Q: A population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of change of the population at t = 10$?

$$P'(10) = 5e$$^{0.5}$$ \times 100$$

Question 1

Given the function $$y = x^3$(x^2$$ - $$4x)^3$$, use the chain rule to find $$\frac{dy}{dx}$. Which of the following is the correct derivative?

Accepted Answer

$$\frac{dy}{dx} = 3x^2$(x^2$$ - $$4x)^3$$ + $$x^3$$ \cdot $$3(x^2$$ - $$4x)^2$$ \cdot (2x - 4)$

Question 2

For the function $$y = x^3$(x^2$$ - $$4x)^3$$, at which $$x$-values does the curve have horizontal tangent lines?

Accepted Answer

$$x = 0, x = 4, x = \frac{8}{3}$$

Question 3

Suppose you are asked to graph $$y = x^3$(x^2$$ - $$4x)^3$ and show the points where the curve has horizontal tangent lines. Which of the following ordered pairs correctly represents one of these points?

Accepted Answer

$$(0, 0)$$

Question 4

Given $$y = 5u^2$ \cdot 3u$$ where $$u = x^2$ + 1$$, find $$\frac{dy}{dx}$$ using the chain rule.

Accepted Answer

$$\frac{dy}{dx} = 30u^2$ x$$

Question 5

Let $$f(x) = \frac{x^2$ + 1}{x - 1}$$ and $$g(x) = \sqrt{x}. $$What is $$f(g(4))$$?

Accepted Answer

$$f(g(4)) = \frac{5}{2}$$

Question 6

Which of the following best describes the chain rule for differentiation?

Accepted Answer

The derivative of a composite function $$f(g(x)) is$$ $$f'(g(x)) \cdot g'(x).$$

Question 7

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = (2t + 1)^4$. What is the rate of change of concentration with respect to time at t = 2$?

Accepted Answer

$$C'(2) = 8(2t + 1)^3$

Question 8

A business models its profit as $$P(x) = x^2$ e$$^{3x}$$. What is the derivative P$$'(x)$$?

Accepted Answer

$$P'(x) = 2x e$$^{3x}$$ + 3x^2$ e$$^{3x}$$

Question 9

If $$h(x) = \sin(x^2$)$$, what is $$h'(x)$$?

Accepted Answer

$$h'(x) = 2x \cos(x^2$)$$

Question 10

A company’s revenue is modeled by $$R(x) = \ln(5x^2$ + 1). $$What is $$R'(x)$$?

Accepted Answer

$$R'(x) = \frac{10x}{5x^2$ + 1}$$

Question 11

Suppose $$y = (3x - 2)^5$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = 15(3x - 2)^4$

Question 12

If $$f(x) = x^2$ and g(x) = e^x$, what is the derivative of f(x)g(x$$)$$?

Accepted Answer

$$2x e^x + x^2$ e^x$$

Question 13

A population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of change of the population at t = 10$?

Accepted Answer

$$P'(10) = 5e$$^{0.5}$$ 	imes 100$$

Question 14

If $$y = \sqrt{2x + 1}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2}{2\sqrt{2x + 1}}$$

Question 15

A company’s cost function is $$C(x) = 4x^3$ - 2x^2$ + 7x - 5. $$What is the marginal cost at $$x = 2$$?

Accepted Answer

$$C'(2) = 48 - 8 + 7$$

Question 16

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2x}{x^2$ + 1}$$

Question 17

A company’s revenue is modeled by $$R(x) = x^2$ \ln(x). $$What is $$R'(x)$$?

Accepted Answer

$$R'(x) = 2x \ln(x) + x$$

Question 18

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

Accepted Answer

$$\frac{dy}{dx} = 2e$$^{2x}$$

Question 19

Suppose $$y = (x^2$ + 3x)^4$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = 4(x^2$ + 3x)^3$ (2x + 3)$$

Business Calculus Quiz Study Notes: Differentiation and Applications

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