Q: Which of the following limits represents the derivative of $$f(x) at$$ $$x = a$$ using the definition of the derivative?

$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: Which of the following statements is true about continuity at $$x = a$$?

$$f(a) is $$defined, $$\displaystyle \lim_{x \to a} f(x)$$ exists, and $$\displaystyle \lim_{x \to a} f(x) = f(a).$$

Question 1

Which of the following best describes the geometric meaning of the derivative $$f'(a) at a $$point $$x = a$$?

Accepted Answer

It is the slope of the tangent line to the curve $$y = f(x) at$$ $$x = a.$$

Question 2

Given the function $$f(x) = x^2$, what is the equation of the tangent line to the curve at x = 3$?

Accepted Answer

$$y = 6x - 9$$

Question 3

Which of the following limits represents the derivative of $$f(x) at$$ $$x = a$$ using the definition of the derivative?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 4

A car's position is given by $$s(t) = 4t^2$ + 2t. $$What is the instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$26$$

Question 5

Evaluate the one-sided limit: $$\displaystyle \lim_{x 	o 2^+} \frac{x^2$ - 4}{x-2}.$$

Accepted Answer

$$4$$

Question 6

Which of the following statements is true about continuity at $$x = a$$?

Accepted Answer

$$f(a) is $$defined, $$\displaystyle \lim_{x 	o a} f(x)$$ exists, and $$\displaystyle \lim_{x 	o a} f(x) = f(a).$$

Question 7

If $$\displaystyle \lim_{x 	o 1^-} f(x) = 2$$ and $$\displaystyle \lim_{x 	o 1^+} f(x) = 3$$, what can be said about $$\displaystyle \lim_{x 	o 1} f(x)$$?

Accepted Answer

The limit does not exist.

Question 8

Which of the following is the correct interpretation of the average rate of change of a function $$f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\displaystyle \frac{f(b) - f(a)}{b - a}$$

Question 9

Suppose $$f(x) is $$continuous everywhere except at $$x = 2$$, where $$f(2) is $$undefined. Which of the following is true?

Accepted Answer

$$f(x) is $$not continuous at $$x = 2.$$

Question 10

A chemical reaction's rate is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is the instantaneous rate of change at t = 0$?

Accepted Answer

$$-1$$

Question 11

Which of the following is NOT a requirement for a function to be differentiable at $$x = a$$?

Accepted Answer

The function must be increasing at $$x = a.$$

Question 12

If $$f(x) = |x|$$, which of the following statements is true about the derivative at $$x = 0$$?

Accepted Answer

The derivative does not exist at $$x = 0.$$

Question 13

Find the derivative of $$f(x) = 3x^3$ - 2x + 5.$$

Accepted Answer

$$9x^2 - 2$

Question 14

A population grows according to $$P(t) = 1000e$$^{0.05t}$$. What is the rate of growth at t = 10$?

Accepted Answer

$$50e^{0.5}$$

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 does not match the limit.

Question 16

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

Accepted Answer

$$1$$

Question 17

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

Accepted Answer

$$f(x) is $$continuous at $$x = a.$$

Question 18

A ball is thrown upward with position $$s(t) = 20t - 5t^2$. What is the velocity when the ball reaches its maximum height?

Accepted Answer

$$0$$

Question 19

Which of the following is the best description of a tangent line to a curve at a point?

Accepted Answer

A line that touches the curve at exactly one point and has the same slope as the curve at that point.

Question 20

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

Accepted Answer

$$0$$

Key Concepts in Limits and Derivatives

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