Q: The half-life of a certain element is 700 years. How long will it take for only 33% of the original amount to remain? (Leave your answer in terms of logs.)

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

Question 1

Suppose a bank account earns 9% interest compounded annually. If you deposit $$1,000 in January 2017, in what year will the account first exceed 10$$,000? (Use $$A = P(1 + r)^t$$ and solve for $$t.)$$

Accepted Answer

$$t = \frac{\log(10)}{\log(1.09)} \approx 25.8$$; so in 2043

Question 2

A bird population on an island doubles every 6 months. If the initial population is 450 birds, how many birds will there be after 18 years? (Do not simplify your answer.)

Accepted Answer

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

Question 3

The half-life of a certain element is 700 years. How long will it take for only 33% of the original amount to remain? (Leave your answer in terms of logs.)

Accepted Answer

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

Question 4

Solve for $$x$$: $$2^x = 5 	imes 7^x. $$Leave logs in your answer.

Accepted Answer

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

Question 5

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

Accepted Answer

$$8.0$$

Question 6

A ferret runs along the $$x-$$axis and is $$f(t) = t^2$ + 3 cm $$from the origin after $$t$$ seconds. What is the instantaneous velocity after 4 seconds?

Accepted Answer

$$8 cm/s$$

Question 7

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

Accepted Answer

When the price is $$50 per set, increasing the price by 1$ decreases weekly sales by 201$ TVs.

Question 8

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

Accepted Answer

$$3$$

Question 9

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

Accepted Answer

$$x = 5$$

Question 10

If $$f(x) = 10^x$$, what is the average rate of change of $$f$$ between $$x = 0.2$$ and $$x = 0.7$$?

Accepted Answer

$$\frac{10$$^{0.7}$$ - 10$$^{0.2}$$}{0.5}$$

Question 11

If $$f(x) is $$the number of people who get a disease when $$x$$ percent are inoculated, what does $$f'(50) = -10^3$ mean?

Accepted Answer

At 50% inoculation, increasing inoculation by 1% decreases the number of cases by 1,000.

Question 12

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

Accepted Answer

$$-4$$

Question 13

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

Accepted Answer

$$x = \frac{\log(5/7)}{4\log(7)}$$

Question 14

If a population doubles every 6 months, how many years will it take for the population to quadruple? (Leave logs in your answer.)

Accepted Answer

$$t = 1 	ext{ year}$$

Question 15

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

Accepted Answer

$$2$$

Question 16

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

$$5.06$$

Question 17

Solve for $$x$$: $$5^{3x+1} + 2 = 9$. Leave logs in your answer.

Accepted Answer

$$x = \frac{\log(7/5)}{3\log(5)}$$

Question 18

If $$f(x) = 10^x$$, what is the rate of change of $$f at$$ $$x = 0.55$$?

Accepted Answer

$$10^{0.55}$$ \ln(10)$$

Question 19

If $$\log(y) = 1.87$$, what is $$y$$?

Accepted Answer

$$y = 10$$^{1.87}$$

Question 20

If $$\log(100 	imes 378) is to be $$computed using log properties, which is the correct expression?

Accepted Answer

$$\log(100) + \log(378)$$

Math 34A Exam 2 Review – Calculus Concepts and Problem-Solving Guidance

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