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

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

Q: Suppose $$f(x) is $$continuous at $$x = 3. $$Which of the following must be true?

$$\lim_{x \to 3} f(x)$$ exists and equals $$f(3)$$

Q: Which of the following is the correct application of L'Hôpital's Rule to $$\lim_{x \to 0} \frac{\sin x}{x}$$?

$$\lim_{x \to 0} \frac{\cos x}{1} = 1$$

Question 1

Which of the following statements best describes the relationship between exponential and logarithmic functions?

Accepted Answer

The logarithmic function is the inverse of the exponential function.

Question 2

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

Accepted Answer

$$4$$

Question 3

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.3t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$C'(2) = -1.5e$$^{-0.6}$$

Question 4

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

Accepted Answer

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

Question 5

Suppose $$f(x) is $$continuous at $$x = 3. $$Which of the following must be true?

Accepted Answer

$$\lim_{x 	o 3} f(x)$$ exists and equals $$f(3)$$

Question 6

A company wants to minimize the cost $$C(x) = x^2$ - 8x + 20$$ for producing $$x$$ units. What is the minimum cost and at how many units is it achieved?

Accepted Answer

Minimum cost is $$4 at$$ $$x = 4$$ units

Question 7

Which of the following is the correct application of the chain rule for $$y = \sin(x^2$)$$?

Accepted Answer

$$y' = 2x \cos(x^2$)$$

Question 8

Evaluate the definite integral $$\$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 9

A car's velocity is given by $$v(t) = 4t$$ meters per second. What is the total distance traveled from $$t = 0 to$$ $$t = 5$$ seconds?

Accepted Answer

$$50$$ meters

Question 10

Which of the following is the correct statement of the Fundamental Theorem of Calculus?

Accepted Answer

If $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a)$$

Question 11

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 12

A tank is being filled with water at a rate given by $$R(t) = 10 - t$$ liters per minute, where $$t is in $$minutes. How much water enters the tank from $$t = 0 to$$ $$t = 5$$ minutes?

Accepted Answer

$$37.5$$ liters

Question 13

Which of the following is the correct application of L'Hôpital's Rule to $$\lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\cos x}{1} = 1$$

Question 14

Find the area between the curves $$y = x^2$ and y = x$ from x = 0$ to x = 1$.

Accepted Answer

$$\frac{1}{6}$$

Question 15

Which of the following is the correct derivative of $$f(x) = \arctan(x)$$?

Accepted Answer

$$\frac{1}{1 + x^2$}$$

Question 16

A bacteria population doubles every hour and is modeled by $$P(t) = 100 \cdot 2^t. $$What is the population's rate of change at $$t = 3$$ hours?

Accepted Answer

$$800 \ln 2$$

Question 17

Which of the following is the correct antiderivative of $$f(x) = \cos(x)$$?

Accepted Answer

$$\sin(x) + C$$

Question 18

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in an $$interval. Which of the following must be true?

Accepted Answer

$$f(x) is $$increasing on that interval

Question 19

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t m/s. $$What is the maximum height reached by the ball?

Accepted Answer

$$20.41$$ meters

Calculus I Syllabus: Key Topics and Study Guide

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