Question 1

Given $$f(x) = x^2$ - x^3$, what is $$\frac{d}{dx} f(x)$$?

Accepted Answer

$$2x - 3x^2$

Question 2

If $$g(x) = 2x$$ and $$f(x) = x^2$ + x$$, what is $$\frac{d}{dx} f(g(x))$$?

Accepted Answer

$$4x + 2$$

Question 3

Find $$\frac{d}{dx} \left[ \sin(x^3$) + 	an(x) + \cos(x) ight].$$

Accepted Answer

$$3x^2$$ \$$cos(x^3$$) + \$$sec^2(x$$) - \sin(x)$$

Question 4

Given $$f(x) = x^2$, g(x) = 1 - x$, and k(x) = x$, what is $$\frac{d}{dx} f(g(k(x)))$$?

Accepted Answer

$$-2(1-x)$$

Question 5

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$2$$

Question 6

Using the limit definition of the derivative, find $$f'(1)$$ for $$f(x) = x^2$ + x.$$

Accepted Answer

$$3$$

Question 7

For the curve $$x^2$$ + $$y^2 + 2y = 9$, what is $$\frac{dy}{dx}$$ at the point (3$$, -2)$$?

Accepted Answer

$$\frac{3}{2}$$

Question 8

A spherical balloon is being inflated at a rate of $$5 	ext{ cm}^3/	ext{s}. $$What is the rate of change of the radius when the radius is $$3 	ext{ cm}$$? (Volume of sphere: $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$\frac{5}{12\pi}$$

Question 9

A designer wants to minimize the average cost $$A(x) = \frac{900 + 42x}{x}$$ for producing $$x$$ bookcases. What is the optimal number of bookcases?

Accepted Answer

$$x = 20$$

Question 10

Let $$F(x) = \frac{x^2$}{x^2$ + 2}. On $$which interval is $$F(x)$$ increasing?

Accepted Answer

$$(0, \infty)$$

Question 11

Find the horizontal asymptote of $$F(x) = \frac{x^2$}{x^2$ + 2}.$$

Accepted Answer

$$y = 1$$

Question 12

Given the graph of $$f on$$ $$[0,2]$$ and rectangles for a left Riemann sum, what is the approximate area under the curve?

Accepted Answer

$$2.5$$

Question 13

Compute the area of the region enclosed by $$y = x^2$ and y$$ = $$x^4$$ between $$x = 0$$ and $$x = 1.$$

Accepted Answer

$$\$$int_0$$^1 (x^2$ - x^4$) dx$$

Question 14

The Road Runner is running at a constant $$10 	ext{ ft/s}$$, and the Coyote starts chasing after him at $$t = 4 s $$with a head start of $$96 ft. $$How long will it take the Coyote to catch up if he runs at $$14 	ext{ ft/s}$$?

Accepted Answer

$$24 s$$

Question 15

If $$y = \$$int_0$$^x \sqrt{1 + t^2$} dt$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sqrt{1 + x^2$}$$

Question 16

For the function $$f(x) = \frac{x^2$}{x^2$ + 2}$$, what is the value of $$f(0)$$?

Accepted Answer

$$0$$

Question 17

Find the equation of the tangent line to the curve $$x^2$$ + $$y^2 + 2y = 9$ at the point (3$$, -2)$$.

Accepted Answer

$$y + 2 = \frac{3}{2}(x - 3)$$

Question 18

If $$f(x) = x^2$ + x$$, what is the slope of the tangent at $$x = 1$$ using the limit definition?

Accepted Answer

$$3$$

Question 19

For the function $$F(x) = \frac{x^2$}{x^2$ + 2}$$, what is the local minimum?

Accepted Answer

$$x = 0$$

Calculus Final Exam Study Guidance

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