Q: Find the derivative of $$f(x) = e$$^{-\tan^2 x}$$.

$$-2 \tan x \sec^2$ x \cdot e$$^{-\tan^2 x}$$

Question 1

Evaluate the following limit using L'Hôpital's Rule: 
$$
\lim_{x 	o 0} \frac{e^x - 1 - x}{x^2$}
$$

Accepted Answer

$$\frac{1}{2}$$

Question 2

A ladder 10 ft long rests against a vertical wall. If the bottom slides away from the wall at $$1 ft/s$$, how fast is the top descending when the bottom is $$6 ft $$from the wall? (Let $$x be $$the distance from the wall to the bottom, $$y$$ the height on the wall.)

Accepted Answer

$$-0.8 ft/s$$

Question 3

Find the derivative of $$f(x) = e$$^{-	an^2 x}$$.

Accepted Answer

$$-2 	an x \sec^2$ x \cdot e$$^{-	an^2 x}$$

Question 4

Which of the following integrals represents the volume generated by revolving $$y = x^2$, 0$$ \leq x \leq 2$$ about the y$-axis?

Accepted Answer

$$\int_{0}^{2} 2\pi x (x^2$) dx$$

Question 5

Find $$\frac{dy}{dx} if$$ $$x^2 - x y$$ + $$y^2 = 3$.

Accepted Answer

$$\frac{2x - y}{2y - x}$$

Question 6

Evaluate $$\int_{1}^{e} \frac{\ln x}{x} dx.$$

Accepted Answer

$$\frac{1}{2}$$

Question 7

A sphere's radius increases at $$4 cm/s $$when the radius is $$6 cm. At $$that instant, what is the rate of change of the volume?

Accepted Answer

$$576\pi cm$^3/s$$

Question 8

Which of the following is the third-order Taylor polynomial for $$f(x) = e$$^{-4x}$$ at x = 0$?

Accepted Answer

$$1 - 4x + 8x^2$ - \frac{32}{3}x^3$

Question 9

Determine whether the series $$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$ converges.

Accepted Answer

It converges because it is a $$p-$$series with $$p = 2 \u003e 1.$$

Question 10

Use the Intermediate Value Theorem to justify that $$x^5 + x - 1 = 0$ has a real solution.

Accepted Answer

Because the function is continuous and changes sign between $$x=0$$ and $$x=1.$$

Question 11

Find the area enclosed between $$y = x^3$ - 3x^2$ + 3x$$ and $$y = x.$$

Accepted Answer

$$2$$

Question 12

Which of the following is the correct linear approximation for $$e^{0.1} at$$ $$x=0$$?

Accepted Answer

$$1 + 0.1$$

Question 13

Set up the integral representing the volume generated by revolving $$y = 1/x$$, $$1 \leq x \leq 5$$ about the $$x-$$axis.

Accepted Answer

$$\int_{1}^{5} \pi \left(\frac{1}{x}\$$right)^2$$ dx$$

Question 14

Find all critical points of $$f(x) = x^3$ - 3x^2$ + 2.$$

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 15

Which of the following is the correct derivative of $$f(x) = \ln(\sin x)$$?

Accepted Answer

$$\cot x$$

Question 16

Find the radius of convergence of the series $$\sum_{n=0}^{\infty} \frac{(x-1)^n}{5^n}.$$

Accepted Answer

$$5$$

Question 17

Evaluate $$\int_{0}^{\pi} \sin x dx.$$

Accepted Answer

$$2$$

Question 18

Which of the following best describes the geometric interpretation of the definite integral $$\int_{a}^{b} f(x) dx$$?

Accepted Answer

The net area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b.$$

Question 19

Suppose $$F(x) = \int_{0}^{x^4$} \sin t dt. $$Compute $$\frac{dF}{dx}.$$

Accepted Answer

$$4x^3$$ \$$sin(x^4$$)$$

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