Question 1

Which of the following statements about the domain of the function $$f(x) = \sqrt{x-2} is $$correct?

Accepted Answer

The domain is all real numbers $$x$$ such that $$x \geq 2.$$

Question 2

Suppose $$f(x) = x^2$ and g(x$$) = \sin(x)$$. What is the derivative of the product f(x)g(x$$)$$ with respect to x$?

Accepted Answer

$$2x \sin(x) + x^2$ \cos(x)$$

Question 3

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

Accepted Answer

$$26$$

Question 4

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 5

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

Accepted Answer

$$1$$

Question 6

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

Accepted Answer

It is the slope of the tangent line to the graph of $$f at$$ $$x=a.$$

Question 7

Let $$f(x) = |x|. $$Which statement about the differentiability of $$f at$$ $$x=0 is $$correct?

Accepted Answer

$$f is $$not differentiable at $$x=0.$$

Question 8

A chemical reaction's concentration $$C(t)$$ changes over time according to $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of concentration at t=0$?

Accepted Answer

Question 9

Which of the following is the correct application of the quotient rule for differentiation?

Accepted Answer

$$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2$}$$

Question 10

If $$\lim_{x 	o a} f(x) = L$$ and $$\lim_{x 	o a} g(x) = M$$, what is $$\lim_{x 	o a} [f(x) + g(x)]$$?

Accepted Answer

$$L + M$$

Question 11

Which of the following is an example of a function that is continuous everywhere but not differentiable at some point?

Accepted Answer

$$f(x) = |x|$$

Question 12

A patient’s temperature $$T(t) is $$modeled by $$T(t) = 37 + 0.5\sin(\pi t/12)$$, where $$t is $$hours after midnight. What is the rate of change of temperature at $$t=6$$?

Accepted Answer

$$\frac{\pi}{24}$$

Question 13

Which of the following is the correct definition of the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$|x - a| \u003c \delta$$, then $$|f(x) - L| \u003c \epsilon.$$

Question 14

Which of the following statements about the function $$f(x) = \frac{1}{x} is $$true?

Accepted Answer

$$f(x) is $$continuous everywhere except at $$x=0.$$

Question 15

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t=0$?

Accepted Answer

$$5$$

Question 16

Which of the following is the correct application of the chain rule for differentiation?

Accepted Answer

$$\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$$

Question 17

Which of the following is the correct formula for the derivative of $$f(x) = x^n$$ where $$n is a $$positive integer?

Accepted Answer

$$f'(x) = nx$$^{n-1}$$

Question 18

Which of the following is true about the function $$f(x) = \sqrt{x}$$?

Accepted Answer

$$f(x) is $$defined for $$x \geq 0.$$

Question 19

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 I: Functions, Limits, and Derivatives – Core Concepts and Techniques

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