Univariate Calculus (part 2)

Problem 1

Suppose s(x) and c(x) are functions for which s'(x)=c(x), c'(x)=s(x), s(0)=0, and c(0)=1.  Find a formula for ((s(x))^2)-((c(x))^2).

Problem 2

Perform the Calculus-based graphing technique on f(x)=(Sqrt[9-(x^2)])-(3/4)x. This includes finding the intervals on which the function is increasing, decreasing, concave up, concave down, classifying critical points, and finding inflection points.

Problem 3

Perform the Calculus-based graphing technique on f(x)=((x^3)/((x^2)-12)), finding asymptotes and including them the graph.

Problem 4

In the graphs below represent the graphs of y=f(x), y=f'(x), y=f''(x), and y=(∫_(3/2)^x)f(t)dt for some function f(x).  Identify which is which, explaining your answer.

f[x_]:= (x^2)Sin[Pi x]/2;
p4b=Plot[ f[x], {x,0,4}, PlotLabel-> "Graph B"];
p4d=Plot[Evaluate[ Integrate[ f[t], {t,3/2,x}]], {x,0,4}, PlotLabel-> "Graph D"];
p4a=Plot[ f'[x], {x,0,4}, PlotLabel->"Graph A"];
p4c=Plot[ f''[x], {x,0,4}, PlotLabel-> "Graph C"];
GraphicsGrid[ {{p4a,p4b},{p4c,p4d}}]

Problem 5

A cylinder with an open top and closed bottom is to have a fixed volume of 27 cubic inches.  What is the smallest surface area of such a cylinder?

Problem 6

You have to design a window which is a rectangle topped by a semi-circle.  If the perimeter of the window needs to be 40 feet, what is the width of the window which yields the maximum area?

Show[ Graphics[ {Circle[ {0,0}, 1, {0, Pi}], Line[ {{-1,0}, {-1,-2},{1,-2}, {1,0}}]}]]

Problem 7

Sand is falling on to a conical pile at the rate of 2 cubic meters per second, and the pile maintains the shape of a cone whose height is the same as the radius of the base.  How fast is the height of the pile increasing when it is 6 meters tall?

Problem 8

Two bicyclists are approaching an intersection, one from the north and one from the west.  At a particular instant the bicyclist coming from north is 200 feet from the intersection and approaching it at 15 feet per second; the cyclist coming from the west is 150 feet from the intersection and approaching it at 20 feet per second.  At that instant in time, how fast is the distance between the two cyclists changing?

Show[Graphics[ {Line[ {{6,0},{0,0},{0,6}} ], PointSize[Medium], Thick, Line[{{0,5},{3,0}}], Point[{{0,5},{3,0}}], Style[Text[θ(t), {.44,3.6}], FontSize-> 14],Style[Text[y(t), {-.5,5}], FontSize-> 14],Style[Text[x(t), {3,-.5}], FontSize-> 14],Style[Text[3, {2,2.5}], FontSize-> 14], Arrow[{{0,5},{0,3}}], Arrow[ {{3,0}, {5,0}}]}], ImageSize-> 250]

Problem 9

One of the graphs below represents the solution to y'=-x|y|, y(0)=1. Which one is it and why?

Quiet[p7d=Plot[Evaluate[DSolveValue[{ y'[x]==-x RealAbs[y[x]],y[1]==1}, y[x],x]], {x,-2,2}, PlotLabel-> "Graph D", PlotRange-> 4];
p7a=Plot[Evaluate[DSolveValue[{ y'[x]==x RealAbs[y[x]],y[1]==1}, y[x],x]], {x,-2,2}, PlotLabel-> "Graph A", PlotRange-> 4];
p7b=Plot[Evaluate[DSolveValue[{ y'[x]==y[x] RealAbs[x],y[1]==1}, y[x],x]], {x,-2,2}, PlotLabel-> "Graph B", PlotRange-> 4];
p7c=Plot[Evaluate[DSolveValue[{ y'[x]==-y[x] RealAbs[x],y[1]==1}, y[x],x]], {x,-2,2}, PlotLabel-> "Graph C", PlotRange-> 4];
GraphicsGrid[ {{p7a,p7b},{p7c,p7d}}]]

Problem 10

Find the area of the region bounded by the curves y=(x^2) and y=7-6x.  Find the net/signed area under y=((sin(x))/((x^2)+1)) from x=-π to x=π.

Problem 11

Let R be the region bounded by y=1 and y=6x-(x^2)-4.  Find the volume of solid (Subscript[S, 1]) you get by rotating R about the x-axis.  Find the volume of the solid you get by rotating R about the y-axis.

Problem 12

Find (∫_0^(π/2))((sin(x))/(((1+cos(x)))^2))dx.   Find (∫_0^∞)x (e^(-2x))dx.

Applications of the Derivative, Integration Basics

Derivatives and graphs (critical numbers, increasing/decreasing, concavity, maxima/minima)

Max/Min problems

Related rate problems

Basic definite integrals

Integrals and area

Volumes of revolution

Integration by substitution

Integration by parts

Improper integrals