Free Printable BC Topics 7.6-7.9 Review

1.

The function R satisfies the logistic differential equation $\frac{dR}{dt}=\frac{1}{50}R\left(1-\frac{R}{400}\right)$ where $R\left(0\right)=10$ . Which of the following statements is/are true?

a)

$\lim_{x\rightarrow\infty}R\left(t\right)=400$

b)

$\frac{dR}{dt}$ has a maximum value when $R=200$

c)

When $R=50$ , $\frac{dR}{dt}>0$ and $\frac{d^2R}{dt^2}<0$

d)

$\lim_{x\rightarrow\infty}R'\left(t\right)=400$

2.

A local camp experiences a poison ivy outbreak and the number of campers that have poison ivy t days after camp starts is modeled by a logistic differential equation with carrying capacity 100. Which of the following tables BEST models the number of campers, P, with poison ivy?

a)

b)

c)

d)

3.

The number of teachers at a local high school that have entered final exam scores at time t hours from 8AM is modeled by the function F, the solution to a logistic differential equation. At 8AM, only 3 of the 80 teachers have entered their final exam scores. Also at 8AM, F is increasing at a rate of 6 teachers per hour. Which of the following could be the logistic differential equation?

a)

$\frac{dF}{dt}=\frac{1}{87}F\left(80-F\right)$

b)

$\frac{dF}{dt}=\frac{2}{77}F\left(80-F\right)$

c)

$\frac{dF}{dt}=\frac{1}{148}F\left(80-F\right)$

d)

$\frac{dF}{dt}=\frac{9}{37}F\left(80-F\right)$

4.

The number of rabbits R living in a small wooded area at time t is increasing according to a logistic differential equation. Which of the following could be the differential equation?

a)

$\frac{dR}{dt}=0.155R$

b)

$\frac{dR}{dt}=0.155t\left(2400-t\right)$

c)

$\frac{dR}{dt}=0.155\left(240-R\right)$

d)

$\frac{dR}{dt}=0.155R\left(2400-R\right)$

5.

The number of people P in a small town that have seen a newly released movie changes at a rate modeled by the differential equation $\frac{dP}{dt}=0.1P\left(3000-P\right)$ where t is measured in days. What are all values of P for which the number of people that have seen the movie is increasing at an increasing rate?

a)

1500

b)

$0<p<1500$

c)

$1500<P<3000$

d)

$0<P<3000$

6.

If $\frac{dy}{dx}=3x^2y^2$ and y=1 when x=2, what is y when x= -1?

a)

$\frac{1}{10}$

b)

$\frac{1}{8}$

c)

$\frac{1}{19}$

d)

$-\frac{1}{8}$

7.

If $\frac{dy}{dx}=3\sqrt{xy}$ and y=4 when x=1, what is y when x=2?

a)

9

b)

$9+4\sqrt{2}$

c)

$9-4\sqrt{2}$

d)

$9-2\sqrt{2}$

8.

Which of the following differential equations are separable?

a)

$\frac{dy}{dx}=x^2-y^2$

b)

$\frac{dy}{dx}=xy-3x+2y-6$

c)

$\frac{dy}{dx}=e^{y-x}$

d)

$\frac{dy}{dx}=xy-3y$

9.

Solve the differential equation $\frac{dy}{dx}=y^4$

a)

$y=-3\left(3x+C\right)^{\frac{1}{3}}$

b)

$y=\left(-\frac{1}{3x+C}\right)^{\frac{1}{3}}$

c)

$y=\left(-\frac{1}{3x}+C\right)^{\frac{1}{3}}$

d)

$y=\left(-\frac{1}{3x}\right)^{\frac{1}{3}}+C$

10.

Solve the differential equation $\frac{dy}{dx}=\left(3x^2+1\right)\left(y^2+16\right)$ passing through the origin.

a)

$y=\tan\left(4x^3+4x\right)$

b)

$y=4\tan\left(x^3+x\right)$

c)

$y=16\tan\left(x^3+x\right)$

d)

$y=4\tan\left(4x^3+4x\right)$

11.

Find the particular solution that satisfies the differential equation $dT+\frac{1}{2}\left(T-100\right)dt=0$ with initial condition $T\left(0\right)=40$ .

a)

$T=100e^{-\frac{t}{2}}-60$

b)

$T=100-40e^{-\frac{t}{2}}$

c)

$T=100-60e^{-\frac{t}{2}}$

d)

$T=100+60e^{-\frac{t}{2}}$

12.

If the solution to $\frac{dy}{dx}=-\sqrt{xy}$ passes through (0,4), find the value of y when x=4.

a)

$\sqrt{\frac{2}{3}}$

b)

$\frac{4}{9}$

c)

$-\frac{2}{3}$

d)

$\frac{16}{9}$

13.

Solve the differential equation $yy'=e^{-x}+1$ passing through the point (0,-1).

a)

$y=\sqrt{2x-\frac{2}{e^x}+3}$

b)

$y=-\sqrt{2x-\frac{2}{e^x}+3}$

c)

$y=\sqrt{2x+\frac{2}{e^x}+1}$

d)

$y=-\sqrt{2x+\frac{2}{e^x}+1}$

14.

If $\frac{dy}{dx}=xy\sec^2\left(x^2\right)$ and $y=\pi$ when $x=\sqrt{\pi}$ , find y when $x=\sqrt{\frac{\pi}{4}}$ .

a)

$\frac{\sqrt{\pi}}{e}$

b)

$\pi\sqrt{e}$

c)

$\sqrt{\pi e}$

d)

$\frac{e}{\sqrt{\pi}}$

15.

If $4f\left(x\right)f'\left(x\right)-6e^x=0$ and $f\left(0\right)=3$ , find $\lim_{x\rightarrow\infty}f\left(x\right)$ .

(This one is a bit strange... was an AP question #30)

a)

6

b)

36

c)

$\sqrt{6}$

d)

$\infty$

16.

Solve the differential equation $\frac{dy}{dt}=\frac{ty+5t}{t^2+1}$ if $y\left(0\right)=3$ .

a)

$y=8e^{\sqrt{t^2+1}}-5$

b)

$y=2e^{\sqrt{t^2+1}}-5$

c)

$y=e^{\sqrt{t^2+1}}-5$

d)

$y=e^{\frac{1}{2}\left(t^2+1\right)}-5$

17.

Which of the following differential equations model(s) exponential decay if y=1 when t=0?

a)

$\frac{dy}{dt}=-y$

b)

$\frac{dy}{dt}=e^{-y}$

c)

$\frac{dy}{dt}=-\frac{t}{y}$

18.

Solve $\frac{dy}{dx}=\frac{2x}{e^{2y}}$

a)

$y=\ln\sqrt{2x^2+C}$

b)

$y=\ln\sqrt{2x^2}+C$

c)

$y=\ln\left(2\sqrt{x^2+C}\right)$

d)

$y=\ln\left(2\sqrt{x^2}\right)+C$

19.

If $\frac{dy}{dt}=-0.3y$ and y=41 when t=0, solve for y.

a)

$y=41e^{-0.3t}$

b)

$y=e^{-0.3t}+40$

c)

$y=-0.3t+41$

d)

$y=41e^{-0.3y}$

20.

During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 5,000 people are infected when the epidemic is first discovered, and 6,000 people are infected 3 days later, how many people are infected 20 days after the

epidemic is first discovered?

a)

16,859

b)

18,492

c)

13,974

d)

19,871

BC Topics 7.6-7.9 Review

Total questions: 20

Worksheet time: 1hrs 13mins

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