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

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

Q: A function $$g(x) is $$continuous at $$x = 5 if $$which of the following conditions are satisfied?

$$\displaystyle \lim_{x \to 5} g(x)$$ exists, $$g(5)$$ exists, and $$\displaystyle \lim_{x \to 5} g(x) = g(5)$$

Question 1

Suppose the position of a car along a straight road is given by $$s(t) = 3t^2$ + 2t + 1$$, where $$s is in $$meters and $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 2$$ seconds?

Accepted Answer

$$v = 14 m/s$$

Question 2

Which of the following best describes the difference between the average rate of change and the instantaneous rate of change of a function $$f(x)$$?

Accepted Answer

The average rate of change is the slope of the secant line, while the instantaneous rate of change is the slope of the tangent line.

Question 3

Evaluate $$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$2$$

Question 4

A function $$f(x) is $$defined as follows: $$f(x) = \frac{1}{x-3}. $$Which of the following statements is true?

Accepted Answer

There is a vertical asymptote at $$x = 3$$ because the denominator is zero at this point.

Question 5

Which of the following is the correct definition of the derivative of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 6

If $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x} is $$evaluated, what is the result and why is this limit important?

Accepted Answer

$$1$$, because it is a fundamental limit used in calculus, especially in derivatives of trigonometric functions.

Question 7

A function $$g(x) is $$continuous at $$x = 5 if $$which of the following conditions are satisfied?

Accepted Answer

$$\displaystyle \lim_{x 	o 5} g(x)$$ exists, $$g(5)$$ exists, and $$\displaystyle \lim_{x 	o 5} g(x) = g(5)$$

Question 8

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

Accepted Answer

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

Question 9

Which of the following scenarios describes a removable discontinuity?

Accepted Answer

A function has a hole at $$x = a$$ because the limit exists but $$f(a) is $$either undefined or not equal to the limit.

Question 10

Suppose $$h(x) = \sqrt{x+4}. $$What is the average rate of change of $$h(x)$$ from $$x = 0 to$$ $$x = 5$$?

Accepted Answer

$$\frac{\sqrt{9} - \sqrt{4}}{5 - 0} = \frac{3 - 2}{5} = 0.2$$

Question 11

If $$\displaystyle \lim_{x 	o 1} f(x) = 3$$ but $$f(1) is $$undefined, which of the following is true?

Accepted Answer

The limit exists at $$x = 1$$, but the function is not continuous at $$x = 1.$$

Question 12

Which of the following is the first step when trying to compute any type of limit?

Accepted Answer

Directly substitute the value into the function.

Question 13

A chemical reaction's concentration is modeled by $$C(t) = \frac{10}{t+2}. $$What is the horizontal asymptote of $$C(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$y = 0$$

Question 14

Which of the following best describes the Squeeze Theorem?

Accepted Answer

If $$f(x) \leq g(x) \leq h(x)$$ near $$x = a$$ and $$\displaystyle \lim_{x 	o a} f(x) = \lim_{x 	o a} h(x) = L$$, then $$\displaystyle \lim_{x 	o a} g(x) = L.$$

Question 15

A function $$p(x)$$ has a nonzero number over zero as $$x$$ approaches $$a. $$What does this indicate about $$\displaystyle \lim_{x 	o a} p(x)$$?

Accepted Answer

The limit does not exist and is infinite (vertical asymptote).

Question 16

Which of the following is NOT a type of discontinuity?

Accepted Answer

Horizontal discontinuity

Question 17

A ball is thrown upward and its height is given by $$h(t) = -16t^2$ + 32t + 5. $$What is the instantaneous rate of change of height at $$t = 1$$ second?

Accepted Answer

$$h'(1) = -32(1) + 32 = 0 ($$units depend on context)

Question 18

Which of the following best explains why $$\displaystyle \lim_{x 	o a} f(x) = L is $$not equivalent to $$f(a) = L$$?

Accepted Answer

The limit describes the behavior as $$x$$ approaches $$a$$, not necessarily the value at $$a.$$

Calculus I: Rates of Change, Limits, and Introduction to Derivatives

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