Q: Which of the following correctly expresses $$f(x) = 23(2x-7)^3$ as a composition of two simpler functions?

$$f(x) = h(g(x))$$ \\ where $$g(x) = 2x-7$$, $$h(x) = 23x^3$

Q: Which of the following is a correct composite function for $$f(x) = \ln(4x^2$ + 9)$$?

$$f(x) = h(g(x))$$ \\ where $$g(x) = 4x^2$ + 9$$, $$h(x) = \ln(x)$$

Question 1

Given $$f(x) = e$$^{x^2}$$ + 3$$ and $$g(x) = -x^2$ + 5x - 6$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$e^{(-x^2 + 5x - 6)^2} + 3$

Question 2

Which of the following correctly expresses $$f(x) = 23(2x-7)^3$ as a composition of two simpler functions?

Accepted Answer

$$f(x) = h(g(x))$$ \ where $$g(x) = 2x-7$$, $$h(x) = 23x^3$

Question 3

If $$y = f(g(x))$$ and $$f(x) = x^2$, g(x) = 3x+1$, what is $$\frac{dy}{dx}$$ using the chain rule?

Accepted Answer

$$2(3x+1) \cdot 3$$

Question 4

Find the derivative of $$f(x) = \ln(e$$^{x^2}$$ + x)$$ with respect to $$x.$$

Accepted Answer

$$\frac{2x e$$^{x^2}$$ + 1}{e$$^{x^2}$$ + x}$$

Question 5

Given $$f(x) = (x^2$ + 1) \log(5x + 8)$$, what is $$f'(x)$$?

Accepted Answer

$$2x \log(5x + 8) + \frac{5(x^2$ + 1)}{5x + 8}$$

Question 6

A yeast culture at room temperature (68°F) is placed in a refrigerator at 38°F. The temperature after $$t$$ hours is $$T = 30e$$^{-0.214t}$$ + 38. $$What is the rate of change of temperature at $$t=1$$ hour?

Accepted Answer

$$\frac{dT}{dt} = -6.42 e$$^{-0.214}$$

Question 7

Which of the following is the correct application of the chain rule for $$y = f(g(x))$$?

Accepted Answer

$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

Question 8

Given $$f(x) = \sqrt{4 - 7x} + 9$$, what is $$f'(x)$$?

Accepted Answer

$$-\frac{7}{2\sqrt{4 - 7x}}$$

Question 9

If $$g(x) = \ln(4x^2$ + 9)$$, what is $$g'(x)$$?

Accepted Answer

$$\frac{8x}{4x^2$ + 9}$$

Question 10

Find the derivative of $$f(x) = -2e$$^{x^2}$$.

Accepted Answer

$$-4x e$$^{x^2}$$

Question 11

Given $$f(x) = e$$^{-(x+7)^4}$$, what is f$$'(x)$$?

Accepted Answer

$$-4(x+7)^3$$ $$e^{-(x+7)^4}$$

Question 12

A yeast culture's temperature is $$T = 30e$$^{-0.214t}$$ + 38. $$What is the rate of change of temperature at $$t=4$$ hours?

Accepted Answer

$$\frac{dT}{dt} = -6.42 e$$^{-0.856}$$

Question 13

Which of the following is a correct composite function for $$f(x) = \ln(4x^2$ + 9)$$?

Accepted Answer

$$f(x) = h(g(x))$$ \ where $$g(x) = 4x^2$ + 9$$, $$h(x) = \ln(x)$$

Question 14

If $$f(x) = \frac{e$$^{x^2 + x}$$}{\sqrt{x^2$ + 1}}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{e$$^{x^2 + x}$$(2x + 1)\sqrt{x^2$ + 1} - e$$^{x^2 + x}$$ \frac{x}{\sqrt{x^2$ + 1}}}{x^2$ + 1}$$

Question 15

Which scenario best illustrates the use of the chain rule in a real-world context?

Accepted Answer

Calculating how the temperature of a yeast culture changes over time when the temperature depends on an exponential function of time.

Question 16

If $$f(x) = \sqrt{4 - 7x} + 9$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x \leq \frac{4}{7}$$

Question 17

Which of the following is NOT a correct application of the chain rule?

Accepted Answer

$$\frac{d}{dx} f(g(x)) = f'(x) \cdot g'(x)$$

Composition of Functions and the Chain Rule in Business Calculus

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