Q: Which of the following statements about continuity is TRUE?

A function is continuous at $$x = a if$$ $$\lim_{x \to a} f(x) = f(a)$$

Q: If $$f(x) = \dfrac{1}{x}$$, which of the following statements is TRUE?

$$\displaystyle \lim_{x \to 0^+} f(x) = +\infty$$

Question 1

Which of the following best describes the domain of the function $$f(x) = \sqrt{5 - 2x}$$?

Accepted Answer

$$(-\infty, 2.5]$$

Question 2

Given $$f(x) = x^2$ - 3x$$ and $$g(x) = 2x + 1$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$(f \circ g)(x) = (2x + 1)^2$ - 3(2x + 1)$$

Question 3

Which transformation is applied to the graph of $$y = |x| to $$obtain $$y = -2|x + 3| + 1$$?

Accepted Answer

Shift left by 3, reflect over $$x-$$axis, stretch vertically by 2, shift up by 1

Question 4

If $$f(x) = 3x + 7$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x - 7}{3}$$

Question 5

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

Accepted Answer

$$2$$

Question 6

Which of the following statements about continuity is TRUE?

Accepted Answer

A function is continuous at $$x = a if$$ $$\lim_{x 	o a} f(x) = f(a)$$

Question 7

Find the derivative of $$f(x) = x^3$ - 5x^2$ + 4x - 7.$$

Accepted Answer

$$f'(x) = 3x^2$ - 10x + 4$$

Question 8

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$C'(2) = -5e$$^{-1}$$

Question 9

If $$y = \sin(x^2$)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

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

Question 10

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

Accepted Answer

$$y = 4x - 1$$

Question 11

A spherical balloon is being inflated so that its radius increases at a rate of $$2 cm/$$min. At the moment when the radius is $$5 cm$$, how fast is the volume increasing? (Recall: $$V = \dfrac{4}{3}\pi r^3$)

Accepted Answer

$$200\pi cm$^3/$$min

Question 12

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

Accepted Answer

$$\dfrac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Question 13

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

Accepted Answer

$$\dfrac{3}{5}$$

Question 14

If $$f(x) = \dfrac{1}{x}$$, which of the following statements is TRUE?

Accepted Answer

$$\displaystyle \lim_{x 	o 0^+} f(x) = +\infty$$

Question 15

Find $$\dfrac{dy}{dx} if$$ $$x^2$$ + $$y^2 = 25$ using implicit differentiation.

Accepted Answer

$$\dfrac{dy}{dx} = -\dfrac{x}{y}$$

Question 16

Which of the following is NOT a type of discontinuity?

Accepted Answer

Oscillating discontinuity

Question 17

Suppose $$f(x) is $$continuous everywhere except at $$x = 2$$, where $$\lim_{x 	o 2^-} f(x) = 3$$, $$\lim_{x 	o 2^+} f(x) = 5$$, and $$f(2) = 4. $$What type of discontinuity does $$f$$ have at $$x = 2$$?

Accepted Answer

Jump discontinuity

Question 18

A car's position is given by $$s(t) = t^3$ - 6t^2$ + 9t (in $$meters, $$t in $$seconds). What is the car's velocity at $$t = 2$$ seconds?

Accepted Answer

$$v(2) = -3 m/s$$

Question 19

Find the linearization $$L(x) of$$ $$f(x) = \sqrt{x} at$$ $$x = 4.$$

Accepted Answer

$$L(x) = 2 + \dfrac{1}{4}(x - 4)$$

Question 20

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

Accepted Answer

$$f'(x) = 1 - \dfrac{1}{x^2$}$$

Calculus I: Core Concepts and Study Guide

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