Q: A function $$g(x) is $$defined as follows: $$g(x) = \begin{cases} x^2$ & x \u003c 1 \\ 2x + 1 & x \geq 1 \end{cases} Is$$ $$g(x)$$ continuous at $$x=1$$?

No, because $$\displaystyle \lim_{x \to 1} g(x) \neq g(1).$$

Q: Suppose $$f(x)$$ has a removable discontinuity at $$x=2. $$Which of the following is most likely true?

$$\displaystyle \lim_{x \to 2} f(x)$$ exists, but $$f(2) is $$either not defined or not equal to the limit.

Q: A chemical reaction's temperature (in $$^\circC) at $$time $$t (in $$minutes) is given by $$T(t) = 100e$$^{-0.05t}$$. What is the rate of change of temperature at t=10$ minutes?

$$-5e^{-0.5}$$ $$^\circC$/\text{min}$$

Question 1

Let $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What is $$\displaystyle \lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 2

Which of the following best describes why the function $$f(x) = \frac{1}{x-3} is $$discontinuous at $$x=3$$?

Accepted Answer

$$f(3)$$ does not exist because the denominator is zero.

Question 3

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

Accepted Answer

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

Question 4

A function $$g(x) is $$defined as follows:

$$g(x) = \begin{cases} x^2$ & x \u003c 1 \ 2x + 1 & x \geq 1 \end{cases}

Is$$ $$g(x)$$ continuous at $$x=1$$?

Accepted Answer

No, because $$\displaystyle \lim_{x 	o 1} g(x) 
eq g(1).$$

Question 5

Which of the following is the correct derivative of $$f(x) = x^3$ e^x$$?

Accepted Answer

$$3x^2 e^x$$ + $$x^3 e^x$

Question 6

Suppose the position of a car is given by $$s(t) = 4t^2$ - 3t + 2 (in $$meters) at time $$t (in $$seconds). What is the instantaneous velocity at $$t=2$$ seconds?

Accepted Answer

$$13$$ meters/second

Question 7

Which of the following statements about the function $$f(x) = |x| is $$true?

Accepted Answer

$$f(x) is $$continuous everywhere but not differentiable at $$x=0.$$

Question 8

Find the equation of the tangent line to $$f(x) = x^2$ + 2x at$$ $$x=1.$$

Accepted Answer

Question 9

Suppose $$f(x)$$ has a removable discontinuity at $$x=2. $$Which of the following is most likely true?

Accepted Answer

$$\displaystyle \lim_{x 	o 2} f(x)$$ exists, but $$f(2) is $$either not defined or not equal to the limit.

Question 10

Which of the following best describes the horizontal asymptote(s) of $$f(x) = \frac{2x}{x+1}$$?

Accepted Answer

$$y = 2$$

Question 11

Let $$f(x) = \sin(x). $$What is $$f'(\pi)$$?

Accepted Answer

$$-1$$

Question 12

A chemical reaction's temperature (in $$^\circC) at $$time $$t (in $$minutes) is given by $$T(t) = 100e$$^{-0.05t}$$. What is the rate of change of temperature at t=10$ minutes?

Accepted Answer

$$-5e^{-0.5}$$ $$^\circC$/	ext{min}$$

Question 13

Use linear approximation to estimate $$\sqrt{4.1}.$$

Accepted Answer

$$2.025$$

Question 14

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

Accepted Answer

$$\sec^2$(x)$$

Question 15

Suppose $$f(x) is $$differentiable and $$f'(a) = 0. $$Which of the following statements is always true?

Accepted Answer

The tangent line to $$f(x) at$$ $$x=a is $$horizontal.

Question 16

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 17

A ball is thrown upward with position $$s(t) = 20t - 5t^2$ (in meters). What is the maximum height reached by the ball?

Accepted Answer

$$20$$ meters

Question 18

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

Accepted Answer

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

Question 19

Suppose $$f(x) is $$continuous everywhere except at $$x=1$$, where $$\displaystyle \lim_{x 	o 1^-} f(x) = 2$$ and $$\displaystyle \lim_{x 	o 1^+} f(x) = 5. $$What type of discontinuity does $$f(x)$$ have at $$x=1$$?

Accepted Answer

Jump discontinuity

Calculus I Final Exam Study Guide: Limits, Continuity, Derivatives, and Applications

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