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Calculus Independent Study Path

Unit 7 Practice Test

1. Find \frac{dy}{dx} :

y = (x^{2} + 2)^{2−x}.

2. Find

lim_{x→∞} \frac{x^{p}}{e^{x}} ,

where p > 1.

3. Find

∫ (e^{x} + \frac{x + 1}{x}) dx.

4. Find

∫ tan x dx = ∫ \frac{sin x}{cos x}dx

5. Find r such that y = e^{rx} satisfies

\frac{d^{4}y}{dx^{4}} − 13\frac{d^{2}y}{dx^{2}} + 36y = 0.

6. In the Law of Logistic Growth, it is assumed that at time t, the rate of growth, f′(t), of a quantity, f(t), is given by f′(t) = Af(t)[B − f(t)] where A and B are constants. If f(0) = C, show that

f(t) = \frac{BC}{C + (B − C)e^{−ABt}}

7. A sociologist claims that the population of a certain country is growing at the rate of 2% per year. If the present population of the country is 200 million people, and the growth rate remains at 2% per year, what would be the population be in the year 2020?