Question 1

Given the function $$f(x) = \begin{cases} -3x + 7, & x \u003c 1 \ 4x - 1, & x \geq 1 \end{cases}$$, what is $$\lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$-3(1) + 7 = 4$$

Question 2

For the function $$f(x) = x^2$ - 9$$ for $$x \u003c 5$$ and $$x^2 + c$ for x$$ \geq 5$$, what value of c$ makes $$\lim_{x 	o 5} f(x)$$ exist?

Accepted Answer

$$c = 41$$

Question 3

Suppose the total number of people infected with a virus after $$t$$ weeks is given by $$P(t). If$$ $$P(0) = 220$$ and as $$t$$ increases, $$P$$ approaches $$1540$$, what is $$\lim_{t 	o \infty} P(t)$$?

Accepted Answer

$$1540$$

Question 4

Find $$\lim_{x 	o 2} \frac{x^2$ + 4x - 28}{x - 2}.$$

Accepted Answer

$$8$$

Question 5

Given $$f(x) = x^2$ - 4x$$, what is the simplified form of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$2x + h - 4$$

Question 6

A city’s population grows according to $$P(t) = 250{,}000 + 3{,}000t. $$What is the rate of change of the population at $$t = 10$$?

Accepted Answer

$$60{,}000$$

Question 7

Find the derivative of $$y = x^2$ - 3x^2$ + 10x = 100.$$

Accepted Answer

$$y' = 2x - 6x + 10$$

Question 8

Given $$f(x) = 12x\sqrt{x} - 3x$$, find $$f'(x).$$

Accepted Answer

$$f'(x) = 18\sqrt{x} - 3$$

Question 9

For the function $$f(x) = x^3$ - x^2$ - 3x + 1$$, at which $$x$$ values is the tangent line horizontal?

Accepted Answer

$$x = 3$$ and $$x = 1$$

Question 10

Given $$R(x) = 800\sqrt{2x - 5}$$ and $$C(x) = 172\sqrt{x} + 200$$, what is the rate of change of total profit $$P(x)$$ when $$x$$ items have been produced and sold?

Accepted Answer

$$P'(x) = 800 \cdot \frac{1}{\sqrt{2x - 5}} \cdot 2 - 172 \cdot \frac{1}{2\sqrt{x}}$$

Question 11

Find the derivative of $$y = (x^2$ + 2)^5$.

Accepted Answer

$$y' = 5(x^2$ + 2)^4$ \cdot 2x$$

Question 12

Suppose the revenue from selling $$x$$ units is $$R(x) = 0.01x^2$ + 1000x. $$What is the average rate of revenue per unit when $$x$$ changes from 300 to 303?

Accepted Answer

$$994$$

Question 13

Given $$f(x) = x^2$ - 4x$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

Question 14

Find the derivative of $$f(x) = x^2$\sqrt{x} - 8\sqrt{x}.$$

Accepted Answer

$$f'(x) = 2x\sqrt{x} + x^2$ \cdot \frac{1}{2\sqrt{x}} - 4x$$^{-1/2}$$

Question 15

For the function $$u(t) = 7.35 + 1.97t - 0.0076t^2$ + 0.0000252t^3$, what is the rate of change of weight with respect to time at t = 13$ months?

Accepted Answer

$$u'(13) = 1.91$$ lbs/month

Question 16

Given $$T(t) = \frac{9t}{t^2$ + 6t} + 98.6$$, what is $$T'(1)$$, the rate of change of temperature at $$t = 1$$ hour?

Accepted Answer

$$0.84$$ degrees per hour

Question 17

Suppose the distance of a car at time $$t is$$ $$s(t) = 10t^2$. What is the acceleration at t = 4$ seconds?

Accepted Answer

$$a(4) = 20$$

Question 18

Find the derivative of $$f(x) = \frac{x^2$ + 4x + 2}{x - 3}$$ using the quotient rule.

Accepted Answer

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

Business Calculus Study Guide: Limits, Derivatives, and Applications

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