Mini Exam 8 — Timed Mini Exams

Questions in this quiz (10)

1. What are all values of $$\theta$$, for $$0\le\theta$$<$$2\pi$$, where $$3\sin^2\theta=\sin\theta$$?

   • $$0,\pi,\frac{\pi}{6},\frac{5\pi}{6}$$
   • $$0,\pi,\frac{\pi}{2},\frac{3\pi}{2}$$
   • $$0,\pi,\frac{\pi}{6},\frac{5\pi}{6},\frac{3\pi}{2}$$
   • $$0,\pi,\sin^{-1}\left(\frac{1}{3}\right),\pi-\sin^{-1}\left(\frac{1}{3}\right)$$

2. Consider the graph of the polar function $$r=f(\theta)$$, where $$f(\theta)=2+3\cos\theta$$ for $$0\le\theta\le2\pi$$. Which statement is true about the distance from the origin?

   • The distance is increasing for $$0\le\theta\le\pi$$ because $$f(\theta)$$ is increasing.
   • The distance is decreasing for $$0\le\theta\le\pi$$ because $$f(\theta)$$ is decreasing.
   • The distance is increasing for $$\pi\le\theta\le2\pi$$ because $$f(\theta)$$ is negative.
   • The distance remains constant for all $$\theta$$.

3. In a simulation, a population follows a geometric sequence. The population on day $$3$$ is $$2000$$ and on day $$7$$ is $$32000$$. What is the population on day $$5$$?

   • $$8000$$
   • $$10000$$
   • $$16000$$
   • $$12000$$

4. The rate of cars passing a checkpoint is modeled by $$R(t)=0.02t^3-0.6t^2+4t+2$$ for $$0\le t\le15$$. At what value of $$t$$ does the rate change from increasing to decreasing?

   • $$2.11$$
   • $$4.22$$
   • $$7.9$$
   • $$10.0$$

5. The figure shows the graph of a function $$g$$ in the $$xy$$-plane with four labeled points. It is known that a relative maximum of $$g$$ occurs at $$A$$, and the only point of inflection of the graph of $$g$$ is $$C$$. Of the following points, at which is the rate of change of $$g$$ the least?

   • $$A$$
   • $$B$$
   • $$C$$
   • $$D$$

6. The figure shows the graph of an exponential decay function $$f$$. The coordinates of two of the points are labeled. If $$y=f(x)$$, what is the $$y$$-coordinate of the point on the graph where $$x=0$$?

   • $$40$$
   • $$30$$
   • $$20$$
   • $$15$$

7. A physical therapy center has a bicycle that patients use for exercise. The height, in inches, of the bicycle pedal above level ground periodically increases and decreases when used. The figure gives the position of the pedal $$P$$ at a height of $$12$$ inches above the ground at time $$t=0$$ seconds. The pedal's $$8$$-inch arm defines the circular motion of the pedal. If a patient pedals $$1$$ revolution per second, which of the following could be an expression for $$h(t)$$, the height, in inches, of the bicycle pedal above level ground at time $$t$$ seconds?

   • $$8-12\sin t$$
   • $$12-8\sin t$$
   • $$8-12\sin(2\pi t)$$
   • $$12-8\sin(2\pi t)$$

8. The function $$f$$ is defined by $$f(x)=a\cos(b(x+c))+d$$, where $$a,b,c,d$$ are constants. The points $$(1,3)$$ and $$(3,7)$$ represent a minimum and a maximum value, respectively, on the graph of $$f$$. What are the values of $$a$$ and $$d$$?

   • $$a=2,\ d=5$$
   • $$a=4,\ d=5$$
   • $$a=2,\ d=4$$
   • $$a=4,\ d=3$$

9. The function $$S$$ is given by $$S(t)=\frac{600{,}000}{1+0.5e^{-kt}}$$, where $$k$$ is a constant. If $$S(2)=300{,}000$$, what is the value of $$S(10)$$?

   • $$450{,}000$$
   • $$400{,}000$$
   • $$300{,}000$$
   • $$35{,}294.12$$

10. Two function models $$k$$ and $$m$$ represent sales (in thousands) after $$t$$ weeks, for $$t\ge2$$. If $$k(t)=20-3\ln t$$ and $$m(t)=-t+20$$, at what value of $$t$$ will the logarithmic model first exceed the linear model by $$0.2$$ thousand units?

    • $$3.912$$
    • $$4.876$$
    • $$5.067$$
    • $$6.021$$