Q: A $$150,000$$ deposit earns 6.3% interest compounded continuously. What is the account balance after 10 years?

$$A(10) = 150000e$$^{0.063 \times 10}$$

Q: How much do you need to invest at 6.5% interest compounded continuously to have $$1$$ million after $$40$$ years?

$$P = \frac{1000000}{e$$^{0.065 \times 40}$$}$$

Question 1

A bacteria culture contains 250 cells and grows at a rate proportional to its size. After one hour, the population increases to 400. What is the expression for the number of bacteria after t hours?

Accepted Answer

$$P(t) = 250e$$^{0.182t}$$

Question 2

If a population doubles every 16 days, what is the growth constant $$k$$ for the exponential model $$P(t) = P_0$ e$$^{kt}$$?

Accepted Answer

$$k = \frac{\ln 2}{16}$$

Question 3

A savings account earns 4.7% interest compounded continuously. How long will it take for the investment to double?

Accepted Answer

$$t = \frac{\ln 2}{0.047}$$

Question 4

A radioactive isotope has a half-life of 50 years. What is the decay constant $$k$$ for the exponential decay model $$M(t) = M_0$ e$$^{-kt}$$?

Accepted Answer

$$k = \frac{\ln 2}{50}$$

Question 5

A person is injected with 150 milligrams of penicillin. The amount in the bloodstream after t hours is $$F(t) = 150e$$^{-0.391t}$$. What is the biological half-life of penicillin?

Accepted Answer

$$t = \frac{\ln 2}{0.391}$$

Question 6

Given $$f(x) = -4x^3$ + 4x^2$ - 5x + 10$$, what are the intervals of concavity?

Accepted Answer

Concave up on $$(-\infty, 0.5)$$, concave down on $$(0.5, \infty)$$

Question 7

For the function $$f(x) = x^4$ - 24x^2$ - 5x - 10$$, what are the inflection points?

Accepted Answer

$$x = -2$$ and $$x = 2$$

Question 8

A $$150,000$$ deposit earns 6.3% interest compounded continuously. What is the account balance after 10 years?

Accepted Answer

$$A(10) = 150000e$$^{0.063 	imes 10}$$

Question 9

If the world population in 1950 was 2550 million and in 1980 was 4300 million, what is the exponential model for population growth?

Accepted Answer

$$P(t) = 2550e$$^{0.016t}$$

Question 10

A food product contains Iodine-131 with a half-life of 8 days. If the product has 5 times the allowable limit, how long until it reaches the allowable limit?

Accepted Answer

$$t = \frac{\ln(1/5)}{-0.0866}$$

Question 11

Given $$f(x) = x^2$ + 6x - 12x^2$ - 12$$, find the absolute max and min values on the interval $$[2,9].$$

Accepted Answer

Absolute max at $$x=9$$, absolute min at $$x=4$$

Question 12

For the function $$f(x) = x^3$ - 3x^2$ + 12x + 9$$, what are the critical numbers and their classification?

Accepted Answer

Critical numbers at $$x=0 ($$local min), $$x=3 ($$local max)

Question 13

A $$10,000$$ deposit earns 4.1% interest compounded continuously. How long until the investment doubles?

Accepted Answer

$$t = \frac{\ln 2}{0.041}$$

Question 14

A $$4000$$ deposit earns 5.5% interest compounded continuously. How long until the investment is worth $$8000$$?

Accepted Answer

$$t = \frac{\ln 2}{0.055}$$

Question 15

A $$7500$$ deposit grows to $$9000 in 7 $$years with continuous compounding. What is the interest rate?

Accepted Answer

$$r = \frac{\ln(9000/7500)}{7}$$

Question 16

How much do you need to invest at 6.5% interest compounded continuously to have $$1$$ million after $$40$$ years?

Accepted Answer

$$P = \frac{1000000}{e$$^{0.065 	imes 40}$$}$$

Question 17

A plant fiber contains 75% of the original Carbon-14. If the half-life of Carbon-14 is 5730 years, what is the age of the plant fiber?

Accepted Answer

$$t = \frac{\ln(0.75)}{-\ln(0.5)/5730}$$

Question 18

Given $$f(x) = -3x^5$ + 40x^3$ + 5x - 1$$, what is the inflection point?

Accepted Answer

$$x = 0$$

Business Calculus Study Guide: Exponential Growth, Decay, Derivatives, and Optimization

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