Section II Part A — Full Practice Exam 5

Questions in this quiz (2)

1. Examine the function $$f$$ defined as $$f(x)=\frac{x+\sqrt{x}}{4},\ 0\le x\le 8$$. a) Use a Riemann sum with four equal subintervals evaluated at the midpoint to approximate the area under $$f$$ from $$x=0$$ to $$x=8$$. b) Again using four equal subintervals, use the trapezoidal rule to approximate the area under $$f$$ from $$x=0$$ to $$x=8$$. c) Using your result from part (b), approximate the average value of the function $$f$$ from $$x=0$$ to $$x=8$$. d) Determine the actual average value of the function $$f$$ from $$x=0$$ to $$x=8$$.

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2. Let $$v(t)$$ be the velocity, in meters per second, of a car at time $$t$$ seconds, $$t\ge0$$. At time $$t=0$$, the car is traveling at $$30$$ meters per second. The driver applies the brakes so that the car's velocity satisfies the differential equation $$\frac{dv}{dt}=-\frac{1}{10}t-3$$ a) Find an expression for $$v$$ in terms of $$t$$, where $$t$$ is measured in seconds. b) How long does it take for the car to come to a complete stop? c) Write an equation for the tangent line to the velocity curve at $$t=5$$ seconds. d) Find the total distance traveled from $$t=0$$ until the car stops.

   • a) $$v(t)=-\frac{1}{20}t^2-3t+30$$ b) $$t=-30+10\sqrt{15}\approx8.73$$ c) $$v-13.75=-\frac{7}{2}(t-5)$$ d) $$\approx144.1$$