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 $$f(x) = \begin{cases} x^2$ & x \leq 1 \\ 2x + 1 & x \u003e 1 \end{cases}. Is$$ $$f(x)$$ 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 $$f(x) = \begin{cases} x^2$ & x \leq 1 \ 2x + 1 & x \u003e 1 \end{cases}. Is$$ $$f(x)$$ 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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