Lab 4: Student Assignment

Week 4, February 1-7

MAT 229, Spring 2021

Exercises to submit

These must be done by hand on paper (or typed into Mathematica). Use Mathematica to check your work. Submit as pdf files in Canvas.

Exercise 1

Let f(x)=(x^(3x)). Its domain is (0,∞).

f[x_]:=x^(3x)
GraphicsGrid[{{
Plot[f[x],{x,0,1.5}],
Plot[f[x],{x,0.3,0.5}]
}}]

End behavior:

Is (lim_(x->(SuperPlus[0])))f(x) a determinate limit or indeterminate limit? If it is determinate, give its value. If it is indeterminate, write it in a form on which you can use L’Hospital’s rule and evaluate it.

f[x_]:=E^(3 Log[x]/(1/x))
GraphicsGrid[{{
Plot[f[x],{x,0,1.5}],
Plot[f[x],{x,0.3,0.5}]
}}]
Simplify[3/x/(-1/x^2)]
(* So the function tends to E^0=1 as x -> 0 *)

Is (lim_(x->∞))f(x) a determinate limit or indeterminate limit? If it is determinate, give its value. If it is indeterminate, write it in a form on which you can use L’Hospital’s rule and evaluate it.

It’s determinate -- (∞^∞) is infinite....

Find any critical numbers. Determine if each produces a maximum point, a minimum point, or neither.

f'[x]
Solve[f'[x]==0,x]

Sketch the graph that shows the end behavior and the critical points.

Show[
Plot[f[x],{x,0,1.5}],
ListPlot[{{1/E,f[1/E]}}]
]

Exercise 2

Consider the indefinite integral ∫((x+3)/(Sqrt[(x^2)+6x +7]))d x.

Find an appropriate substitution.

f[x_]:=x^2+6x +7
f'[x] (* The derivative is twice that numerator we see in the integrand *)
Integrate[1/2 f'[x]/Sqrt[f[x]],x]
Integrate[1/2 /Sqrt[u],u]

Using your substitution, evaluate the integral.

Exercise 3

Consider the indefinite integral ∫(((((x^2)-10))^2)/x)d x.

Rewrite it in a form that you can use the power/log rule.

integrand[x_]:=(x^2-10)^2/x
Expand[integrand[x]]

Using your rewrite, evaluate the integral.

Integrate[Expand[integrand[x]],x]

Exercise 4

Let g(x)=cos(x)(e^(2sin(x))). This is a periodic function, and we want to find the area of one of the bounded regions above the x-axis and y=g(x).

You might notice, first of all, that this rather strange looking function is periodic of at most period 2 π. I’m generally inclined to graph a function right away (kind of cheating!), but trying to imagine what it looks like first:

g[x_]:=Cos[x] E^(2 Sin[x])
Plot[g[x],{x,-Pi,Pi}]

Find the two x-intercepts closes to the origin.

g[x] is only zero when cos(x) is zero -- so at Pi/2 and -Pi/2

Set up the integral that gives the area and find an appropriate substitution.

Evaluate the integral to determine the area.

Integrate[g[x],{x,-Pi/2,Pi/2}]
N[%] (* says give a decimal value to the preceding command. *)

Exercise 5

Consider the region in the first quadrant bounded by y=(3/(Sqrt[4+(x^2)])) and y=1.

y[x_]:=3/Sqrt[4+x^2]
Plot[{y[x],1},{x,0,3},PlotRange->{0,2.0},Filling->1]

Provide a rough sketch from Mathematica.

(* Thanks to Matt Longwell *)
xleft=0
xright=Sqrt[5]
z[x_]:=1
RevolutionPlot3D[{{y[x]},{z[x]}},{x,xleft,xright},RevolutionAxis->{1,0,0},PlotLabel->"Looks like some kind of aircraft turbine"]

Find any intersection points for these two curves.

Solve[y[x]==1,x]

The solid of revolution obtained by rotating the bounded region between these two curves has cross sections that are washers. Set up the integral that gives the volume.

Integrate[Pi(y[x]^2-1^2),{x,xleft,xright}]

Evaluate that integral to determine the solid’s volume.

NIntegrate[Pi(y[x]^2-1^2),{x,xleft,xright}]