Section II Part B — Full Practice Exam 1

Questions in this quiz (4)

1. The differential equation $$\frac{dy}{dx}=\frac{y+2}{x-1}$$ is used in parts $$a$$ through $$c$$.$$a.$$ On the coordinate axes provided, sketch the slope field for the differential equation at the indicated points.$$b.$$ Identify the family of curves that represent solutions to this differential equation. Justify your answer mathematically.$$c.$$ Given that the solution passes through the point $$(3,1)$$, find the particular solution.

   • Check Answer Explanation
   • Check Answer Explanation

2. Consider the curve defined implicitly by $$x^{2}+y^{3}=4x.$$$$(a)$$ Find $$\frac{dy}{dx}.$$$$(b)$$ Determine the equation(s) of any horizontal tangent lines, if they exist.$$(c)$$ Determine the equation(s) of any vertical tangent lines, if they exist.$$(d)$$ The line $$y=\frac{4}{3}x-\frac{2}{3}$$ is tangent to the curve. Find the point of tangency $$(x,y).$$

   • Check Answer Explanation
   • Check Answer Explanation

3. Let $$R$$ represent the first-quadrant region bounded by the $$y$$-axis and the curves $$y=2^x$$ and $$y=8\cos\frac{\pi x}{6},$$ as shown in the graph.$$a.$$ Find the area of region $$R.$$$$b.$$ Set up, but do not evaluate, integrals in terms of a single variable for:$$(i)$$ the volume of the solid formed when $$R$$ is rotated around the $$x$$-axis$$(ii)$$ the volume of the solid whose base is $$R,$$ if all cross sections in planes perpendicular to the $$x$$-axis are squares

   • Check Answer Explanation
   • Check Answer Explanation

4. Given the function $$f(x)=e^{2x}(x^2-2):$$$$(a)$$ For what values of $$x$$ is $$f$$ decreasing?$$(b)$$ Does this decreasing arc reach a local or a global minimum? Justify your answer.$$(c)$$ Does $$f$$ have a global maximum? Justify your answer.