Lab 3: Student Assignment

Week 3, January 25-31

MAT 229, Spring 2021

Special Constants

({{Standard notation, Mathematica notation}, {π≈3.14159, Pi}, {e≈2.71828, E}})

Commands

({{Functionality, Mathematica notation}, {plot the graph of a function, Plot[...]}, {square root of something, (Sqrt[…]), Sqrt[...]}, {absolute value of something, |…|, Abs[...]}, {sine of something (radian mode),sin(…), Sin[...]}, {cosine of something (radian mode),cos(…), Cos[...]}, {tangent of something (radian mode),tan(…), Tan[...]}, {natural logarithm of something, ln(…), Log[...]}, {inverse sine of something, (sin^(-1))(…), ArcSin[...]}, {inverse cosine of something, (cos^(-1))(…), ArcCos[...]}, {inverse tangent of something, (tan^(-1))(…), ArcTan[...]}, {integrate something, ∫(…), Integrate[...]}})

Exercises to submit

Exercise 1

The linear approximation for a function f(x) at x=a is the m x+b that comes from the tangent line y=m x+b to f(x) at x=a. It provides a simple approximation to f(x) for values of x near a.
	f(x)≈m x+b
Let f(x)=(tan^(-1))(4x)-2x+3.

Define this function in Mathematica.

Define x0=1/2 (if you use a semi-colon after your definition, nothing will print -- but the parameter x0 will be defined. Sometimes you don’t want to see so much stuff....).

Determine the linear approximation for f(x) at x=x0.

Define this linear function as g(x) in Mathematica.

Plot both f(x) and g(x) on [-1,2].

What is the absolute value (decimal) of the difference between f(1) and g(1)? In other words, how bad is the approximation at x=1?

What is the absolute value (decimal) of the difference between f(2) and g(2)? In other words, how bad is the approximation at x=2?

Exercise 2

Let f(x)=(sin^(-1))((x/2))+ln((x^2)+1)-2.

Define this function in Mathematica.

Use your function to determine the y-intercept of y=f(x). Get a decimal value.

Get all real-valued critical numbers for f(x) as decimal values. Name them xleft and xright, where xleft<xright.

Evaluate f(x) at your critical numbers to determine the y-values of the critical points.

Evaluate f''(x) at your critical numbers to determine if each is a local maximum or a local minimum.

Local minimum points:

Local maximum points:

Determine the inflection points for this function, and name it xinf.

Make a plot of the graph of f(x) to verify your calculations for the y-intercept, the local max/min points, and inflection points. If you’d defined everything correctly, then you’ll see everything looking beautiful below:

Show[
Plot[f[x],{x,-3,1}],
ListPlot[{{xleft,f[xleft]}, {xright, f[xright]},{xinf,f[xinf]}}]
]

Exercise 3

Let g(x)=(1/(a+(x^2))) and h(x)=((x^2)/(b+x)). Define these in Mathematica. Then set a to 1 and b to 2.

Plot the graphs of g(x) and of h(x)  together on the same axes for various ranges of x-values until you have one that clearly shows the region bounded by these two graphs. Make sure to zoom in on the action!

Get decimal numbers for the x-values of the intersection points. Name them xleft and xright.
	Intersection points: x≈

Modify the “Show” command from Exercise 2, to include the intersection points. You’ll know if you’ve got them right!

Find the area of the region bounded by y=g(x) and y=h(x).
	Area:

Does your area make sense? Can you approximate it, and are you close?

Exercise 4

Let p(x)=ln((3/(1+(x^2))))

Plot this function to see the region bounded by the x-axis and y=p(x).

Get the x-intercepts.
	Intersection point: x≈

Sketch (you could use Mathematica) the solid obtained by rotating the region bounded by y=p(x) and the x-axis about the line y=-1.

Find the volume of this solid:
	Volume: