Free Printable BC Topics 7.1-7.5 Review

1.

The rate of change of G with respect to t is proportional to the square of G.

a)

$\frac{dG}{dt}=kG^2$

b)

$\frac{dG}{dt}=kt^2$

c)

$\frac{dG}{dt}=\frac{k}{G^2}$

d)

$\frac{dG}{dt}=\frac{k}{t^2}$

2.

The rate of change of H with respect to t is proportional to the square root of H.

a)

$\frac{dH}{dt}=k\sqrt{H}$

b)

$\frac{dH}{dt}=k\sqrt{t}$

c)

$\frac{dH}{dt}=k+\sqrt{H}$

d)

$\frac{dH}{dt}=\frac{k}{\sqrt{H}}$

3.

The rate of change of J with respect to t is inversely proportional to the square of J.

a)

$\frac{dJ}{dt}=\frac{k}{J^2}$

b)

$\frac{dJ}{dt}=kJ^2$

c)

$\frac{dJ}{dt}=k\sqrt{J}$

d)

$\frac{dJ}{dt}=\frac{k}{\sqrt{J}}$

4.

The rate of change of M with respect to t is inversely proportional to the cube root of M.

a)

$\frac{dM}{dt}=\frac{k}{M^{\frac{1}{3}}}$

b)

$\frac{dM}{dt}=kM^{\frac{1}{3}}$

c)

$\frac{dM}{dt}=\frac{k}{M^3}$

d)

$\frac{dM}{dt}=\frac{k}{t^{\frac{1}{3}}}$

5.

Write as a differential equation: The rate of change of a population of bees, b, over time is proportional to the population of bees.

a)

$\frac{db}{dt}=kb$

b)

$\frac{db}{dt}=kt$

c)

$\frac{db}{dt}=\frac{k}{b}$

d)

$\frac{db}{dt}=\frac{k}{t}$

6.

The rate of growth of a population of zombies, z, over time is proportional to the population of zombies.

a)

$\frac{dz}{dt}=kz$

b)

$\frac{dz}{dt}=kt$

c)

$\frac{dz}{dt}=\frac{k}{z}$

d)

$\frac{dz}{dt}=\frac{k}{t}$

7.

Newton’s Law of Cooling states that the rate of cooling an object is proportional to the temperature difference between the object and the surrounding air. A cup of hot chocolate is put on a table in a room that is 70º. Let H be the temperature of the hot chocolate after t minutes.

a)

$\frac{dH}{dt}=k\left(H-70\right)$

b)

$\frac{dH}{dt}=k\left(t-70\right)$

c)

$\frac{dH}{dt}=\frac{k}{H-70}$

d)

$\frac{dH}{dt}=\frac{k}{t-70}$

8.

In a community of F farmers, the number x of farmers who own a certain combine changes with respect to time t at a rate that is jointly proportional to the number of farmers who own the combine and to the number of farmers who do not own the combine.

a)

$\frac{dx}{dt}=kx\left(F-x\right)$

b)

$\frac{dx}{dt}=kF\left(F-x\right)$

c)

$\frac{dx}{dt}=kF\left(x-t\right)$

d)

$\frac{dx}{dt}=kx\left(F-t\right)$

9.

The rate of change in the height h of a tree with respect to its age a is inversely proportional to the tree’s height.

a)

$\frac{dh}{da}=\frac{k}{h}$

b)

$\frac{da}{dh}=\frac{k}{h}$

c)

$\frac{dh}{dt}=\frac{k}{a}$

d)

$\frac{da}{dt}=\frac{k}{h}$

10.

Suppose a virus is spreading in a network of computers. There are M computers in the network. The rate of change in the number of infected computers is jointly proportional to the number of computers already infected, j, and the number of computers not yet infected.

a)

$\frac{dj}{dt}=kj\left(M-j\right)$

b)

$\frac{dj}{dt}=kM\left(M-j\right)$

c)

$\frac{dM}{dt}=kj\left(M-j\right)$

d)

$\frac{dM}{dt}=kM\left(M-j\right)$

11.

The rate of change of a level of response R with respect to the level of a stimulus s is a joint proportionality between the level of the response and the inverse of the level of the stimulus.

a)

$\frac{dR}{ds}=k\frac{R}{s}$

b)

$\frac{ds}{dR}=k\frac{R}{s}$

c)

$\frac{ds}{dR}=kRs$

d)

$\frac{dR}{ds}=kRs$

12.

Let S represent the total sales of a computer product t years after the product was introduced. The total sales of the computer product are growing in inverse proportion to $\ln(t+1.4)$ .

a)

$\frac{dR}{dt}=k\ln\left(t+1.4\right)$

b)

$\frac{dR}{dt}=\frac{k}{\ln\left(t+1.4\right)}$

c)

$\frac{dR}{dt}=ke^{t+1.4}$

d)

$\frac{dR}{dt}=\frac{k}{e^{t+1.4}}$

13.

Barometric pressure p (measured in inches of mercury) changes with respect to altitude a (measured in feet) at a rate that is directly proportional to the altitude.

a)

$\frac{dp}{dt}=ka$

b)

$\frac{dp}{da}=ka$

c)

$\frac{dp}{da}=\frac{k}{t}$

d)

$\frac{dp}{dt}=\frac{k}{a}$

14.

Which of the following is a solution to the differential equation $y''-4y=0$ ?

a)

$y=e^{2x}$

b)

$y=2e^{2x}$

c)

$y=\sin\left(2x\right)$

d)

$y=\cos\left(2x\right)$

15.

Which of the following is a solution to the differential equation $y''-10y'+9y=0$ ?

a)

$y=2\sin\left(3x\right)$

b)

$y=5e^x$

c)

$y=Ce^{9x}$ where C is a constant

16.

For what value of k, if any, will $y=3ke^{2x}+\cos\left(4x\right)$ be a solution to the differential equation $\frac{y''}{2}+8y=15e^{2x}$ ?

a)

$\frac{1}{2}$

b)

$\frac{8}{15}$

c)

$\frac{5}{6}$

d)

There is no such value of k

17.

For what value of k, if any, will $y=k\sin\left(-x\right)+2\cos\left(3x\right)$ be a solution to the differential equation $2y''+18y=32\sin\left(-x\right)$ ?

a)

2

b)

4

c)

8

d)

There is no such value of k

18.

For what value of k, if any, will $y=e^{-3x}+ke^{4x}$ be a solution to the differential equation $3y'+y''=-14e^{4x}$ ?

a)

$-\frac{1}{2}$

b)

$\frac{4}{7}$

c)

$-\frac{3}{4}$

d)

There is no such value of k

19.

For what value of k, if any, will $y=e^{3x}+ke^{-2x}$ be a solution to the differential equation $y''-2y'-3y=4e^{-2x}$ ?

a)

$\frac{4}{5}$

b)

$\frac{2}{3}$

c)

$\frac{4}{7}$

d)

There is no such value of k

20.

Which of the following functions are solutions to the differential equation $y''+y=0$ ?

a)

$y=3\cos x$

b)

$y=e^{-x}$

c)

$y=\sin x$

d)

$y=-\cos x$

e)

$y=\sin\left(3x\right)$

21.

Is $y=4x$ a solution to the differential equation $\frac{dy}{dx}=\frac{4y}{x}$ ?

a)

Yes

b)

No

22.

Is $y=x^4$ a solution to the differential equation $\frac{dy}{dx}=\frac{4y}{x}$ ?

a)

Yes

b)

No

23.

The function $y=e^{2x}-3x+4$ is a solution to which of the following differential equations?

a)

$y"-y'+6=0$

b)

$y"-2y'-6=0$

c)

$y"-y'+3=0$

d)

$2y"-y'+3=0$

24.

Which differential equation below is represented by the given slope field?

a)

$\frac{dy}{dx}=x+y$

b)

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

c)

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

d)

$\frac{dy}{dx}=x^2+xy$

25.

Which differential equation below is represented by the given slope field?

a)

$\frac{dy}{dx}=x+y$

b)

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

c)

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

d)

$\frac{dy}{dx}=x^2+xy$

26.

Which differential equation below is represented by the given slope field?

a)

$\frac{dy}{dx}=x+y$

b)

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

c)

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

d)

$\frac{dy}{dx}=x^2+xy$

27.

Which differential equation below is represented by the given slope field?

a)

$\frac{dy}{dx}=x+y$

b)

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

c)

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

d)

$\frac{dy}{dx}=x^2+xy$

28.

Which slope field represents the differential equation $\frac{dy}{dx}=\frac{x}{y}$ ?

a)

b)

c)

d)

29.

Which slope field represents the differential equation $\frac{dy}{dx}=-\frac{x}{y}$ ?

a)

b)

c)

d)

30.

Which slope field represents the differential equation $\frac{dy}{dx}=\frac{y}{x}$ ?

a)

b)

c)

d)

31.

Which slope field represents the differential equation $\frac{dy}{dx}=-\frac{y}{x}$ ?

a)

b)

c)

d)

32.

Let $y=f\left(x\right)$ be the solution to the differential equation $\frac{dy}{dx}=x+y^2$ with initial condition $f\left(3\right)=-1$ . Using Euler's method with a step size of 0.5, what is the approximation for $f\left(4\right)$ ?

a)

2.25

b)

3.25

c)

4.5

d)

5.5

33.

Let $y=f\left(x\right)$ be the solution to the differential equation $\frac{dy}{dx}=y^x$ with initial condition $f\left(0\right)=k-1$ where k is a constant and $k\ne0$ . Using Euler's method with 3 steps of equal size and getting the approximation $f\left(3\right)\approx0$ , find the value of k.

a)

$-\frac{1}{2}$

b)

$\frac{1}{2}$

c)

$1$

d)

$-1$

34.

Approximate the value of $f\left(1.5\right)$ for $\frac{dy}{dx}=y^2-x^2$ passing through $\left(0,-1\right)$ . Use Euler's method with three equal steps.

a)

$-0.5$

b)

$-0.875$

c)

$-0.75$

d)

$-0.37$

35.

Given that $y\left(1\right)=-3$ and $\frac{dy}{dx}=2x+y$ , what is the approximation for $y\left(2\right)$ if Euler's method is used with a step size of 0.5?

a)

-5

b)

-4.25

c)

-4

d)

-3.75

36.

Let $y=f\left(x\right)$ be the solution to the differential equation $\frac{dy}{dx}=x-y-1$ with initial condition $f\left(1\right)=-2$ . What is the approximation for $f\left(1.4\right)$ if Euler's method is used with two steps of equal size?

a)

0.2

b)

-1.2

c)

-1.24

d)

-0.64

37.

Let $y=f\left(x\right)$ be the solution to the differential equation $\frac{dy}{dx}=2y-x$ with initial condition $f\left(1\right)=2$ . What is the approximation for $f\left(0\right)$ obtained by using Euler's method with two steps of equal length?

a)

$-\frac{5}{4}$

b)

$-1$

c)

$\frac{1}{4}$

d)

$\frac{1}{2}$

38.

Using the table and Euler's Method with a step size of 1, approximate f(4.5) given f(1.5)=4.

a)

4.1

b)

6.0

c)

5.3

d)

8.29

39.

Given $\frac{dy}{dx}=3x-2y$ and $y\left(0\right)=k$ . Starting with $x=0$ and a step size of 1, find k if $y\left(2\right)=4.5$

a)

1

b)

2

c)

1.5

d)

2.5

40.

Given $\frac{dy}{dx}=x+3y$ and $y\left(2\right)=3$ . Use Euler's Method with a step size of 1 to approximate $y\left(5\right)$

a)

11

b)

181

c)

45

d)

240

41.

If $\frac{dy}{dx}=x-2$ and $y\left(0\right)=5$ , approximate $y\left(0.8\right)$ using two equal steps in Euler's method.

a)

3.91

b)

4.32

c)

3.56

d)

5.30

42.

Use the table and Euler's method with two equal steps to approximate the value of f(2.6).

a)

1.78

b)

0.18

c)

1.86

d)

1.9

43.

Which functions below will have the same slope field? (choose all that apply)

a)

y = x² - 3

b)

y = x² + 4

c)

y = 2x - 5

d)

y = 2x²

44.

Which differential equation could be represented by the slope field?

a)

A

b)

B

c)

C

d)

D

45.

Which differential equation could be represented by the slope field? (Dashes on x-axis)

a)

A

b)

B

c)

C

d)

D

46.

If a function is y = 2x, then its slope field would...

a)

have slopes of 2

b)

have slopes of x²

c)

have slopes x³

d)

have slopes of 0

47.

Which differential equation could be represented by the slope field?

a)

A

b)

B

c)

C

d)

D

48.

The table above shows values of f', the derivative of f, for selected values of x. If $f\left(-0.3\right)=2$ , what is the approximation for f(0.6) obtained by using Euler's method with a step side of 0.3 starting at $x=-0.3$ .

a)

2.36

b)

2.66

c)

3.02

d)

3.14

49.

The equation $y=e^x-2$ is a particular solution to which of the following differential equations?

a)

y' - y = -2

b)

y' - y = 2

c)

y' + y = -2

d)

y' + y = 2

50.

Which of the following is a solution to the differential equation y'' + y = 0?

a)

y = ln(x)

b)

y = e^x

c)

y = x

d)

y = sin(x)

51.

In which quadrant(s) is the differential equation $\frac{dy}{dx}=\frac{xy^2}{e^y}$ always negative?

a)

Quadrants I and II only

b)

Quadrants II and III only

c)

Quadrants I and III only

d)

Quadrants II, III, and IV only

52.

Which differential equation does this slope field belong to?

a)

dy/dx = xy

b)

dy/dx = xy+y

c)

dy/dx = xy+x

d)

dy/dx = xy-y

53.

y varies jointly with x and the square of z

a)

y= kxz²

b)

y= kx√z

c)

y= kxz

d)

kx²z

54.

Which slope field is represented by dy/dx = x²?

a)

b)

c)

d)

BC Topics 7.1-7.5 Review

Total questions: 54

Worksheet time: 3hrs 42mins

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