Q: Which of the following is the correct application of L'Hôpital's Rule to $$\displaystyle \lim_{x \to 0} \frac{1 - \cos 6x}{2x}$$?

$$\lim_{x \to 0} \frac{6 \sin 6x}{2}$$

Question 1

Suppose you want to estimate $$\sqrt{101}$$ using linear approximation at $$a=100. $$Which of the following is the correct linear approximation for $$f(x) = \sqrt{x} at$$ $$x=100$$?

Accepted Answer

$$f(x) \approx f(100) + f'(100)(x-100)$$

Question 2

Given $$f(x) = 2x^3$ - 3x^2$ - 36x + 12$$, which of the following $$x-$$values is a critical point?

Accepted Answer

$$x = -3$$

Question 3

For $$g(x) = x e$$^{-x}$$, at which x$-value does the function have an inflection point?

Accepted Answer

$$x = 2$$

Question 4

Which of the following best describes the intervals where $$f(x) = x^9$ - x is $$increasing?

Accepted Answer

$$x \u003e 1$$ and $$x \u003c -1$$

Question 5

Use L'Hôpital's Rule to evaluate $$\displaystyle \lim_{x 	o \infty} \frac{-3x^3$ - 4x^2$ + 7}{7x^3$ + 5x - 8}. $$What is the value?

Accepted Answer

$$-\frac{3}{7}$$

Question 6

Given $$f(x) = \ln(x+e)$$, what is the linearization at $$a = e$$?

Accepted Answer

$$L(x) = 1 + \frac{1}{e+e}(x-e)$$

Question 7

For the function $$f(x) = x^2$ - x - 2x^2$ - 4$$, which of the following statements is true about its horizontal asymptotes?

Accepted Answer

There are no horizontal asymptotes.

Question 8

A window is constructed by adjoining a semicircle to the top of a rectangle. If the perimeter is 16 ft, which equation expresses the relationship between the rectangle's width $$w$$ and height $$h$$ for optimization?

Accepted Answer

$$16 = 2h + w + \pi \frac{w}{2}$$

Question 9

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{e$$^{-2x}$$ - 1 + 2x}{x^2$}$$ using L'Hôpital's Rule.

Accepted Answer

$$2$$

Question 10

Which of the following is the correct antiderivative of $$f(x) = 3x^2$ - 4x + 5$$?

Accepted Answer

$$x^3$$ - $$2x^2 + 5x + C$

Question 11

For $$g(x) = \sin(2x) + 3 on$$ $$[-\pi, \pi]$$, at which $$x-$$values does $$g(x)$$ attain its maximum?

Accepted Answer

$$x = \frac{\pi}{4}$$ and $$x = -\frac{3\pi}{4}$$

Question 12

Given $$f(x) = e^x x - 7$$, use the second derivative test to determine the nature of the critical point at $$x = 0.$$

Accepted Answer

It is a local minimum.

Question 13

Which of the following is the correct application of L'Hôpital's Rule to $$\displaystyle \lim_{x 	o 0} \frac{1 - \cos 6x}{2x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{6 \sin 6x}{2}$$

Question 14

For the function $$f(x) = x$$^{16}$$ - x^2$, at which x$-values does the function change concavity?

Accepted Answer

$$x = 0$$ and $$x = \pm 1$$

Question 15

Which of the following is the correct setup to find the points on $$f(x) = 4 - x^2$ closest to (0$$,2)$$?

Accepted Answer

Minimize $$d^2$$ = $$x^2 + (4$$ - $$x^2$$ - $$2)^2$$

Question 16

Given $$f(x) = x$$^{33}$$ - \ln x$$, use the first derivative test to determine the nature of the critical point at $$x = 1.$$

Accepted Answer

It is a local minimum.

Question 17

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{x^3$ - 125}{x^2$ + 25}.$$

Accepted Answer

$$-5$$

Question 18

Which of the following is the correct antiderivative of $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$\ln|x| + C$$

Question 19

For $$g(x) = 4 \cos(\pi x) - 1 on$$ $$[0,2]$$, at which $$x-$$value does the function attain its minimum?

Accepted Answer

$$x = 1$$

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