DE's and Mathematical Modelling Quiz 1

Shown above is a slope field for which differential equation?

• $\frac{dy}{dx}=\ xy$

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

• $\frac{\text{d}y}{\text{d}x}=xy-y$

• $\frac{\text{d}y}{\text{d}x}=xy+x$

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

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?

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)?

• 4t + 4

•  $2t^2+4$  

•  $e^{\frac{t}{2}\ln9}+3$  

•  $4e^{\frac{t}{2}\ln3}$  

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.

• P=.05e^1000t

• P=1000e^5t

• P=1000e^.05t

• p=lne^1000t

Tags

CCSS.HSF.BF.A.1

CCSS.HSF.LE.A.1

CCSS.HSF.LE.A.2

CCSS.HSF.LE.B.5

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

• 𝑦 = ln(15t)

• 𝑦 =  $16t^3$  

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

• 𝑦 =  $\left(2t^3-16\right)$  

∫ 2x cos(x²) dx

• sin( $x^2$ ) + C

• 2 sin ( 2x ) + C

• $\frac{1}{2}$ sin( $x^2$ ) + C

• 4 cos(2x ) + C

• ln(2x²+6) + C
• ln(2x²+6)(4x) + C
• 1/(2x²+6) + C
• 1/(2x²+6)² + C

Tags

CCSS.HSA.APR.D.6

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

∫(3x⁴+2)⁴(12x³)dx

• ½(3x⁴+2)⁴+C
• (3x⁴+2)⁴+C
• ⅕(3x⁴+2)⁵+C
• ⅓(3x⁴+2)⁶+C

Tags

CCSS.HSA.APR.A.1

 $\int100x^3\sqrt{5x^4+3}dx$  

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

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

•  $\sqrt{\left(5x^4+3\right)^3}+C$  

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