Q: If $\lim_{x \to 2} f(x)$ exists, which of the following must be true?

$f(x)$ approaches a finite value as $x \to 2$.

Q: If $f$ is continuous at $x = -10$, which of the following must be true?

$\lim_{x \to -10} f(x) = f(-10)$

Question 1

Which of the following statements is TRUE about the function $f(x) = \frac{2x-1}{6x-12}$?

Accepted Answer

The domain is all real numbers except $x = 2$.

Question 2

Let $f(x) = x^2 - 8x + 7$. What is the vertex of the parabola?

Accepted Answer

The vertex is at $\left(4, -9\right)$.

Question 3

Suppose $\lim_{x 	o -4} H(x)$ exists and $H(-4)$ is defined, but $\lim_{x 	o -4} H(x) 
eq H(-4)$. What can you conclude about the continuity of $H$ at $x = -4$?

Accepted Answer

$H$ is not continuous at $x = -4$.

Question 4

Evaluate $\lim_{x 	o \infty} \frac{5x^2 - 9x + 8}{4x^2 - 8x + 1}$.

Accepted Answer

$\frac{5}{4}$

Question 5

An investor puts $\$9900$ into an account that pays $9\%$ interest compounded continuously. How much will be in the account after 4 years? (Use $A = Pe^{rt}$)

Accepted Answer

$\$14060.12$

Question 6

Let $f(x) = 2x^2 - 9x + 79$. What is $f'(x)$?

Accepted Answer

$4x - 9$

Question 7

Given $f(x) = 3x^2 - 9x - 66$, find the equation of the tangent line to $f(x)$ at the point $(6, -12)$.

Accepted Answer

$y = 27x - 174$

Question 8

Jimmy’s Music Shop has cost function $C(x) = 77x + 1900$ and revenue function $R(x) = 146x$. What is the profit from selling $x = 80$ keyboards?

Accepted Answer

$5520$

Question 9

Given the function $P(t) = 0.6t^2 + 0.4t + 75$, where $t$ is the number of years since 2010, what is the average rate of change of $P$ between 2015 and 2024?

Accepted Answer

$10.8$

Question 10

Which of the following is TRUE about the function $f(x) = x^2 - 8x + 7$?

Accepted Answer

The parabola opens upward.

Question 11

If $\lim_{x 	o 2} f(x)$ exists, which of the following must be true?

Accepted Answer

$f(x)$ approaches a finite value as $x 	o 2$.

Question 12

If $f$ is continuous at $x = -10$, which of the following must be true?

Accepted Answer

$\lim_{x 	o -10} f(x) = f(-10)$

Question 13

Which of the following equations can be represented as a function of $x$?

Accepted Answer

$y = x^2 + 1$

Question 14

If $f(-10) = -3$ and $f(2) = -3$, which of the following is TRUE?

Accepted Answer

$f$ is a function.

Question 15

If $\lim_{x 	o a} f(x) = f(a)$, which of the following is TRUE?

Accepted Answer

$f$ is continuous at $x = a$.

Question 16

Given $C(x) = 77x + 1900$ and $R(x) = 146x$, what is the average unit profit function $P(x)$?

Accepted Answer

$P(x) = 69 - \frac{1900}{x}$

Question 17

If $\lim_{x 	o 8} f(x) = f(8)$, which of the following is TRUE?

Accepted Answer

$f$ is continuous at $x = 8$.

Question 18

For the function $f(x) = x^2 - 8x + 7$, what are the $x$-intercepts?

Accepted Answer

$x = 1$ and $x = 7$

Business Calculus: Functions, Limits, Derivatives, and Applications Study Guide

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