Section II Part B — Full Practice Exam 1

Questions in this quiz (4)

1. The table gives selected values of $$P(t)$$, the rate at which people are arriving, in people per hour.$$a.$$ Use the data in the table to estimate the rate at which the number of people arriving is changing at $$t=5$$. Show the computation that leads to your answer and indicate units of measure.$$b.$$ Use a midpoint Riemann sum with five intervals of equal size to estimate the total number of people arriving during the $$10$$-hour period.$$c.$$ Suppose the rate, in people per hour, at which people are vaccinated is modeled by the function$$r(t)=350\sin\left(\frac{\pi(t+2)}{12}\right)$$for $$0\leq t\leq10$$. How many people could receive vaccines from $$10$$ a.m. $$\left(t=2\right)$$ to $$6$$ p.m. $$\left(t=10\right)$$?$$d.$$ At what time during the interval $$0\leq t\leq10$$ is the rate $$r(t)$$ greatest? Estimate the maximum number of people vaccinated per hour. Is this rate higher or lower than the number of people arriving at that time?

   • $$a.$$ $$150$$ people per hour per hour$$b.$$ $$8350$$ people$$c.$$ $$\frac{6300}{\pi}\approx2005$$ people$$d.$$ $$t=4,\ 350$$ people per hour, lower than the arrival rate
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2. An object moving along a curve in the $$xy$$-plane is at position $$(x(t),y(t))$$ at time $$t$$, where $$\frac{dx}{dt}=6\cos\left(\frac{\pi t}{4}\right)$$ and $$\frac{dy}{dt}=12\sin\left(\frac{\pi t}{4}\right)$$ for $$t\geq0$$. At time $$t=1$$, the object is at position $$(1,-1)$$.$$(a)$$ Write an equation for the line tangent to the curve at time $$t=1$$.$$(b)$$ Find the acceleration vector and the speed of the object at time $$t=1$$.$$(c)$$ Write an integral expression that can be used to find the total distance traveled by the object over the time interval $$0\leq t\leq2$$.$$(d)$$ Is there a time $$t$$>$$1$$ at which the object is on the $$x$$-axis? Explain your reasoning.

   • $$(a)$$ $$y=2x-3$$$$(b)$$ $$\left\langle-\frac{3\pi\sqrt{2}}{4},\frac{3\pi\sqrt{2}}{2}\right\rangle,\ 3\sqrt{10}$$$$(c)$$ $$\int_{0}^{2}\sqrt{\left(6\cos\left(\frac{\pi t}{4}\right)\right)^2+\left(12\sin\left(\frac{\pi t}{4}\right)\right)^2}dt$$$$(d)$$ Yes
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3. The twice-differentiable function $$f$$ is defined for all real numbers and satisfies $$f(2)=3$$, $$f'(2)=2$$ and $$f''(2)=-1$$.$$(a)$$ The function $$g$$ is given by $$g(x)=f(x)\ln(x)$$. Find $$g'(2)$$ and $$g''(2)$$.$$(b)$$ The function $$h$$ is given by $$h(x)=f(x)\cos(\pi x)$$. Find $$h'(x)$$ and write an equation for the line tangent to the graph of $$h$$ at $$x=2$$.

   • $$(a)$$ $$g'(2)=2\ln(2)+\frac{3}{2}$$$$g''(2)=\frac{5}{4}-\ln(2)$$$$(b)$$ $$h'(x)=f'(x)\cos(\pi x)-\pi f(x)\sin(\pi x)$$Tangent line: $$y=2x-1$$
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4. The derivative of a function $$f$$ is given by $$f'(x)=(x-3)e^{x-2}$$ and $$f(2)=4$$.$$(a)$$ The function has a single critical point. Find this point. Is this a relative maximum, a relative minimum, or neither?$$(b)$$ On what intervals, if any, is the function decreasing and concave down?$$(c)$$ Find the value of $$f(4)$$.

   • $$(a)$$ Critical point at $$x=3$$, relative minimum$$(b)$$ Decreasing and concave down on $$(-\infty,2)$$$$(c)$$ $$f(4)=6$$
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