Q: A bank pays 9% interest compounded annually. If you deposit $$1,000 in January 2017, in what year will your account first reach 10$$,000? (Assume no withdrawals.)

When $$n$$ satisfies $$1000 \times (1.09)^n = 10000$$, so $$n = \frac{\log(10)}{\log(1.09)} \approx 27$$ years, thus in 2044.

Q: The half-life of a certain element is 700 years. How long will it take to have 33% of the original amount remaining? (Leave any exponents or logs in your answer.)

$$t = 700 \times \frac{\log(0.33)}{\log(0.5)}$$

Q: If a population doubles every 6 months, how many years would it take for the population to increase from 450 to 1900? (Leave logs in your answer.)

$$t = \frac{\log(1900/450)}{\log(2)} \times 0.5$$ years

Question 1

A bank pays 9% interest compounded annually. If you deposit $$1,000 in January 2017, in what year will your account first reach 10$$,000? (Assume no withdrawals.)

Accepted Answer

When $$n$$ satisfies $$1000 	imes (1.09)^n = 10000$$, so $$n = \frac{\log(10)}{\log(1.09)} \approx 27$$ years, thus in 2044.

Question 2

A bird population on an island doubles every 6 months. The initial size is 450 birds. How many birds will there be in 18 years?

Accepted Answer

$$450 	imes 2$$^{36}$$

Question 3

The half-life of a certain element is 700 years. How long will it take to have 33% of the original amount remaining? (Leave any exponents or logs in your answer.)

Accepted Answer

$$t = 700 	imes \frac{\log(0.33)}{\log(0.5)}$$

Question 4

If $$f(x) = 2x^3$ - x + 1$$, what is the average rate of change between $$x = 1$$ and $$x = 2$$?

Accepted Answer

$$\frac{f(2) - f(1)}{2 - 1} = \frac{15 - 2}{1} = 13$$

Question 5

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

Accepted Answer

$$x = 5$$

Question 6

If $$Q(p) is $$the number of TVs sold each week at price $$p$$, what does $$Q'(50) = -201$$ mean?

Accepted Answer

For each $$1 increase in price at p = 50$, 201 fewer TVs are sold per week.

Question 7

Compute $$\log_4$(64).$$

Accepted Answer

$$3$$

Question 8

If $$f(x) is $$the number of people who get a disease when $$x$$ percent are inoculated, what does $$f'(50) = -103$$ mean?

Accepted Answer

At $$x = 50$$, for each 1% increase in inoculation, 103 fewer people get the disease.

Question 9

Find the instantaneous velocity of a ferret after 4 seconds if its position is $$f(t) = t^2$ + 3 cm $$from the origin.

Accepted Answer

$$8 cm/s$$

Question 10

Solve for $$x$$: $$2^x = 5 	imes 7^x.$$

Accepted Answer

$$x = \frac{\log(5)}{\log(2) - \log(7)}$$

Question 11

Compute $$\log_2$\left(\frac{1}{16}\right).$$

Accepted Answer

$$-4$$

Question 12

If $$f(x) = 2x^3$ - x + 1$$, estimate $$f'(1)$$ using the average rate of change between $$x = 1$$ and $$x = 1.01.$$

Accepted Answer

$$\frac{f(1.01) - f(1)}{1.01 - 1} \approx 5.12$$

Question 13

Solve for $$x$$: $$7^{4x+1} - 3 = 2$ (leave logs in your answer).

Accepted Answer

$$x = \frac{\log(5) - 1}{4}$$

Question 14

Compute $$\ln(e^2$).$$

Accepted Answer

$$2$$

Question 15

If a population doubles every 6 months, how many years would it take for the population to increase from 450 to 1900? (Leave logs in your answer.)

Accepted Answer

$$t = \frac{\log(1900/450)}{\log(2)} 	imes 0.5$$ years

Question 16

What does $$f(0) = 106$$ mean in the context of disease inoculation?

Accepted Answer

If 0% of people are inoculated, 106 people get the disease.

Question 17

Compute $$\log_{25}(5).$$

Accepted Answer

$$0.5$$

Question 18

Find the average rate of change of $$10^x$$ between $$x = 0.2$$ and $$x = 0.7.$$

Accepted Answer

$$\frac{10$$^{0.7}$$ - 10$$^{0.2}$$}{0.7 - 0.2}$$

Question 19

If $$f(x) = 2x^3$ - x + 1$$, what is the average rate of change between $$x = 1$$ and $$x = 1.5$$?

Accepted Answer

$$\frac{f(1.5) - f(1)}{1.5 - 1} = \frac{6.25 - 2}{0.5} = 8.5$$

Math 34A Exam 2 Review – Calculus Concepts and Applications

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