Q: The population of a city grows according to $$P(t) = 600,000 + 9000t^2$, where t$ is in years. What is the instantaneous growth rate at t = 10$?

$$P'(10) = 18,000 \times 10 = 180,000$$

Q: A rock is dropped and its position is given by $$s(t) = 4.9t^2$. What is the velocity of the rock at t = 5$ seconds?

$$s'(5) = 9.8 \times 5 = 49 m/s$$

Question 1

A company tracks the number of annual trips taken on its mass transportation system. In 1995, there were 8.8 million trips, and in 2000, there were 7.8 million trips. What is the average rate of change in the number of trips per year from 1995 to 2000?

Accepted Answer

$$\frac{7.8 - 8.8}{2000 - 1995} = -0.2$$ million trips per year

Question 2

Given $$f(x) = x^2$ - 2x - 1$$, find the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$2x - 2 + h$$

Question 3

An object dropped from rest from the top of a tall building falls a distance $$D(t) = 16t^2$ feet in the first t$ seconds. What is the average velocity from t = 3$ to t = 6$ seconds?

Accepted Answer

$$\frac{D(6) - D(3)}{6 - 3} = \frac{576 - 144}{3} = 144 ft/sec$$

Question 4

Find $$\lim_{x 	o 5} \frac{x^2$ - 9}{5 - x}.$$

Accepted Answer

$$-2$$

Question 5

If $$f(x) = 4x^2$, use the definition of the derivative to find f$$'(5)$$.

Accepted Answer

$$40$$

Question 6

The population of a city grows according to $$P(t) = 600,000 + 9000t^2$, where t$ is in years. What is the instantaneous growth rate at t = 10$?

Accepted Answer

$$P'(10) = 18,000 	imes 10 = 180,000$$

Question 7

Given $$y = (5x + 8)^3$, find $$\frac{dy}{dx}$$ using the chain rule.

Accepted Answer

$$3(5x + 8)^2$ 	imes 5$$

Question 8

Find the derivative of $$y = 6x^7$.

Accepted Answer

$$42x^6$$

Question 9

A rock is dropped and its position is given by $$s(t) = 4.9t^2$. What is the velocity of the rock at t = 5$ seconds?

Accepted Answer

$$s'(5) = 9.8 	imes 5 = 49 m/s$$

Question 10

Given $$y = \sqrt{x} - (x - 4)^5$, find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{1}{2\sqrt{x}} - 5(x - 4)^4$

Question 11

If $$f(x) = \frac{x^2$ + 3x - 40}{x^2$ - 25}$$, find $$\lim_{x 	o 5} f(x).$$

Accepted Answer

$$\frac{28}{10} = 2.8$$

Question 12

Which of the following statements about continuity is true?

Accepted Answer

A function $$f(x) is $$continuous at $$x = a if$$ $$\lim_{x 	o a} f(x) = f(a)$$

Question 13

Given $$y = \frac{9x - 1}{2x + 9} + x^3$, find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{9(2x + 9) - (9x - 1)2}{(2x + 9)^2$} + 3x^2$

Question 14

If $$y = 8\sqrt{x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{4}{\sqrt{x}}$$

Question 15

Find the derivative of $$y = 4x^3$ - 13x^2$ + 35x + 7.$$

Accepted Answer

$$12x^2 - 26x + 35$

Question 16

Given $$y = (5 - x)^7$, find $$\frac{dy}{dx}$$ using the chain rule.

Accepted Answer

$$-7(5 - x)^6$

Question 17

If $$f(x) = \frac{4x^2$ - 9}{3x^2$ + 1}$$, find $$f'(x)$$ using the quotient rule.

Accepted Answer

$$\frac{8x(3x^2$ + 1) - 6x(4x^2$ - 9)}{(3x^2$ + 1)^2$}$$

Question 18

The temperature of a person during an illness is given by $$T(t) = \frac{2t}{t^2$ + 3} + 98.6. $$What is the rate of change of temperature with respect to time at $$t = 1$$ hour?

Accepted Answer

$$T'(1) = \frac{(t^2$ + 3) \cdot 2 - 2t \cdot 2t}{(t^2$ + 3)^2$}$$ evaluated at $$t = 1$$ gives $$T'(1) = \frac{8 - 4}{16} = 0.25$$

Question 19

Which rule should be used to differentiate $$y = (2x^2$ - 5$$)^{14}$$?

Accepted Answer

The chain rule

Business Calculus Study Guide: Limits, Continuity, Rates of Change, and Derivatives

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