} \frac{1}{2} (1 + \sin(\$$theta))^2$$ d	heta$$

Question 14

A chemical reaction's rate is modeled by $$y = e$$^{-kt}$$, where k$$ \u003e 0$$. If y$ is expanded as a Maclaurin series, what is the coefficient of t^2$?

Accepted Answer

$$-\frac{k^2$}{2}$$

Question 15

A circle of radius $$3 is $$described by $$r = 3 in $$polar coordinates. What is the length of the curve traced as $$	heta$$ goes from $$0 to$$ $$2\pi$$?

Accepted Answer

$$6\pi$$

Question 16

Which of the following parametric equations describes a line passing through $$(1,2)$$ with slope $$-3$$?

Accepted Answer

$$\begin{cases} x(t) = 1 + t \ y(t) = 2 - 3t \end{cases}$$

Question 17

A medical imaging device traces the curve $$x = 2\cos(t)$$, $$y = 2\sin(t)$$ for $$0 \leq t \leq \pi. $$What is the length of the path traced?

Accepted Answer

$$2\pi$$

Question 18

Convert the polar point $$\left(5, \frac{3\pi}{4}\right) to $$Cartesian coordinates.

Accepted Answer

$$\left(-\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)$$

Question 19

Which of the following is the correct Cartesian equation for the polar equation $$r = 4\sin(	heta)$$?

Accepted Answer

$$x^2$$ + $$y^2 = 4y$

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