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

Factor the equation.$$3\sin^2\theta-\sin\theta=0$$ $$\sin\theta(3\sin\theta-1)=0$$So,$$\sin\theta=0$$ or $$\sin\theta=\frac{1}{3}$$Solve $$\sin\theta=0$$.$$\theta=0,\pi$$Solve $$\sin\theta=\frac{1}{3}$$.$$\theta=\sin^{-1}\left(\frac{1}{3}\right)$$ and $$\pi-\sin^{-1}\left(\frac{1}{3}\right)$$Final answer: $$0,\pi,\sin^{-1}\left(\frac{1}{3}\right),\pi-\sin^{-1}\left(\frac{1}{3}\right)$$

Q: 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 from the origin is $$|r|$$.For $$0\le\theta\le\pi$$,$$\cos\theta$$ decreases.So, $$f(\theta)=2+3\cos\theta$$ decreases.Thus, the distance is decreasing on $$0\le\theta\le\pi$$.Final answer: $$\text{B}$$

Q: 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 $$5$$?

Use the geometric formula.$$a_n=a_1r^{n-1}$$Compute the ratio.$$\frac{a_7}{a_3}=r^4=\frac{32000}{2000}=16$$So,$$r^4=16$$$$r=2$$Now find $$a_5$$.$$a_5=a_3r^2$$$$a_5=2000\cdot2^2$$$$a_5=2000\cdot4$$$$a_5=8000$$Final answer: $$8000$$

Q: 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?

Find the derivative.$$R'(t)=0.06t^2-1.2t+4$$Set equal to zero.$$0.06t^2-1.2t+4=0$$Divide by $$0.06$$.$$t^2-20t+66.67=0$$Solve.$$t=\frac{20\pm\sqrt{400-266.67}}{2}$$$$t=\frac{20\pm\sqrt{133.33}}{2}$$$$t\approx\frac{20\pm11.55}{2}$$Critical values:$$t\approx4.22,\;15.78$$Within the interval $$[0,15]$$,$$t\approx4.22$$Final answer: $$4.22$$

Q: 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?

The rate of change is the slope of the tangent line.At $$A$$, there is a relative maximum.So, the slope at $$A$$ is $$0$$At $$D$$, the graph is rising.So, the slope at $$D$$ is positive.Between $$A$$ and $$C$$, the graph is decreasing and concave down.The slopes become more negative as the graph moves toward the inflection point.At the inflection point $$C$$, the slope is the most negative.So, the least rate of change occurs at $$C$$Final answer: $$\text{C}$$

Q: 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 $$x=0$$?

From the graph,$$f(2)=10$$ and $$f(3)=5$$For an exponential decay function, equal increases in $$x$$ multiply the output by a constant ratio.So, the ratio is $$\frac{5}{10}=\frac{1}{2}$$Going backward from $$x=2$$ to $$x=1$$ doubles the output.$$f(1)=20$$Going backward from $$x=1$$ to $$x=0$$ doubles the output again.$$f(0)=40$$Final answer: $$40$$

Q: 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?

The pedal arm has length $$8$$ So the amplitude is $$8$$ At $$t=0$$, the pedal is at height $$12$$ This is the midline. So, the vertical shift is $$12$$ The bicycle makes $$1$$ revolution per second. So, the period is $$1$$ For a sinusoidal model, $$\frac{2\pi}{b}=1$$ So, $$b=2\pi$$ At $$t=0$$, the pedal is on the midline and then moves downward. A matching model is $$h(t)=12-8\sin(2\pi t)$$ Final answer: $$12-8\sin(2\pi t)$$

Q: 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 $$d$$?

The minimum value is $$3$$ The maximum value is $$7$$ The amplitude is $$a=\frac{7-3}{2}=2$$ The midline is $$d=\frac{7+3}{2}=5$$ Final answer: $$a=2,\ d=5$$

Question 1

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

Accepted Answer

Factor the equation.$$3\sin^2	heta-\sin	heta=0$$ $$\sin	heta(3\sin	heta-1)=0$$So,$$\sin	heta=0$$ or $$\sin	heta=\frac{1}{3}$$Solve $$\sin	heta=0$$.$$	heta=0,\pi$$Solve $$\sin	heta=\frac{1}{3}$$.$$	heta=\sin^{-1}\left(\frac{1}{3}ight)$$ and $$\pi-\sin^{-1}\left(\frac{1}{3}ight)$$Final answer: $$0,\pi,\sin^{-1}\left(\frac{1}{3}ight),\pi-\sin^{-1}\left(\frac{1}{3}ight)$$

Question 2

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

Accepted Answer

The distance from the origin is $$|r|$$.For $$0\le	heta\le\pi$$,$$\cos	heta$$ decreases.So, $$f(	heta)=2+3\cos	heta$$ decreases.Thus, the distance is decreasing on $$0\le	heta\le\pi$$.Final answer: $$	ext{B}$$

Question 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 $$5$$?

Accepted Answer

Use the geometric formula.$$a_n=a_1r^{n-1}$$Compute the ratio.$$\frac{a_7}{a_3}=r^4=\frac{32000}{2000}=16$$So,$$r^4=16$$$$r=2$$Now find $$a_5$$.$$a_5=a_3r^2$$$$a_5=2000\cdot2^2$$$$a_5=2000\cdot4$$$$a_5=8000$$Final answer: $$8000$$

Question 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?

Accepted Answer

Find the derivative.$$R'(t)=0.06t^2-1.2t+4$$Set equal to zero.$$0.06t^2-1.2t+4=0$$Divide by $$0.06$$.$$t^2-20t+66.67=0$$Solve.$$t=\frac{20\pm\sqrt{400-266.67}}{2}$$$$t=\frac{20\pm\sqrt{133.33}}{2}$$$$t\approx\frac{20\pm11.55}{2}$$Critical values:$$t\approx4.22,\;15.78$$Within the interval $$[0,15]$$,$$t\approx4.22$$Final answer: $$4.22$$

Question 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?

Accepted Answer

The rate of change is the slope of the tangent line.At $$A$$, there is a relative maximum.So, the slope at $$A$$ is $$0$$At $$D$$, the graph is rising.So, the slope at $$D$$ is positive.Between $$A$$ and $$C$$, the graph is decreasing and concave down.The slopes become more negative as the graph moves toward the inflection point.At the inflection point $$C$$, the slope is the most negative.So, the least rate of change occurs at $$C$$Final answer: $$	ext{C}$$

Question 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 $$x=0$$?

Accepted Answer

From the graph,$$f(2)=10$$ and $$f(3)=5$$For an exponential decay function, equal increases in $$x$$ multiply the output by a constant ratio.So, the ratio is $$\frac{5}{10}=\frac{1}{2}$$Going backward from $$x=2$$ to $$x=1$$ doubles the output.$$f(1)=20$$Going backward from $$x=1$$ to $$x=0$$ doubles the output again.$$f(0)=40$$Final answer: $$40$$

Question 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?

Accepted Answer

The pedal arm has length $$8$$ So the amplitude is $$8$$ At $$t=0$$, the pedal is at height $$12$$ This is the midline. So, the vertical shift is $$12$$ The bicycle makes $$1$$ revolution per second. So, the period is $$1$$ For a sinusoidal model, $$\frac{2\pi}{b}=1$$ So, $$b=2\pi$$ At $$t=0$$, the pedal is on the midline and then moves downward. A matching model is $$h(t)=12-8\sin(2\pi t)$$ Final answer: $$12-8\sin(2\pi t)$$

Question 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 $$d$$?

Accepted Answer

The minimum value is $$3$$ The maximum value is $$7$$ The amplitude is $$a=\frac{7-3}{2}=2$$ The midline is $$d=\frac{7+3}{2}=5$$ Final answer: $$a=2,\ d=5$$

Mini Exam 8

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