Section II Part B — Full Practice Exam 2

Questions in this quiz (3)

1. The velocity of a cyclist moving along a straight road is given by the function $$v(t)$$, measured in meters per second. Selected values of $$v(t)$$ are shown in the table below. The cyclist starts from rest at time $$t=0$$. For $$0$$<$$t\le10$$, $$v(t)$$ is differentiable and increasing. (a) Use the data in the table to estimate the cyclist's acceleration at time $$t=3$$ seconds. Show the computations that lead to your answer. Indicate units of measure.(b) Use a right Riemann sum with the four subintervals indicated by the data in the table to approximate $$\int_0^{10} v(t) dt$$. Indicate units of measure. Is this approximation an overestimate or underestimate?(c) Another cyclist starts 3 meters ahead of the first cyclist at time $$t=0$$. The second cyclist's velocity is given by $$w(t)=\frac{120}{t+2}-30e^{-t}$$. Write but do not evaluate an expression involving an integral that gives the second cyclist's distance from the starting line at time $$t=10$$ seconds.(d) Write an expression for the second cyclist's acceleration in terms of $$t$$

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2. In a wildlife reserve, the population of deer increases over time. At time $$t=0$$, where $$t$$ is measured in years, the population is found to be 30 deer. (a) One ecologist suggests a population model $$P$$ that satisfies the differential equation $$\frac{dP}{dt}=\frac{1}{5}(300-P).$$ Use separation of variables to solve this differential equation for $$P$$ with the initial condition $$P(0)=30$$.(b) A second ecologist suggests a population model $$Q$$ that satisfies $$\frac{dQ}{dt}=\frac{1}{600}Q(300-Q).$$ Find the value of $$\frac{dQ}{dt}$$ at the time when $$Q$$ grows most rapidly.(c) For the population model $$Q$$ introduced in part (b), use Euler's method, starting at $$t=0$$ with two steps of equal size, to approximate $$Q(8)$$. Show the computations that lead to your answer.

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3. The Maclaurin series for a function $$f$$ is given by$$\frac{x}{2}+\frac{x^2}{3}+\frac{x^3}{4}+...+\frac{x^n}{n+1}+...$$ (a) Use the ratio test to find the interval of convergence of the Maclaurin series for $$f$$.(b) Let $$g$$ be the function given by $$g(x)=f(3x)$$. Find the first three terms and the general term of the Maclaurin series for $$g$$.(c) The first two terms of the Maclaurin series for $$f$$ are used to approximate $$f(0.2)$$. Given that $$|f^{''}(x)|\le3$$ for $$0\lex\le0.2$$, use the Lagrange error bound to show that this approximation differs from $$f(0.2)$$ at most $$\frac{1}{500}$$.

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