Section II Part B — Full Practice Exam 3

Questions in this quiz (4)

1. The number of fish in a lake increases at a rate proportional to the difference between $$800$$ and the current population. There are $$120$$ fish at time $$t=0$$. If $$P(t)$$ is the number of fish at time $$t$$ hours, then$$\frac{dP}{dt}=\frac{1}{8}\left(800-P\right)$$$$\left(a\right)$$ If $$y=P(t)$$ is the particular solution to the differential equation for $$t\geq0,$$ find $$y=P(t).$$$$\left(b\right)$$ Find the value of $$\lim_{t\to\infty}P(t)$$$$\left(c\right)$$ Is the graph of $$y=P(t)$$ concave up?$$\left(d\right)$$ Find $$\frac{d^2P}{dt^2}$$ in terms of $$P.$$ Is the value of $$\frac{d^2P}{dt^2}$$ always positive? Explain your reasoning.

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2. Let $$f$$ be the function given by $$f(x)=\frac{1}{1+3x}$$ and recall that the Maclaurin series for$$\frac{1}{1+x}=1-x+x^2-x^3+x^4-\cdots+\left(-1\right)^nx^n+\cdots$$$$\left(a\right)$$ Write the first four nonzero terms and the general term of the Maclaurin series for $$f.$$$$\left(b\right)$$ Find the radius and interval of convergence for the Maclaurin series for $$f.$$$$\left(c\right)$$ Let $$P_3(x)$$ be the third-degree Taylor polynomial for $$g$$ about $$x=0.$$ If $$g(x)=\int_0^xf(t)\,dt$$ find the value of $$P_3\left(\frac{1}{6}\right).$$$$\left(d\right)$$ Find the exact value of $$g\left(\frac{1}{6}\right).$$$$\left(e\right)$$ Let $$P_3(x)$$ be the third-degree Taylor polynomial for $$g$$ about $$x=0.$$ Use the Lagrange error bound to show that$$\left|P_3\left(\frac{1}{6}\right)-g\left(\frac{1}{6}\right)\right|$$<$$\frac{1}{150}.$$

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3. A continuous function $$g$$ is defined on the closed interval $$-8\le x\le6.$$ The graph of $$g,$$ shown above, consists of three line segments and a quarter of a circle centered at the point $$\left(0,2\right).$$ Let $$f$$ be the function given by $$f(x)=\int_{-8}^{x}g(t)\,dt$$$$\left(a\right)$$ Find all values of $$x$$ in the interval $$-8<x<6$$ at which $$f$$ has a critical point. Classify each critical point as the location of a local minimum, a local maximum, or neither. Justify your answers.$$\left(b\right)$$ Find $$f(0).$$$$\left(c\right)$$ Find $$\lim_{x\to-4}\frac{f(x)}{x^2+4x}$$$$\left(d\right)$$ Let $$h$$ be the function defined by $$h(x)=\frac{g(x)}{x^2+1}.$$ Find $$h'(1).$$

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4. Consider the differential equation:$$\frac{dy}{dx}=\left(y-4\right)\left(x^2+2\right)$$$$\left(a\right)$$ Find $$y=g(x),$$ the particular solution to the given differential equation with initial condition $$g(0)=6.$$$$\left(b\right)$$ For the particular solution $$y=g(x)$$ found in part $$\left(a\right),$$ find $$\lim_{x\to\infty}g(x)$$$$\left(c\right)$$ Let $$y=f(x)$$ be the particular solution to the given differential equation with initial condition $$f(1)=5.$$ Find the value of $$\frac{d^2y}{dx^2}$$ at the point $$\left(1,5\right).$$ Is the graph of $$y=f(x)$$ concave up or concave down at the point $$\left(1,5\right)?$$ Give a reason for your answer.

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