Q: A medication decays in the bloodstream according to $$C(t) = 50e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is time in hours. What is the rate of change of concentration at t = 3$ hours?

$$C'(3) = -0.2 \times 50e$$^{-0.2 \times 3}$$

Q: A bacteria culture grows according to $$N(t) = 200e$$^{0.5t}$$, where N$ is the number of bacteria and t$ is in hours. What is the instantaneous growth rate at t = 2$?

$$N'(2) = 0.5 \times 200e$$^{0.5 \times 2}$$

Question 1

A company’s profit function is given by $$P(x) = 100x - 0.5x^2$, where x$ is the number of units sold. What is the instantaneous rate of change of profit when x = 20$ units?

Accepted Answer

$$60$$

Question 2

Find the equation of the tangent line to the function $$f(x) = e$$^{2x}$$ at the point where x = 0$.

Accepted Answer

$$y = 2x + 1$$

Question 3

Given $$h(x) = (3x^2$ + 2x)(x^3$ - 4)$$, find $$h'(x)$$ using the product rule.

Accepted Answer

$$h'(x) = (6x + 2)(x^3$ - 4) + (3x^2$ + 2x)(3x^2$)$$

Question 4

If $$f(x) = \frac{5x^2$ + 1}{x - 3}$$, what is $$f'(x)$$ using the quotient rule?

Accepted Answer

$$f'(x) = \frac{(10x)(x - 3) - (5x^2$ + 1)(1)}{(x - 3)^2$}$$

Question 5

Find the derivative of $$f(x) = \ln(2x^2$ + 1).$$

Accepted Answer

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

Question 6

Convert the exponential equation $$3^x = 7 to $$its equivalent logarithmic form.

Accepted Answer

$$x = \log_3$ 7$$

Question 7

Solve for $$x$$: $$e^{2x} = 10$.

Accepted Answer

$$x = \frac{1}{2} \ln 10$$

Question 8

Which of the following is the correct application of the chain rule for $$f(x) = (5x^2$ + 1)^4$?

Accepted Answer

$$f'(x) = 4(5x^2$ + 1)^3$ \cdot 10x$$

Question 9

A medication decays in the bloodstream according to $$C(t) = 50e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is time in hours. What is the rate of change of concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -0.2 	imes 50e$$^{-0.2 	imes 3}$$

Question 10

Find all critical numbers for $$f(x) = x^3$ - 6x^2$ + 9x + 1.$$

Accepted Answer

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

Question 11

Which interval is $$f(x) = x^3$ - 3x^2$ increasing?

Accepted Answer

$$(-\infty, 0)$$ and $$(2, \infty)$$

Question 12

If $$f(x) = \ln(x^2$ + 4)$$, what is the equation of the tangent line at $$x = 0$$?

Accepted Answer

$$y = \frac{1}{4}x + \ln 4$$

Question 13

Solve for $$x$$: $$\ln(x - 2) = 3.$$

Accepted Answer

$$x = e^3$ + 2$$

Question 14

A bacteria culture grows according to $$N(t) = 200e$$^{0.5t}$$, where N$ is the number of bacteria and t$ is in hours. What is the instantaneous growth rate at t = 2$?

Accepted Answer

$$N'(2) = 0.5 	imes 200e$$^{0.5 	imes 2}$$

Question 15

Which of the following is the correct sum property of logarithms?

Accepted Answer

$$\ln(ab) = \ln a + \ln b$$

Question 16

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

Accepted Answer

$$f'(x) = 2xe$$^{x^2}$$

Question 17

A company’s revenue is $$R(x) = 500 \ln(x + 1)$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 9$$?

Accepted Answer

$$R'(9) = \frac{500}{10}$$

Question 18

Which of the following is the correct difference property of logarithms?

Accepted Answer

$$\ln\left(\frac{a}{b}\right) = \ln a - \ln b$$

Question 19

Find the derivative of $$f(x) = \frac{\ln x}{x}$$ using the quotient rule.

Accepted Answer

$$f'(x) = \frac{1 - \ln x}{x^2$}$$

Business Calculus Study Guide: Applications of Derivatives, Exponential & Logarithmic Functions, and First Derivative Test

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