Q: Suppose the graph of $f(x)$ has a jump at $x = 2$ such that $\lim_{x \to 2^-} f(x) = 3$ and $\lim_{x \to 2^+} f(x) = 5$. What can you conclude about $\lim_{x \to 2} f(x)$?

$\lim_{x \to 2} f(x)$ does not exist

Q: Interpret the meaning of the marginal revenue $R'(1600) = 60$ in the context of the shoe company.

Selling one more pair when 1600 pairs are sold increases revenue by approximately $60.

Q: Find the derivative of $y = \frac{5x^3}{4} - \frac{2}{5}x^2$.

$\frac{15x^2}{4} - \frac{4x}{5}$

Q: Which of the following best describes the relationship between one-sided limits and the two-sided limit at $x = a$?

The two-sided limit exists if and only if both one-sided limits exist and are equal.

Question 1

Suppose the graph of $f(x)$ has a jump at $x = 2$ such that $\lim_{x 	o 2^-} f(x) = 3$ and $\lim_{x 	o 2^+} f(x) = 5$. What can you conclude about $\lim_{x 	o 2} f(x)$?

Accepted Answer

$\lim_{x 	o 2} f(x)$ does not exist

Question 2

Evaluate the following limit using algebraic methods: $\lim_{x 	o 2} \frac{2x^2 - 3x - 2}{x^2 + x - 6}$.

Accepted Answer

$1$

Question 3

Given $f(x) = \frac{x-4}{x^2 + 2x}$, which of the following is NOT a partition number for $f(x)$?

Accepted Answer

$x = 2$

Question 4

Solve the inequality $\frac{x-4}{x^2 + 2x} < 0$ and express your answer in interval notation.

Accepted Answer

$(-\infty, -2) \cup (0, 4)$

Question 5

Using the definition of the derivative, find $f'(x)$ for $f(x) = 2x^2 + 8$.

Accepted Answer

$4x$

Question 6

Find the derivative of $f(x) = 4\sqrt{x} + 2$ using differentiation formulas.

Accepted Answer

$2x^{-1/2}$

Question 7

Given the price-demand equation $x = 4000 - 40p$, express the price $p$ in terms of demand $x$.

Accepted Answer

$p = 100 - 0.025x$

Question 8

For the price-demand equation $x = 4000 - 40p$, what is the domain of the price function $p(x)$?

Accepted Answer

$0 \leq x \leq 4000$

Question 9

Given $f(x) = \frac{2x^5 - 4x^3 + 2x}{x^3}$, find $f'(x)$ using differentiation formulas.

Accepted Answer

$10x^2 - 12x + 2x^{-2}$

Question 10

Suppose the revenue function for a product is $R(x) = x \left(100 - 0.025x\right)$. Find the marginal revenue at $x = 1600$.

Accepted Answer

$R'(1600) = 60$

Question 11

Interpret the meaning of the marginal revenue $R'(1600) = 60$ in the context of the shoe company.

Accepted Answer

Selling one more pair when 1600 pairs are sold increases revenue by approximately $60.

Question 12

Find the derivative of $y = \frac{5x^3}{4} - \frac{2}{5}x^2$.

Accepted Answer

$\frac{15x^2}{4} - \frac{4x}{5}$

Question 13

Evaluate $\lim_{x 	o 0} \frac{2x^2 - 3x - 2}{x^2 + x - 6}$.

Accepted Answer

$\frac{-2}{-6} = \frac{1}{3}$

Question 14

Which of the following best describes the relationship between one-sided limits and the two-sided limit at $x = a$?

Accepted Answer

The two-sided limit exists if and only if both one-sided limits exist and are equal.

Question 15

Find the derivative of $f(x) = 3x - \frac{5}{x} + 2\sqrt{x}$.

Accepted Answer

$3 + 5x^{-2} + x^{-1/2}$

Question 16

If $f(x) = 4x - x^2$, use the definition of the derivative to find $f'(x)$.

Accepted Answer

$4 - 2x$

Question 17

Given $f(x) = x^2 + 2x$, which of the following is the correct sign chart for $f(x)$?

Accepted Answer

Negative for $x < -2$, positive for $x > -2$

Question 18

Suppose the revenue function is $R(x) = x(100 - 0.025x)$. For what value of $x$ is the marginal revenue zero?

Accepted Answer

$x = 2000$

Question 19

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

Accepted Answer

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

Business Calculus Exam 1 Study Guide: Limits, Derivatives, and Marginal Analysis

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