Q: Which of the following is the correct definition of the derivative?

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

Q: If $$f(x)$$ has a jump discontinuity at $$x = 2$$, which of the following is true?

$$\lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x)$$

Question 1

Evaluate the following limit: $$\lim_{x 	o 0} |x| \cos\left(\frac{1}{x}ight).$$

Accepted Answer

$$0$$

Question 2

Evaluate the limit: $$\lim_{x 	o \infty} \frac{\sqrt{x^2$ + 3x + 1}}{3x}.$$

Accepted Answer

$$\frac{1}{3}$$

Question 3

Evaluate the limit: $$\lim_{x 	o 5^-} \frac{2}{x-5}.$$

Accepted Answer

$$-\infty$$

Question 4

Based on the graph provided, which of the following $$x-$$values is a removable discontinuity for $$f(x)$$?

Accepted Answer

$$x = 6$$

Question 5

Which interval is $$f(x)$$ continuous on, based on the graph?

Accepted Answer

$$(3, 6)$$

Question 6

Evaluate $$\lim_{x 	o 2^+} f(x)$$ using the graph.

Accepted Answer

$$4$$

Question 7

Using the limit definition, what is $$f'(x)$$ for $$f(x) = 3x^2$?

Accepted Answer

$$6x$$

Question 8

Using the Power Rule, what is the derivative of $$f(x) = 3x^2$?

Accepted Answer

$$6x$$

Question 9

What is the equation of the tangent line to $$y = 3x^2$ at x = 1$?

Accepted Answer

$$y = 6x - 3$$

Question 10

Given $$s(t) = \frac{3t - 1}{t + 1}$$, what is the average velocity on $$[0, 3]$$?

Accepted Answer

$$1$$

Question 11

Find the instantaneous velocity $$v(t)$$ for $$s(t) = \frac{3t - 1}{t + 1}.$$

Accepted Answer

$$v(t) = \frac{4}{(t+1)^2$}$$

Question 12

Compute $$v(3)$$ for $$s(t) = \frac{3t - 1}{t + 1}.$$

Accepted Answer

$$0.25$$

Question 13

Which theorem justifies the existence of a solution to $$s(t) = 0 on an $$interval of length 1?

Accepted Answer

Intermediate Value Theorem

Question 14

Which of the following is the correct definition of the derivative?

Accepted Answer

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

Question 15

Which rule would you use to compute $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$$?

Accepted Answer

Quotient Rule

Question 16

If $$f(x)$$ has a jump discontinuity at $$x = 2$$, which of the following is true?

Accepted Answer

$$\lim_{x 	o 2^-} f(x) 
eq \lim_{x 	o 2^+} f(x)$$

Question 17

Which of the following is the correct Power Rule for differentiation?

Accepted Answer

$$\frac{d}{dx} x^n = n x$$^{n-1}$$

Question 18

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees there is a $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Calculus I Midterm Study Guide: Limits, Continuity, Derivatives, and Applications

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