Q: Given the function $$p(t) = 0.858t^2$ - 18.864t + 78.354$$ representing the price of a Super Bowl ticket $$t$$ years after 1967, what is the rate of change of the ticket price with respect to the year in 2020?

$$p'(53) = 0.858 \times 2 \times 53 - 18.864$$

Q: A particle moves along a line with position $$s(t) = \sqrt{40 + 4t}. $$What is the rate of change of the particle’s position at $$t = 6$$ seconds?

$$s'(6) = \dfrac{2}{\sqrt{40 + 4 \times 6}}$$

Question 1

Given the function $$p(t) = 0.858t^2$ - 18.864t + 78.354$$ representing the price of a Super Bowl ticket $$t$$ years after 1967, what is the rate of change of the ticket price with respect to the year in 2020?

Accepted Answer

$$p'(53) = 0.858 	imes 2 	imes 53 - 18.864$$

Question 2

If $$C(x) = 200 + 100\sqrt[4]{x} is $$the cost of producing $$x$$ hundred geraniums, what is the average cost function $$A(x)$$?

Accepted Answer

$$A(x) = \dfrac{200 + 100\sqrt[4]{x}}{x}$$

Question 3

Differentiate $$f(x) = 7x^4$ - 3x + 5$$ using the sum-difference rule.

Accepted Answer

$$f'(x) = 28x^3$ - 3$$

Question 4

Find the second derivative of $$y = x^5$ - 3x^4$ + x.$$

Accepted Answer

$$y'' = 20x^3$ - 36x^2$

Question 5

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

Accepted Answer

$$\dfrac{dy}{dx} = \dfrac{1}{2}(1 + x^3$)^{-1/2}$$ \cdot $$3x^2$

Question 6

Given $$f(x) = x^2$ e^x$$, find $$f'(x).$$

Accepted Answer

$$f'(x) = 2x e^x + x^2$ e^x$$

Question 7

If $$y = \ln(x^2$ - 5)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = \dfrac{2x}{x^2$ - 5}$$

Question 8

A particle moves along a line with position $$s(t) = \sqrt{40 + 4t}. $$What is the rate of change of the particle’s position at $$t = 6$$ seconds?

Accepted Answer

$$s'(6) = \dfrac{2}{\sqrt{40 + 4 	imes 6}}$$

Question 9

Differentiate $$f(x) = \dfrac{x^2$ - 3x}{x - 1}$$ using the quotient rule and simplify the numerator.

Accepted Answer

$$f'(x) = \dfrac{(x - 1)(2x - 3) - (x^2$ - 3x)(1)}{(x - 1)^2$}$$

Question 10

If $$y = e$$^{8x}$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = 8e$$^{8x}$$

Question 11

Suppose $$N(a) = 1000 + 200 \ln a$$ models the number of units sold as a function of advertising $$a (in $$thousands of dollars). What is $$N'(a)$$?

Accepted Answer

$$N'(a) = \dfrac{200}{a}$$

Question 12

If $$y = \dfrac{1}{x}$$, what is $$\dfrac{d^2$y}{dx^2$}$$?

Accepted Answer

$$\dfrac{d^2$y}{dx^2$} = \dfrac{2}{x^3$}$$

Question 13

Given $$f(x) = (3x - 5)^4$ (7 - x$$)^{10}$$, which rule(s) must be used to differentiate f(x$$)$$?

Accepted Answer

Both the product rule and the chain rule

Question 14

If $$f(x) = \ln\left(\dfrac{x^3$ + 4}{x}\right)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \dfrac{3x^2$}{x^3$ + 4} - \dfrac{1}{x}$$

Question 15

A stone is dropped from a cliff and its distance fallen after $$t$$ seconds is $$s(t) = 4.905t^2$. What is its velocity after 5 seconds?

Accepted Answer

$$v(5) = 9.81 	imes 5$$

Differentiation Techniques and Applications in Calculus

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