Q: Which of the following is the correct definition of the derivative of a function $$f(x) at a $$point $$x = a$$?

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: A business uses the function $$C(x) = 1000 + 5x to $$model its costs. If the company produces 200 units, what is the average cost per unit?

$$C_{avg} = \frac{C(200)}{200} = \frac{1000 + 5 \times 200}{200} = 10$$

Question 1

Which of the following best describes the main objective of Business Calculus as outlined in the course syllabus?

Accepted Answer

To understand functions, limits, and derivatives with their applications of differentiation in the real world.

Question 2

A company’s profit $$P(x) is $$modeled by the function $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. What is the maximum profit, and how many units should be sold to achieve it?

Accepted Answer

Maximum profit is $$300 at$$ $$x = 10$$ units.

Question 3

Which of the following is the correct definition of the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 4

Suppose the cost function for producing $$x$$ items is $$C(x) = 500 + 20x + 0.5x^2$. What is the marginal cost when x = 10$?

Accepted Answer

$$30$$

Question 5

Which of the following statements about exponential functions is correct?

Accepted Answer

An exponential function has the form $$f(x) = a^x$$, where $$a \u003e 0$$ and $$a 
eq 1.$$

Question 6

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

Accepted Answer

$$f'(x) = \frac{1}{x}$$

Question 7

A business’s revenue function is $$R(x) = 100x - x^2$. What is the rate of change of revenue when x = 20$?

Accepted Answer

$$R'(x) = 100 - 2x$$, so at $$x = 20$$, $$R'(20) = 60.$$

Question 8

Which of the following best describes the Chain Rule for differentiation?

Accepted Answer

If $$y = f(g(x))$$, then $$y' = f'(g(x)) \cdot g'(x).$$

Question 9

A company’s demand function is $$p = 50 - 2x. $$What is the elasticity of demand when $$x = 10$$ and $$p = 30$$?

Accepted Answer

Elasticity $$E = \left| \frac{p}{x} \cdot \frac{dx}{dp} \right| = \frac{30}{10} \cdot \frac{1}{-2} = 1.5 (in $$absolute value)

Question 10

Which of the following is NOT a learning objective for this Business Calculus course?

Accepted Answer

Students are able to memorize all calculus proofs.

Question 11

Given $$f(x) = x^3$ - 6x^2$ + 9x + 15$$, find all critical points and determine if they are maxima, minima, or neither.

Accepted Answer

Critical points at $$x = 1 ($$max), $$x = 3 ($$min)

Question 12

If $$y = e$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2e^{2x}$$

Question 13

A business uses the function $$C(x) = 1000 + 5x to $$model its costs. If the company produces 200 units, what is the average cost per unit?

Accepted Answer

$$C_{avg} = \frac{C(200)}{200} = \frac{1000 + 5 	imes 200}{200} = 10$$

Question 14

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

Accepted Answer

If $$y = f(x)g(x)$$, then $$y' = f'(x)g(x) + f(x)g'(x)$$

Question 15

Suppose the revenue function is $$R(x) = 50x$$ and the cost function is $$C(x) = 20x + 200. $$What is the break-even point?

Accepted Answer

$$x = 8$$

Question 16

Which of the following is a correct interpretation of the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L.$$

Question 17

A company’s cost function is $$C(x) = 100 + 10x. If $$the company wants to minimize average cost, what should they do as $$x$$ increases?

Accepted Answer

Produce more units, since average cost decreases as $$x$$ increases.

Question 18

If $$f(x) = x^2$ and g(x) = 3x + 1$, what is the derivative of h(x) = f(g(x$$))$$?

Accepted Answer

$$h'(x) = 2g(x) \cdot g'(x) = 2(3x+1) \cdot 3 = 6(3x+1)$$

Riley Business Calculus Syllabus and Course Structure Overview

Guida di studio - Domande di pratica

Study guide - Flashcard