Q: A function $$f(x) is $$continuous at $$x = a if$$:

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

Q: Which of the following limits does NOT exist?

$$\displaystyle \lim_{x \to 0} \frac{1}{x}$$

Question 1

Which of the following statements best describes the meaning of $$\displaystyle \lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ becomes arbitrarily close to $$L.$$

Question 2

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

Accepted Answer

$$4$$

Question 3

A function $$f(x) is $$continuous at $$x = a if$$:

Accepted Answer

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

Question 4

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

Accepted Answer

$$3x^2 - 10x$

Question 5

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

Accepted Answer

$$1$$

Question 6

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

Accepted Answer

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

Question 7

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \u003c 1 \ 2x + 1 & x \geq 1 \end{cases}. $$For which value of $$a is$$ $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

$$a = 3$$

Question 8

Evaluate $$\displaystyle \int x^2$ e$$^{x^3}$$ dx.$$

Accepted Answer

$$\displaystyle \frac{1}{3} e$$^{x^3}$$ + C$$

Question 9

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

Accepted Answer

$$\displaystyle (fg)' = f'g + fg'$$

Question 10

A hospital monitors the rate at which a patient's medication is metabolized, modeled by $$M(t) = 5e$$^{-0.2t}$$. What is the rate of change of medication in the body at t = 3$ hours?

Accepted Answer

$$-1e^{-0.6}$$

Question 11

Evaluate $$\displaystyle \$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 12

Find the derivative of $$f(x) = \sqrt{x^2$ + 1}.$$

Accepted Answer

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

Question 13

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

Accepted Answer

$$1$$

Question 14

Which of the following is the correct antiderivative of $$\displaystyle \frac{1}{1 + x^2$}$$?

Accepted Answer

$$\arctan x + C$$

Question 15

A chemical reaction's rate is given by $$R(t) = 4t^3$ - 2t. $$What is the total change in concentration from $$t = 0 to$$ $$t = 2$$?

Accepted Answer

$$12$$

Question 16

Which of the following limits does NOT exist?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} \frac{1}{x}$$

Question 17

Find $$\displaystyle \frac{d}{dx} \left( \frac{1}{x} \right).$$

Accepted Answer

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

Question 18

Evaluate $$\displaystyle \int \cos(2x) dx.$$

Accepted Answer

$$\frac{1}{2} \sin(2x) + C$$

Question 19

Suppose $$f(x) is $$not continuous at $$x = a$$ because $$\displaystyle \lim_{x 	o a} f(x)$$ does not exist. Which type of discontinuity is this?

Accepted Answer

Essential (infinite or jump) discontinuity

Calculus I Final Exam Review: Limits, Derivatives, and Integrals

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드