Q: Which of the following is the correct limit definition of the derivative of a function $$f at$$ $$x$$?

$$f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$$

Q: If $$\lim_{x \to \infty} f(x) = 0$$ and $$\lim_{x \to -\infty} f(x) = 0$$, which of the following could be the graph of $$f(x)$$?

A function that approaches the $$x-$$axis as $$x \to \pm\infty$$

Question 1

A ball is thrown vertically upward with a velocity of 28 ft/s. Its height (in feet) after $$t$$ seconds is given by $$y = 28t - 16t^2$. What is the velocity of the ball after 1 second?

Accepted Answer

$$-4 ft/s$$

Question 2

Given the function $$h(x) = \dfrac{3x - 3}{x^2$ - 2}$$, what is the horizontal asymptote of $$h(x)$$?

Accepted Answer

$$y = 0$$

Question 3

For the function $$h(x) = \dfrac{3x - 3}{x^2$ - 2}$$, what is the vertical asymptote?

Accepted Answer

$$x = \sqrt{2}$$ and $$x = -\sqrt{2}$$

Question 4

Given the piecewise function $$k(x) = \begin{cases} -\sqrt{x} & x \u003c 0 \ 3 - x & 0 \leq x \u003c 3 \ (x-3)^2$ & x \geq 3 \end{cases}$$, at what value(s) of $$x is$$ $$k(x)$$ discontinuous?

Accepted Answer

$$x = 0$$ and $$x = 3$$

Question 5

Find $$\displaystyle \lim_{x 	o 3^-} k(x)$$ for the piecewise function $$k(x) = \begin{cases} -\sqrt{x} & x \u003c 0 \ 3 - x & 0 \leq x \u003c 3 \ (x-3)^2$ & x \geq 3 \end{cases}.$$

Accepted Answer

$$0$$

Question 6

Given $$g(x) = \dfrac{3}{x}$$, use the limit definition of the derivative to find $$g'(x). $$Which of the following is correct?

Accepted Answer

$$g'(x) = -\dfrac{3}{x^2$}$$

Question 7

A function $$f$$ satisfies $$f(0) = 0$$, $$f(3) = 2$$, $$\lim_{x 	o 0} f(x) = 3$$, and $$\lim_{x 	o 3} f(x) = 4. $$Which of the following statements is true?

Accepted Answer

$$f is $$not continuous at $$x = 0$$ and $$x = 3$$

Question 8

Suppose $$f(x) is a $$function such that $$\lim_{x 	o a} f(x)$$ exists, but $$f(a) is $$not defined. What can you conclude about the continuity of $$f at$$ $$x = a$$?

Accepted Answer

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

Question 9

Given $$y = 28t - 16t^2$, at what time does the ball hit the ground?

Accepted Answer

$$t = 1.75$$ seconds

Question 10

For the function $$y = 28t - 16t^2$, what is the impact velocity (velocity when the ball hits the ground)?

Accepted Answer

$$-28 ft/s$$

Question 11

Find the equation of the tangent line to the curve $$g(x) = \dfrac{3}{x} at $$the point $$(1, 3). $$Which is correct?

Accepted Answer

$$y = -3x + 6$$

Question 12

Which of the following is the correct limit definition of the derivative of a function $$f at$$ $$x$$?

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \dfrac{f(x+h) - f(x)}{h}$$

Question 13

Suppose $$f(x) is $$defined as $$f(x) = \dfrac{(2x + h)^2$ - 64}{h}$$ for $$h 
eq 0. $$For which value of $$a$$ does $$\lim_{h 	o 0} \dfrac{f(a+h) - f(a)}{h}$$ exist?

Accepted Answer

$$a = 4$$

Question 14

If $$\lim_{x 	o \infty} f(x) = 0$$ and $$\lim_{x 	o -\infty} f(x) = 0$$, which of the following could be the graph of $$f(x)$$?

Accepted Answer

A function that approaches the $$x-$$axis as $$x 	o \pm\infty$$

Question 15

Which of the following best describes a removable discontinuity?

Accepted Answer

A point where the limit exists but the function is not defined or is defined differently

Question 16

Given the piecewise function $$k(x) = \begin{cases} -\sqrt{x} & x \u003c 0 \ 3 - x & 0 \leq x \u003c 3 \ (x-3)^2$ & x \geq 3 \end{cases}$$, what is $$\lim_{x 	o 0^+} k(x)$$?

Accepted Answer

$$3$$

Question 17

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$

Calculus Study Notes: Limits, Asymptotes, Piecewise Functions, and Tangents

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