Question 1

Which of the following best describes the graph shown in the first image?

Accepted Answer

It is the graph of $$y = |x| + 2.$$

Question 2

Given the piecewise function $$f(x)$$ shown in the second image, what is $$f(1)$$?

Accepted Answer

$$f(1) = 2$$

Question 3

For the piecewise function in the second image, does $$\lim_{x 	o 1} f(x)$$ exist?

Accepted Answer

No, because the left and right limits are not equal.

Question 4

What is the slope of the line $$5x + y + 2 = 0$$?

Accepted Answer

$$-1/5$$

Question 5

Solve for $$x$$: $$x^2 + 2x - 48 = 0$.

Accepted Answer

$$x = 6 or$$ $$x = -8$$

Question 6

Simplify $$\frac{x^5$ y$$^{-7}$$}{x$$^{-2}$$ y}$$ and express using positive exponents.

Accepted Answer

$$x^7$$ $$y^{-8}$$

Question 7

Given $$x = \log_b 7$$ and $$y = \log_b 2$$, express $$\log_b 98 in $$terms of $$x$$ and $$y.$$

Accepted Answer

$$x + 2y$$

Question 8

Find the derivative $$f'(x)$$ for $$f(x) = 3x^5$ - 7x^2$ + 6x - 5.$$

Accepted Answer

$$15x^4 - 14x + 6$

Question 9

If $$B = 4400e$$^{0.05t}$$ models the number of books in a library, how many books will there be after 3 years? (Round to the nearest whole number.)

Accepted Answer

$$5107$$

Question 10

Solve for $$y in$$ $$y^2 = -5y + 5$ and express your answer exactly.

Accepted Answer

$$y = \frac{-5 \pm \sqrt{25 - 20}}{2}$$

Question 11

Find the equation of the tangent line to $$f(x) = 3x^2$ - 2 at $$the point $$(-2, 10).$$

Accepted Answer

$$y = -12x - 14$$

Question 12

For $$y = \frac{3x^2$ - 5}{2x^3$ + 3}$$, what is $$y'$$?

Accepted Answer

$$\frac{6x(2x^3$ + 3) - (3x^2$ - 5)6x^2$}{(2x^3$ + 3)^2$}$$

Question 13

If $$f(x) = \frac{1}{x+4}$$ for $$x \u003e -4$$ and $$f(x) = x^2$ + 4x$$ for $$x \leq -4$$, is $$f$$ continuous at $$x = -4$$?

Accepted Answer

No, because the left and right limits are not equal.

Question 14

Find the marginal revenue $$R'(x) if$$ $$p(x) = 40.5 - 8\ln x$$ and $$R(x) = x p(x).$$

Accepted Answer

$$40.5 - 8\ln x - 8$$

Question 15

Solve the exponential equation $$e^{0.01t} = 9$ for t$. Round to three decimal places.

Accepted Answer

$$t = 219.722$$

Question 16

Differentiate $$f(x) = e$$^{x^2 + 4x}$$.

Accepted Answer

$$e^{x^2 + 4x}(2x + 4$$)$$

Question 17

If $$g(x) = \ln(8x)$$, what is $$g'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 18

Find all values of $$x$$ where the tangent line to $$y = \frac{1}{3}x^3$ - 5x + 8 is $$horizontal.

Accepted Answer

$$x = \pm \sqrt{5}$$

Math 115 Practice Midterm – Step-by-Step Calculus Guidance

스터디 가이드 - 연습 문제