Question 1

Evaluate the following limit using L'Hôpital's Rule, if it is in an indeterminate form and applicable:  
$$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{\sqrt{x} - \sqrt{2}}$$

Accepted Answer

$$4\sqrt{2}$$

Question 2

Determine whether Rolle's Theorem applies to the function $$f(x) = x^2$ - 4x + 3 on $$the interval $$[1, 3]. If so$$, find the point(s) guaranteed by Rolle's Theorem.

Accepted Answer

Rolle's Theorem applies, and the point(s) guaranteed are $$x = 2$$

Question 3

A rectangular pen is built with one side against a barn, using 48 ft of fencing to cover the other three sides of the pen. What dimensions maximize the area of the pen?

Accepted Answer

Width $$= 12 ft$$, Length $$= 24 ft$$

Question 4

Determine the intervals on which the function $$g(x) = -x^3$ - 5x^2$ - 3x is $$concave up or concave down. Select the correct answer.

Accepted Answer

Concave up on $$(-\infty, -\frac{10}{3})$$, concave down on $$(-\frac{10}{3}, \infty)$$

Question 5

Find the absolute maximum/minimum of $$f(x) = x^3$ - 3x^2$ + 1 on$$ $$[0, 3].$$

Accepted Answer

Absolute maximum at $$x = 3$$, absolute minimum at $$x = 2$$

Question 6

Given the graph of a function on $$[a, b]$$, which of the following statements is true about the absolute extrema?

Accepted Answer

Absolute maximum occurs at a critical point or endpoint

Question 7

Does the Mean Value Theorem apply to $$f(x) = |x| on$$ $$[-1, 1]$$? If not, why?

Accepted Answer

No, $$f(x) is $$not differentiable at $$x = 0$$

Question 8

Write the formula for Newton's method for $$f(x) = x^3$ - 2x + 1$$ and give the next approximation $$x_1 if$$ $$x_0 = 1$.

Accepted Answer

Question 9

Find the intervals on which $$f(x) = x^3$ - 6x^2$ + 9x + 1 is $$increasing and decreasing.

Accepted Answer

Increasing on $$(-\infty, 1)$$, decreasing on $$(1, 3)$$, increasing on $$(3, \infty)$$

Question 10

Which of the following is the correct graph of $$f(x) = x^3$ - 3x$$?

Accepted Answer

A graph with inflection point at $$x = 0$$ and local extrema at $$x = \pm 1$$

Question 11

Find the linear approximation $$L(x) of$$ $$f(x) = \sqrt{x} at$$ $$x = 4.$$

Accepted Answer

$$L(x) = 2 + \frac{1}{4}(x - 4)$$

Question 12

A police officer uses a radar gun to measure the speed of a car. If the officer knows the car's position at two times, how can the officer use the Mean Value Theorem to conclude the car was speeding?

Accepted Answer

If the average speed between two points exceeds the speed limit, then at some time the car was speeding

Question 13

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

Accepted Answer

$$4$$

Question 14

Which of the following is the correct graph of $$f(x) = x^4$ - 4x^2$?

Accepted Answer

A graph with local minima at $$x = \pm \sqrt{2}$$ and a local maximum at $$x = 0$$

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