Q: Which of the following is a correct application of the Sandwich (Squeeze) Theorem?

To show $$\displaystyle \lim_{x \to 0} x^2$ \sin\left(\frac{1}{x}\right) = 0$$

Q: If $$f(x) is $$continuous at $$x = 3$$ and $$f(3) = 7$$, which of the following must be true?

$$\displaystyle \lim_{x \to 3} f(x) = 7$$

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

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

Q: Which of the following limits does NOT exist?

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

Q: Which of the following is a necessary condition for a function $$f(x) to be $$continuous at $$x = a$$?

Both $$f(a) is $$defined and $$\displaystyle \lim_{x \to a} f(x)$$ exists, and they are equal

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 average rate of change of the car's position from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$17$$ meters/second

Question 2

Find the equation of the tangent line to the curve $$y = x^3$ - 2x at $$the point where $$x = 2.$$

Accepted Answer

$$y = 10x - 12$$

Question 3

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

Accepted Answer

$$4$$

Question 4

Which of the following statements is TRUE about the limit $$\displaystyle \lim_{x 	o 0} \frac{\sin(x)}{x}$$?

Accepted Answer

The limit equals $$1 by $$the key trigonometric limit.

Question 5

Given $$f(x) = \frac{1}{x}$$, which of the following best describes $$\displaystyle \lim_{x 	o 0^-} f(x)$$?

Accepted Answer

$$-\infty$$

Question 6

Which of the following functions is NOT continuous at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{x-2}{x-2}$$

Question 7

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

Accepted Answer

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

Question 8

A chemical reaction's temperature (in $$^\circC) at $$time $$t (in $$minutes) is given by $$T(t) = 100 - 20e$$^{-0.5t}$$. What is the instantaneous rate of change of temperature at t = 0$?

Accepted Answer

$$10$$ $$^\circC/$$min

Question 9

Which of the following is a correct application of the Sandwich (Squeeze) Theorem?

Accepted Answer

To show $$\displaystyle \lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight) = 0$$

Question 10

If $$f(x) is $$continuous at $$x = 3$$ and $$f(3) = 7$$, which of the following must be true?

Accepted Answer

$$\displaystyle \lim_{x 	o 3} f(x) = 7$$

Question 11

Find $$\displaystyle \lim_{x 	o 0^+} \ln(x).$$

Accepted Answer

$$-\infty$$

Question 12

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

Accepted Answer

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

Question 13

Given $$f(x) = x^2$ - 4x + 3$$, find the slope of the tangent line at $$x = 1.$$

Accepted Answer

$$-6$$

Question 14

Which of the following limits does NOT exist?

Accepted Answer

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

Question 15

A medication's concentration in the bloodstream is modeled by $$C(t) = \frac{10}{t+2}$$, where $$t is in $$hours. What is $$\displaystyle \lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 16

Which of the following is a necessary condition for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

Both $$f(a) is $$defined and $$\displaystyle \lim_{x 	o a} f(x)$$ exists, and they are equal

Question 17

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

Accepted Answer

$$1$$

Question 18

Which of the following best describes the end behavior of the rational function $$f(x) = \frac{2x^3$ + 1}{x^3$ - 5} as$$ $$x 	o -\infty$$?

Accepted Answer

$$f(x) 	o 2$$

Question 19

If $$f(x) = |x|$$, which of the following statements is TRUE?

Accepted Answer

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

Calculus I Exam 1 Study Guide: Limits, Continuity, and Derivatives

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