Section II Part B — Full Practice Exam 2

Questions in this quiz (4)

1. Consider the curve defined by $$y=x^4-2x^3$$.a. Find the equation of the tangent line to the curve at $$x=1$$.b. Find the coordinates of the absolute minimum of the curve.c. Find the coordinates of the point(s) of inflection.

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2. Water is being poured into a conical tank with height $$10$$ inches and base radius $$5$$ inches at a rate of $$\frac{dV}{dt}=3$$ in$$^3$$/sec.a. Given that the volume of water in the cone is $$V=\frac{1}{3}\pi r^2h$$, find the rate at which the water level is rising when the water is $$4$$ inches deep.b. Find an expression for the radius $$r$$ of the water’s surface in terms of the height $$h$$.c. How fast is the area of the circular surface of the water changing in in$$^2$$/sec when the water is $$4$$ inches deep?

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3. Let $$R$$ be the region in the first quadrant bounded by $$y=\sqrt{x}$$ and $$y=x^2$$.a. Find the area of region $$R$$.b. Find the volume of the solid generated when $$R$$ is revolved about the $$x$$-axis.c. The cross-section of a solid cut by any plane perpendicular to the $$x$$-axis is a square whose base lies in region $$R$$. Find the volume of the solid.

   • a. $$\frac{1}{3}$$ b. $$\frac{3\pi}{10}$$ c. $$\frac{9}{70}$$
   • a. $$\frac{1}{3}$$ b. $$\frac{3\pi}{10}$$ c. $$\frac{9}{70}$$

4. An object moves along a line with velocity $$v(t)=3t^2-6t-4$$.a. Write a polynomial expression for the position of the particle at time $$t\ge0$$, given that the particle is at the origin when $$t=0$$.b. At what time(s) is the particle changing direction?c. Find the total distance traveled by the particle from $$t=0$$ to $$t=3$$.

   • a. $$s(t)=t^3-3t^2-4t$$ b. $$t=1+\frac{\sqrt{21}}{3}$$ c. $$\frac{28\sqrt{21}}{9}$$
   • a. $$s(t)=t^3-3t^2-4t$$ b. $$t=1+\frac{\sqrt{21}}{3}$$ c. $$\frac{28\sqrt{21}}{9}$$