Q: Suppose the cost $C(x)$ to produce $x$ units of a product is given by $C(x) = 5x^2 + 20x + 100$. What is the average rate of change of cost as production increases from 10 to 15 units?

$\frac{C(15) - C(10)}{15 - 10} = \frac{(5 \times 225 + 20 \times 15 + 100) - (5 \times 100 + 20 \times 10 + 100)}{5}$

Q: A business’s profit function is $P(x) = 50x - 0.2x^2$. What is the marginal profit when $x = 100$ units?

$P'(100) = 50 - 0.2 \times 2 \times 100 = 10$

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

If $\lim_{x \to a} f(x) = f(a)$, then $f$ is continuous at $x = a$ if $f(a)$ and $\lim_{x \to a} f(x)$ both exist.

Q: A function $f(x)$ is defined as $f(x) = \begin{cases} x^2 & x \leq 1 \\ 2x + 1 & x > 1 \end{cases}$. Is $f(x)$ continuous at $x = 1$?

No, because $\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$.

Question 1

Which of the following statements correctly describes the definition of continuity for a function $f$ at $x = a$?

Accepted Answer

The function $f$ is continuous at $x = a$ if $f(a)$ exists, $\lim_{x 	o a} f(x)$ exists, and $\lim_{x 	o a} f(x) = f(a)$.

Question 2

Suppose the cost $C(x)$ to produce $x$ units of a product is given by $C(x) = 5x^2 + 20x + 100$. What is the average rate of change of cost as production increases from 10 to 15 units?

Accepted Answer

$\frac{C(15) - C(10)}{15 - 10} = \frac{(5 	imes 225 + 20 	imes 15 + 100) - (5 	imes 100 + 20 	imes 10 + 100)}{5}$

Question 3

Given $f(x) = x^3 - 2x^2 + 4x$, what is $f'(x)$ using the power, constant multiple, and sum-difference rules?

Accepted Answer

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

Question 4

A company’s revenue function is $R(x) = 100x - 0.5x^2$. Find the equation of the tangent line to $R(x)$ at $x = 20$.

Accepted Answer

Tangent line: $y = 80x - 400$

Question 5

Which of the following is the correct application of the Product Rule for differentiation?

Accepted Answer

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

Question 6

If $f(x) = \frac{2x^2 + 3x}{x}$, what is $f'(x)$ using the Quotient Rule?

Accepted Answer

$f'(x) = \frac{(x)(4x + 3) - (2x^2 + 3x)(1)}{x^2}$

Question 7

For the function $f(x) = (3x^2 + 2x)^4$, which rule should be used to find $f'(x)$, and what is the derivative?

Accepted Answer

Use the Chain Rule: $f'(x) = 4(3x^2 + 2x)^3 \cdot (6x + 2)$

Question 8

A medication’s concentration in the bloodstream is modeled by $C(t) = 10e^{-0.2t}$. What is the instantaneous rate of change of concentration at $t = 5$?

Accepted Answer

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

Question 9

Which of the following functions is NOT differentiable at $x = 0$?

Accepted Answer

$f(x) = |x|$

Question 10

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

Accepted Answer

$f''(x) = 12x - 10$

Question 11

A business’s profit function is $P(x) = 50x - 0.2x^2$. What is the marginal profit when $x = 100$ units?

Accepted Answer

$P'(100) = 50 - 0.2 	imes 2 	imes 100 = 10$

Question 12

If $f(x) = 7$, what is $f'(x)$?

Accepted Answer

$f'(x) = 0$

Question 13

Given $f(x) = x^2 + 3x$, find the simplified difference quotient $\frac{f(x+h) - f(x)}{h}$.

Accepted Answer

$2x + 3 + h$

Question 14

Which of the following statements about limits is TRUE?

Accepted Answer

If $\lim_{x 	o a} f(x) = f(a)$, then $f$ is continuous at $x = a$ if $f(a)$ and $\lim_{x 	o a} f(x)$ both exist.

Question 15

A function $f(x)$ is defined as $f(x) = \begin{cases} x^2 & x \leq 1 \ 2x + 1 & x > 1 \end{cases}$. Is $f(x)$ continuous at $x = 1$?

Accepted Answer

No, because $\lim_{x 	o 1^-} f(x) 
eq \lim_{x 	o 1^+} f(x)$.

Question 16

If $f(x) = x^2$ and $g(x) = 3x$, what is $\frac{d}{dx}[f(x) + g(x)]$?

Accepted Answer

$2x + 3$

Question 17

For $f(x) = (2x + 1)^5$, what is $f'(x)$ using the Extended Power Rule?

Accepted Answer

$f'(x) = 5(2x + 1)^4 \cdot 2$

Question 18

A population of bacteria is modeled by $P(t) = 500e^{0.05t}$. What is the rate of change of the population at $t = 10$ hours?

Accepted Answer

$P'(10) = 25e^{0.5}$

Chapter 1: Differentiation – Study Guide for Business Calculus

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