Question 1

Given the graph of $$g'(x)$$ shown, which of the following statements is true about the critical points of $$g(x) on $$the interval $$[0,3]$$?

Accepted Answer

The critical points occur where $$g'(x) is $$zero or undefined.

Question 2

For the function $$f(x) = x^4$ - 4x^3$ + 4x^2$, on which intervals is f(x$$)$$ increasing?

Accepted Answer

$$(-\infty, 1) \cup (2, \infty)$$

Question 3

A rectangle is constructed with its base on the $$x-$$axis and two vertices on the parabola $$y = 48 - x^2$. What is the maximum possible area of such a rectangle?

Accepted Answer

$$256$$

Question 4

Given $$f(x) = e^x$$, what is the linear approximation to $$f at$$ $$x = 0$$?

Accepted Answer

$$L(x) = 1 + x$$

Question 5

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x^3$ + 2x + 1}.$$

Accepted Answer

$$0$$

Question 6

Find the indefinite integral $$\int 6x$$^{1/3}$$ dx.$$

Accepted Answer

$$\frac{9}{2}x$$^{4/3}$$ + C$$

Question 7

Suppose $$f is $$differentiable on $$(-\infty, \infty)$$, $$f(1) = 2$$, and $$f'(x) = 3. $$Use the linear approximation at $$x = 1 to $$estimate $$f(1.1).$$

Accepted Answer

$$2.3$$

Question 8

Evaluate $$\lim_{x 	o e} \frac{\ln x - 1}{x - e}$$ using L'Hôpital's Rule.

Accepted Answer

$$\frac{1}{e}$$

Question 9

Find the solution to the initial value problem: $$g'(x) = 7x^6$ - 4x^3$ + 12$$, $$g(1) = 24.$$

Accepted Answer

Question 10

Use substitution to evaluate $$\int x e$$^{x^2}$$ dx.$$

Accepted Answer

$$\frac{1}{2} e$$^{x^2}$$ + C$$

Question 11

Which of the following is the correct linear approximation for $$f(x) = \sqrt{2x + 9} at$$ $$a = -4$$?

Accepted Answer

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

Question 12

Evaluate $$\int \frac{8x + 6}{2x^2$ + 3x + 1} dx$$ using substitution.

Accepted Answer

$$4 \ln|2x^2$ + 3x + 1| + C$$

Question 13

Evaluate $$\lim_{x 	o 0} \frac{1 - \cos 3x}{8x^2$}.$$

Accepted Answer

$$\frac{9}{16}$$

Question 14

Find the intervals of concavity for $$f(x) = x^4$ - 4x^3$ + 4x^2$.

Accepted Answer

$$(-\infty, 1) \cup (2, \infty)$$ concave up, $$(1, 2)$$ concave down

Question 15

Evaluate $$\int \frac{dy}{1 + x^2$}.$$

Accepted Answer

$$\arctan x + C$$

Question 16

Given $$f(x) = \cos^2$ x on$$ $$[-\pi, \pi]$$, on which intervals is $$f(x)$$ decreasing?

Accepted Answer

$$(-\pi, 0)$$

Question 17

Evaluate $$\int_{0}^{\frac{\pi}{4}} \frac{\sin 	heta}{\cos^3$ 	heta} d	heta$$ using substitution.

Accepted Answer

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

Question 18

A box is formed by cutting squares of side $$x$$ from two corners of an $$18'' 	imes 18''$$ cardboard and rectangles of $$9' 	imes x$$ from the other two corners, then folding. What is the volume of the largest box that can be formed?

Accepted Answer

$$648$$ cubic inches

Calculus Study Guide: Critical Points, Graphing, Optimization, Linear Approximation, Limits, and Integration

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