Differential Equations — Quiz 3

Questions in this quiz (10)

1. If the amount of a certain chemical decreases at a rate proportional to the amount present, then the amount $$A$$ remaining after $$t$$ years, if $$A_0$$ is the initial amount and $$k$$ is a negative constant of proportionality, is given by:

   • $$A=A_0+kt$$
   • $$A=A_0e^{kt}$$
   • $$A=e^{A_0kt}$$
   • $$A=A_0kt$$

2. The slope field is shown on the region:$$[-10,10]\times[-4,4]$$Which differential equation has the slope field shown?

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

3. If a substance decomposes at a rate proportional to the amount of the substance present, and if the amount decreases from $$40$$ g to $$10$$ g in $$2$$ hours, then the constant of proportionality is:

   • $$-\ln2$$
   • $$-\frac{1}{2}$$
   • $$-\frac{1}{4}$$
   • $$\ln\frac{1}{4}$$

4. If $$[f'(x)]^2=f(x)$$ for all real $$x$$ and $$f(0)=0$$, $$f(9)=9$$, then $$f(1)$$ equals:

   • $$\frac{1}{3}$$
   • $$1$$
   • $$3$$
   • $$9$$

5. The solution curve of $$\frac{dy}{dx}=y$$ that passes through the point $$(1,4)$$ is:

   • $$ $$y=4e^x$$
   • $$y=4e^{x-1}$$
   • $$y=e^x+4$$
   • $$y=\frac{e^x}{4}$$

6. The slope field is shown on the region:$$[-10,10]\times[-7,7]$$Which differential equation has the slope field shown?

   • $$\frac{dy}{dx}=x(x+y)$$
   • $$\frac{dy}{dx}=y(x+y)$$
   • $$\frac{dy}{dx}=x(x+1)$$
   • $$\frac{dy}{dx}=x(x^2+y)$$

7. At any point of intersection of a solution curve of the differential equation $$\frac{dy}{dx}=x-y$$and the line $$x-y=0$$ the function $$y$$ at that point:

   • is equal to $$0$$
   • is a local maximum
   • is a local minimum
   • has a point of inflection

8. A bowl of soup at temperature $$200^\circ F$$ is placed in a room at $$70^\circ F$$. The differential equation for the temperature $$y$$ of the soup at time $$t$$ in minutes is$$\frac{dy}{dt}=-0.08(y-70);$$ $$y(0)=200$$After $$10$$ minutes, the temperature in $$^\circ F$$ of the soup is approximately:

   • $$120$$
   • $$129$$
   • $$140$$
   • $$150$$

9. Suppose $$P(t)$$ denotes the size of a fish population at time $$t$$ and its growth is described by the differential equation$$\frac{dP}{dt}=0.003P(800-P)$$If the initial population is $$150$$, then the population is growing fastest:

   • initially
   • when $$P=200$$
   • when $$P=400$$
   • when $$\frac{d^2P}{dt^2}$$ > $$0$$

10. Which of the following statements characterize the logistic growth of a population whose limiting value is $$K$$ and whose initial population is less than $$\frac{K}{2}$$?$$I.$$ The rate of population growth increases during the early stages.$$II.$$ The growth rate reaches its maximum when the population equals $$\frac{K}{2}$$.$$III.$$ The growth rate approaches $$0$$ as the population approaches $$K$$.

    • $$I$$ only
    • $$I$$ and $$II$$ only
    • $$II$$ and $$III$$ only
    • $$I,$$ $$II,$$ and $$III$$