Differential Equations — Quiz 1

Questions in this quiz (10)

1. In this problem, $$a(t)$$ denotes the acceleration function, $$v(t)$$ the velocity function, and $$s(t)$$ the position or height function at time $$t$$. If $$a(t)=6t-2$$ and $$v(1)=5$$ then $$v(t)$$ equals:

   • $$3t^2-2t$$
   • $$3t^2-2t+4$$
   • $$3t^2-2t+3$$
   • $$3t^2-2t+5$$

2. A ball is thrown straight up from the top of a tower with an initial velocity of $$32$$ ft/sec and hits the ground $$5$$ seconds later. The height of the tower, in feet, is:

   • $$240$$
   • $$256$$
   • $$280$$
   • $$320$$

3. In this problem, $$a(t)$$ denotes the acceleration function, $$v(t)$$ the velocity function, and $$s(t)$$ the position or height function at time $$t$$. If $$a(t)=12t^2-4$$, $$s(-1)=1$$ and $$s(1)=5$$ then $$v(t)$$ equals:

   • $$4t^3-4t+1$$
   • $$4t^3-4t+2$$
   • $$4t^3-4t+3$$
   • $$4t^3-4t+4$$

4. A ball is thrown straight up from the top of a tower with an initial velocity of $$48$$ ft/sec. The maximum height of the ball is reached after:

   • $$1$$ second
   • $$1.5$$ seconds
   • $$2$$ seconds
   • $$3$$ seconds

5. A ball is thrown toward a wall so that its velocity is $$v(t)=120-30t$$ $$ft/sec$$, where $$t$$ is measured in seconds. If the ball hits the wall in $$2$$ seconds, then the distance from the point where the ball was thrown to the wall is:

   • $$120$$ feet
   • $$140$$ feet
   • $$160$$ feet
   • $$180$$ feet

6. If the velocity of a train moving along a straight track at time $$t$$ is $$v(t)$$, then the total distance traveled by the train between times $$t=a$$ and $$t=b$$ is:

   • $$\int_a^b|v(t)|dt$$
   • $$\int_a^b v(t)dt$$
   • the net displacement of the train from $$t=a$$ to $$t=b$$
   • the change in the train's position from $$t=a$$ to $$t=b$$

7. If a particle moves along the $$x$$-axis with velocity $$v(t)$$ and $$s'(t)=v(t)$$, then it is true that $$\int_a^b v(t)dt$$ gives:

   • the net displacement of the particle from $$t=a$$ to $$t=b$$
   • the total distance traveled by the particle from $$t=a$$ to $$t=b$$
   • $$\int_a^b |v(t)|dt$$
   • the sum of the speeds of the particle from $$t=a$$ to $$t=b$$

8. Find the domain of the particular solution to the differential equation $$\frac{dy}{dx}=\frac{2x}{y}$$ that passes through the point $$(1,2)$$.

   • $$(-\infty , \infty)$$
   • $$0$$ < $$x \le 2$$
   • $$x$$ < $$\sqrt{3}$$
   • $$|x|$$ < $$\sqrt{3}$$

9. If $$\frac{dy}{dx}=\frac{y}{3x}$$ and $$y=2$$ when $$x=1$$, then:

   • $$\ln y=\frac{1}{3}\ln x$$
   • $$\ln y=\frac{1}{3}\ln x+\ln2$$
   • $$y=2x^{1/3}$$
   • $$y=x^{1/3}$$

10. A function $$f(x)$$ that satisfies the equations $$f(x)f'(x)=2x$$ and $$f(0)=2$$ is:

    • $$f(x)=\sqrt{x^2+4}$$
    • $$f(x)=\sqrt{4-x^2}$$
    • $$f(x)=\sqrt{2x^2+4}$$
    • $$f(x)=e^x$$