Question 1

Which of the following best describes the relationship between finding the instantaneous velocity of an object at a given time and finding the slope of the tangent to the graph of a function at a point?

Accepted Answer

Both are found by taking the limit of the average rate of change as the interval approaches zero.

Question 2

An object moves vertically along a line with position $$s(t) = -16t^2$ + 128t. $$What is the average velocity of the object over the interval $$[1,4]$$?

Accepted Answer

$$48$$

Question 3

Given $$f(x) = 3x^2$ + 2x - 10$$, use the limit definition of the derivative to find $$f'(1).$$

Accepted Answer

$$5$$

Question 4

Evaluate the following limit: $$\lim_{x 	o 2} \frac{3x^2$ - 8x + 4}{x-2}$$

Accepted Answer

$$-2$$

Question 5

Which of the following is NOT a valid way to notate the derivative of a function $$f$$?

Accepted Answer

$$f(x)'$$

Question 6

Suppose $$f(x) is $$continuous everywhere except at $$x=2$$, where $$f(x)$$ has a jump discontinuity. Which statement is true?

Accepted Answer

The discontinuity at $$x=2 is $$non-removable.

Question 7

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 8

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 5x + 6}{x-2}. $$What is the limit as $$x$$ approaches $$2$$?

Accepted Answer

Question 9

Given the function $$f(x) = \sqrt{x^2$ + 1}$$, what is the equation of the tangent line to $$f at$$ $$x=4$$?

Accepted Answer

$$y = \frac{4}{\sqrt{17}}(x-4) + \sqrt{17}$$

Question 10

Use the Intermediate Value Theorem to determine if the equation $$25 - 8x^3$ - x^2$ = 0$$ has a solution in the interval $$[-2,4].$$

Accepted Answer

Yes, because the function changes sign over the interval.

Question 11

Evaluate $$\lim_{x 	o \infty} \frac{2x^2$ + 1}{3x^2$ + 2x + 1}$$ and state the horizontal asymptote.

Accepted Answer

Limit is $$\frac{2}{3}$$, HA: $$y=\frac{2}{3}$$

Question 12

For the function $$f(x) = \frac{x^2$ + 2x}{x^2$ - 1}$$, identify the type and location of any vertical asymptotes.

Accepted Answer

Vertical asymptotes at $$x=1$$ and $$x=-1$$

Question 13

Given the graph of $$f(x)$$ with a sharp corner at $$x=1$$, which statement is true about $$f'(x) at$$ $$x=1$$?

Accepted Answer

$$f'(1)$$ does not exist.

Question 14

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

Accepted Answer

$$-1$$

Question 15

Which of the following intervals is the function $$f(x)$$ continuous if it has jump discontinuities at $$x=-2$$ and $$x=3$$?

Accepted Answer

$$(-\infty, -2) \cup (-2, 3) \cup (3, \infty)$$

Question 16

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

Accepted Answer

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

Question 17

Evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 18

Given $$f(x) = 20 \cos t at$$ $$t = \frac{\pi}{2}$$, what is the instantaneous velocity at $$t = \frac{\pi}{2}$$?

Accepted Answer

$$0$$

Question 19

Which of the following is a removable discontinuity?

Accepted Answer

A hole in the graph where the function is undefined but the limit exists.

Calculus I Study Notes: Limits, Continuity, and Introduction to Derivatives

Guida di studio - Domande di pratica

Study guide - Flashcard