Q: Which of the following statements is TRUE about the function $$f(x) at$$ $$x = a if$$ $$\displaystyle \lim_{x \to a^-} f(x) = 5$$, $$\displaystyle \lim_{x \to a^+} f(x) = 5$$, but $$f(a) is $$undefined?

The limit as $$x \to a$$ exists, but the function is not defined at $$x = a.$$

Question 1

Given the graph of a function $$f(x)$$, at point $$A$$ the tangent line is horizontal. Which of the following best describes the slope of the function at point $$A$$?

Accepted Answer

Zero

Question 2

Evaluate the following limit:

$$\displaystyle \lim_{x 	o 3} (2x^2$ - 5x + 1)$$

Accepted Answer

$$4$$

Question 3

Given the function $$f(x) = x^2$ - 4x + 7$$, find the equation of the tangent line to the curve at $$x = 2.$$

Accepted Answer

$$y = -2x + 11$$

Question 4

Which of the following statements is TRUE about the function $$f(x) at$$ $$x = a if$$ $$\displaystyle \lim_{x 	o a^-} f(x) = 5$$, $$\displaystyle \lim_{x 	o a^+} f(x) = 5$$, but $$f(a) is $$undefined?

Accepted Answer

The limit as $$x 	o a$$ exists, but the function is not defined at $$x = a.$$

Question 5

Find $$\displaystyle \lim_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 6

Given $$f(x) = 3x^5$ - 7x^2$ + 11$$, find $$f'(x).$$

Accepted Answer

$$15x^4 - 14x + 0$

Question 7

Use the limit definition of the derivative to find $$f'(x)$$ for $$f(x) = 2x.$$

Accepted Answer

$$2$$

Question 8

Given $$y = \sqrt[4]{x^9$}$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$\frac{9}{4} x$$^{5/4}$$

Question 9

Given $$g(x) = (x$$^{11}$$ - x + 12)^3$, find g$$'(x)$$.

Accepted Answer

$$3(x^{11} - x$$ + $$12)^2$$ $$(11x^{10} - 1$$)$$

Question 10

Given $$u = 5x - y + x^8$ y^3$, find $$\frac{\partial u}{\partial y}$$.

Accepted Answer

$$-1 + 3x^8$ y^2$

Question 11

Find the third derivative $$f'''(x)$$ for $$f(x) = 2x^5$ - 4x^2$ + 7.$$

Accepted Answer

$$120x^2 - 24$

Question 12

Given $$C(x) = \frac{x^3$}{12} + 4x + 40$$, where $$C is in $$units of $$100 and x$ is units produced, what is the marginal cost function?

Accepted Answer

$$C'(x) = \frac{x^2$}{4} + 4$$

Question 13

Evaluate $$\displaystyle \lim_{x 	o -3} \frac{x + 3}{x^2$ - 9}.$$

Accepted Answer

$$0$$

Question 14

Given $$f(x) = \frac{7x}{x^3$ + 2x + 10}$$, find $$\displaystyle \lim_{x 	o \infty} f(x).$$

Accepted Answer

$$0$$

Question 15

Given $$y = x^4$ + 10x - 12$$, find $$f''(-2).$$

Accepted Answer

$$12$$

Question 16

Suppose the graph of $$f(x)$$ has a sharp corner at $$x = 1. $$Which of the following is TRUE?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 1$$

Question 17

Given $$f(x) = \sqrt{x^5$ + 9}$$, find $$\frac{df}{dx}.$$

Accepted Answer

$$\frac{5x^4$}{2\sqrt{x^5$ + 9}}$$

Question 18

If $$f(x) is $$defined for all $$x$$ and $$\displaystyle \lim_{x 	o a} f(x)$$ exists, but $$f(a) 
eq \lim_{x 	o a} f(x)$$, which of the following is TRUE?

Accepted Answer

$$f(x) is $$not continuous at $$x = a$$

Question 19

Given $$f(x) = \frac{2}{(x^4$ - 1)^3$}$$, find $$\frac{df}{dx}.$$

Accepted Answer

$$-24x^3$$ $$(x^4$$ - $$1)^{-4}$$

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