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

$$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Q: A company’s revenue function is $$R(x) = 5x^2$ + 3x + 10. $$What is the marginal revenue when $$x = 4$$?

$$5 \times 2 \times 4 + 3 = 43$$

Q: A position function is $$s(t) = 4t^3$ - 2t^2$ + t. $$What is the acceleration at $$t = 2$$?

$$s''(2) = 24 \times 2 - 4 = 44$$

Q: A function $$f(x)$$ has a hole at $$x = 2. $$Which statement is TRUE?

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

Q: A business wants to approximate the change in profit for a small increase in sales. If $$P(x) = 4x^2$ + 2x$$, what is the approximate change in profit when sales increase from $$x = 10 to$$ $$x = 10.1$$?

$$P'(10) \cdot 0.1 = (8 \times 10 + 2) \cdot 0.1 = 82 \cdot 0.1 = 8.2$$

Question 1

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

Accepted Answer

$$\lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 2

A company’s revenue function is $$R(x) = 5x^2$ + 3x + 10. $$What is the marginal revenue when $$x = 4$$?

Accepted Answer

$$5 	imes 2 	imes 4 + 3 = 43$$

Question 3

Which of the following is NOT a criterion for continuity at $$x = a$$?

Accepted Answer

$$f'(a)$$ exists

Question 4

Given $$f(x) = 3x^2$ + 2x + 5$$, what is the slope of the tangent line at $$x = 1$$?

Accepted Answer

$$6 	imes 1 + 2 = 8$$

Question 5

If $$f(x) = x^3$ + x^2$ + 5$$, what is $$f'(x)$$ using the power rule?

Accepted Answer

$$3x^2 + 2x$

Question 6

A position function is $$s(t) = 4t^3$ - 2t^2$ + t. $$What is the acceleration at $$t = 2$$?

Accepted Answer

$$s''(2) = 24 	imes 2 - 4 = 44$$

Question 7

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

Accepted Answer

$$f(x) = |x|$$

Question 8

If $$f(x) = x^2$ and g(x) = x + 1$, what is the derivative of h(x) = f(x$$) \cdot g(x)$$ using the product rule?

Accepted Answer

$$2x(x+1) + x^2$

Question 9

A function $$f(x)$$ has a hole at $$x = 2. $$Which statement is TRUE?

Accepted Answer

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

Question 10

Which rule should be used to differentiate $$f(x) = (2x + 3)^5$?

Accepted Answer

Chain Rule

Question 11

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

Accepted Answer

$$\frac{(6x + 2)x - (3x^2$ + 2x) \cdot 1}{x^2$}$$

Question 12

A function $$f(x) is $$continuous everywhere except at $$x = 3$$, where it has a vertical asymptote. Which of the following statements is TRUE?

Accepted Answer

$$f(x) is $$not continuous at $$x = 3$$

Question 13

If $$f(x) = 2x^3$ - 5x^2$ + 4x$$, what is the second derivative $$f''(x)$$?

Accepted Answer

$$12x - 10$$

Question 14

A business wants to approximate the change in profit for a small increase in sales. If $$P(x) = 4x^2$ + 2x$$, what is the approximate change in profit when sales increase from $$x = 10 to$$ $$x = 10.1$$?

Accepted Answer

$$P'(10) \cdot 0.1 = (8 	imes 10 + 2) \cdot 0.1 = 82 \cdot 0.1 = 8.2$$

Question 15

Which of the following best describes the relationship between position, velocity, and acceleration?

Accepted Answer

$$v(t) = s'(t)$$ and $$a(t) = v'(t)$$

Question 16

If $$f(x) = x^2$ and g(x$$) = $$x^3$$, what is the derivative of $$\frac{f(x)}{g(x)}$$ using the quotient rule?

Accepted Answer

$$\frac{2x \cdot x^3$ - x^2$ \cdot 3x^2$}{x^6$}$$

Question 17

A function $$f(x) is $$continuous at $$x = 2$$ but not differentiable there. Which of the following could explain this?

Accepted Answer

There is a corner or cusp at $$x = 2$$

Question 18

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

Accepted Answer

Question 19

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

Accepted Answer

Question 20

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

Accepted Answer

Business Calculus Study Notes: Derivatives, Limits, and Differentiation Techniques

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