Section II Part a — Full Practice Exam 1

Questions in this quiz (2)

1. Let $$R$$ be the region bounded by the graphs of $$y=\sin\left(\frac{\pi x}{2}\right)$$ and $$y=x^2-2$$. Find: $$a.$$ the area of $$R$$ $$b.$$ the value of $$k$$ if the line $$y=k$$ splits $$R$$ into two regions of equal area $$c.$$ the volume of the solid if each cross section perpendicular to the $$x$$-axis is a square $$d.$$ an integral expression for the volume of water in the lake if the depth is given by $$h(x)=12-x^2$$

   • $$a.$$ $$4.016$$$$b.$$ $$k\approx-0.605$$$$c.$$ $$7.708$$$$d.$$ $$\int_{-1}^{a}(12-x^2)\left[\sin\left(\frac{\pi x}{2}\right)-x^2+2\right]dx$$

2. Consider the logistic differential equation $$\frac{dy}{dt}=\frac{y}{5}(15-y)$$ Let $$y=f(t)$$ be the particular solution to the differentialequation with $$f(0)=5$$. $$a.$$ A slope field for this differential equation is givenbelow. Sketch possible solution curves through the points $$(0,5)$$ and$$(2,10)$$. $$b.$$ Use Euler's method, starting at $$t=0$$ with two steps ofequal size, to approximate $$f(1)$$. $$c.$$ Write the third-degree Taylor polynomial for $$f$$ about$$t=0$$, and use it to approximate $$f(1)$$. $$d.$$ Find the particular solution $$y=f(t)$$ with $$f(0)=5$$.

   • $$a.$$ The solution curves through $$(0,5)$$ and $$(2,10)$$ bothincrease and approach $$y=15$$$$b.$$ $$f(1)\approx15$$$$c.$$ $$P_3(t)=5+10t+5t^2-5t^3$$ and$$f(1)\approx15$$$$d.$$ $$f(t)=\frac{15e^{3t}}{2+e^{3t}}$$