Differential Equations — Quiz 4

Questions in this quiz (5)

1. Which of the following differential equations is not logistic?

   • $$\frac{dP}{dt}=0.02P(500-P)$$
   • $$P'=P-0.001P^2$$
   • $$\frac{dx}{dt}=0.5x(200-x)$$
   • $$\frac{dN}{dt}=0.04(300-N)$$

2. The slope field for $$F'(x)=e^{-x^2}$$ is shown above with the particular solution $$F(0)=0$$ superimposed. With a graphing calculator,$$\lim_{x\to\infty}F(x)$$ to $$3$$ decimal places is:

   • $$0.886$$
   • $$0.987$$
   • $$1.000$$
   • $$\infty$$

3. According to Newton's law of cooling, the temperature of an object decreases at a rate proportional to the difference between its temperature and that of the surrounding air. Suppose a cup of tea at a temperature of $$90^\circ C$$ is placed in a room where the temperature is $$20^\circ C$$. The differential equation is:$$\frac{dT}{dt}=-k(T-20)$$where $$T$$ is the temperature of the tea in $$^\circ C$$ and $$t$$ is time in minutes. If the tea cools to $$70^\circ C$$ in $$10$$ minutes, then its temperature is given by the equation:

   • $$T=20+70e^{-0.034t}$$
   • $$T=20+70e^{-0.046t}$$
   • $$T=20+90e^{-0.046t}$$
   • $$T=90e^{-0.046t}$$

4. A bowl of soup at temperature $$190^\circ F$$ is placed on a table in a room at $$70^\circ F$$. The differential equation for its temperature at time $$t$$ in minutes is$$\frac{dy}{dt}=-0.09(y-70);$$ $$y(0)=190$$After $$10$$ minutes, the temperature in $$^\circ F$$ of the soup is approximately:

   • $$112$$
   • $$119$$
   • $$125$$
   • $$140$$

5. $$[-4,4]$$ x $$[-4,4]$$A solution curve has been superimposed on the slope field shown above. The solution is for the differential equation and initial condition:

   • $$\frac{dy}{dx}=\tan x,$$ $$y(0)=0$$
   • $$\frac{dy}{dx}=1+x^2,$$ $$y(0)=0$$
   • $$\frac{dy}{dx}=\frac{1}{1+x^2},$$ $$y\left(\frac{\pi}{4}\right)=1$$
   • $$\frac{dy}{dx}=1+y^2,$$ $$y(0)=0$$