A line $$y = L$$ where $$f(x)$$ approaches $$L as$$ $$x \to \infty or$$ $$x \to -\infty$$

Question 1

Which of the following statements correctly describes a removable discontinuity at $$x = a$$ for a function $$f(x)$$?

Accepted Answer

The function $$f(x) is $$not defined at $$x = a$$, but $$\lim_{x 	o a} f(x)$$ exists and is finite.

Question 2

Given $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what type of discontinuity does $$f(x)$$ have at $$x = 1$$?

Accepted Answer

Removable discontinuity

Question 3

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, which theorem guarantees that $$f(x)$$ has a root in $$(a, b)$$?

Accepted Answer

Intermediate Value Theorem

Question 4

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

Accepted Answer

$$3$$

Question 5

Which of the following best describes a horizontal asymptote for the function $$f(x)$$?

Accepted Answer

A line $$y = L$$ where $$f(x)$$ approaches $$L as$$ $$x 	o \infty or$$ $$x 	o -\infty$$

Question 6

Find the average rate of change of $$f(x) = x^3$ on the interval $$[1, 4]$$.

Accepted Answer

$$21$$

Question 7

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 8

A car's position is given by $$s(t) = 4t^3$ - 2t^2$ + t. $$What is the car's acceleration at $$t = 2$$?

Accepted Answer

$$92$$

Question 9

Which rule should be used to differentiate $$f(x) = x^2$ e^x$$?

Accepted Answer

Product Rule

Question 10

Differentiate $$f(x) = \sin(x^2$).$$

Accepted Answer

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

Question 11

If $$y^2$$ + $$x^2 = 25$, what is $$\frac{dy}{dx}$$ using implicit differentiation?

Accepted Answer

$$-\frac{x}{y}$$

Question 12

Which of the following is the correct derivative of $$f(x) = |x|$$ for $$x 
eq 0$$?

Accepted Answer

$$\frac{x}{|x|}$$

Question 13

Given $$f(x) = \frac{2x}{x^2$ + 1}$$, find $$f'(x).$$

Accepted Answer

$$\frac{2(x^2$ + 1) - 4x^2$}{(x^2$ + 1)^2$}$$

Question 14

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 15

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of concentration at t = 3$ hours?

Accepted Answer

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

Question 16

Which of the following is the tangent line to $$f(x) = x^2$ at x = 2$?

Accepted Answer

$$y = 4x - 4$$

Question 17

If $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 18

Which of the following is the correct application of the Chain Rule to $$f(x) = \sin(3x)$$?

Accepted Answer

$$3 \cos(3x)$$

Question 19

A population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of change at t = 10$?

Accepted Answer

$$5e^{0.5}$$

Calculus Study Guide: Continuity, Limits, and Derivatives

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