Free Printable DE's and Mathematical Modelling Quiz 1

1.

Shown above is a slope field for which differential equation?

a)

$\frac{dy}{dx}=\ xy$

b)

$\frac{\text{d}y}{\text{d}x}=xy\ +\ y$

c)

$\frac{\text{d}y}{\text{d}x}=xy-y$

d)

$\frac{\text{d}y}{\text{d}x}=xy+x$

e)

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

2.

$\int_{ }^{ }\left(x-1\right)\sqrt{x}\ dx\ =$

a)

$\frac{1}{2}x^2+2x^{\frac{3}{2}}-x+c$

b)

$\frac{2}{3}x^{\frac{3}{2}}+\frac{1}{2}x^{\frac{1}{2}}+c$

c)

$\frac{3}{2}\sqrt{x}-\frac{1}{\sqrt{x}}+c$

d)

$\frac{2}{5}x^{\frac{5}{2}}-\frac{2}{3}x^{\frac{3}{2}}+c$

e)

$\frac{1}{2}x^2-x+c$

3.

A rumour spreads among a population of N people at a rate proportional to the product of number of people who have heard the rumour and the number of people who have not heard the rumour. If p denotes the number of people who have heard the rumour, which of the following differential equations could be used to model this situation with respect to time t, where k is a positive constant?

a)

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

b)

$\frac{dp}{dt}=kp\left(N-p\right)$

c)

$\frac{dp}{dt}=kp\left(p-N\right)$

d)

$\frac{dp}{dt}=kt\left(N-t\right)$

e)

$\frac{dp}{dt}=kt\left(t-N\right)$

4.

Let y = f(t) be a solution to the differential equation $\frac{dy}{dt}=ky$ , where k is a constant. Values of f for selected values of t are given in the table above. Which of the following is an expression for f(t)?

a)

4t + 4

b)

$2t^2+4$

c)

$e^{\frac{t}{2}\ln9}+3$

d)

$4e^{\frac{t}{2}\ln3}$

5.

The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year.

Find the general solution.

a)

P=.05e^1000t

b)

P=1000e^5t

c)

P=1000e^.05t

d)

p=lne^1000t

6.

2. Solve the differential equation
$\frac{dy}{dt}=\frac{3t^2}{y}$ with initial condition y(2) = 0.

a)

𝑦 = ln(15t)

b)

𝑦 = $16t^3$

c)

$y=\left(2t^3-16\right)^{\frac{1}{2}}$

d)

𝑦 = $\left(2t^3-16\right)$

7.

∫ 2x cos(x²) dx

a)

sin( $x^2$ ) + C

b)

2 sin ( 2x ) + C

c)

$\frac{1}{2}$ sin( $x^2$ ) + C

d)

4 cos(2x ) + C

8.

a)

ln(2x²+6) + C

b)

ln(2x²+6)(4x) + C

c)

1/(2x²+6) + C

d)

1/(2x²+6)²+ C

9.

∫(3x⁴+2)⁴(12x³)dx

a)

½(3x⁴+2)⁴+C

b)

(3x⁴+2)⁴+C

c)

⅕(3x⁴+2)⁵+C

d)

⅓(3x⁴+2)⁶+C

10.

$\int100x^3\sqrt{5x^4+3}dx$

a)

$\frac{15}{2}\sqrt{\left(5x^4+3\right)^3}+C$

b)

$5\sqrt{\left(5x^4+3\right)^3}+C$

c)

$\sqrt{\left(5x^4+3\right)^3}+C$

d)

$\frac{10}{3}\sqrt{\left(5x^4+3\right)^3}+C$

DE's and Mathematical Modelling Quiz 1

Total questions: 10

Worksheet time: 45mins

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