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

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

Q: If the limit $$\lim_{x \to 2} f(x)$$ does not exist, which of the following could be a reason?

The left and right limits as $$x \to 2$$ are not equal.

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 $$P(x) (in $$thousands of dollars) from selling $$x$$ units is given by $$P(x) = -2x^2$ + 40x - 100. $$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

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

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

A business finds that the demand for its product is given by $$p = 100 - 2x$$, where $$p is $$the price per unit and $$x is $$the number of units sold. What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 100x - 2x^2$

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

If $$y = f(g(x))$$, then $$\dfrac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

Question 9

A company’s revenue is $$R(x) = 50x$$ and its cost is $$C(x) = 30x + 200. $$What is the break-even point?

Accepted Answer

$$x = 20$$

Question 10

Which of the following is NOT a valid application of derivatives in business?

Accepted Answer

Determining the break-even point.

Question 11

If $$f(x) = x^3$ - 6x^2$ + 9x + 15$$, what is the $$x-$$coordinate of the critical point(s)?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 12

Which of the following best describes a function?

Accepted Answer

A relation where each input has exactly one output.

Question 13

Given $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 14

A business wants to minimize its cost function $$C(x) = x^2$ - 8x + 20. $$What value of $$x$$ minimizes the cost?

Accepted Answer

$$x = 4$$

Question 15

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

Accepted Answer

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

Question 16

If the limit $$\lim_{x 	o 2} f(x)$$ does not exist, which of the following could be a reason?

Accepted Answer

The left and right limits as $$x 	o 2$$ are not equal.

Question 17

A company’s cost function is $$C(x) = 1000 + 5x. $$What does the constant 1000 represent?

Accepted Answer

The fixed cost.

Question 18

If $$f(x) = x^2$ and g(x) = 3x + 1$, what is (f$$ \circ g)(x)$$?

Accepted Answer

$$(f \circ g)(x) = (3x + 1)^2$

Question 19

Which of the following is the correct quotient rule for differentiation?

Accepted Answer

$$\dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right] = \dfrac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

Business Calculus Syllabus and Course Structure Overview

Guida di studio - Domande di pratica

Study guide - Flashcard