Lab 2: Student Assignment

Week 2, January 18-24

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[...]}})

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)=(2^x)-2x+3.

Define this function in Mathematica.

Determine the linear approximation for f(x) at x=0.
	m x+b=

What is the absolute value (decimal) of the difference between f(1) and your linear approximation at x=1?

What is the absolute value (decimal) of the difference between f(-1) and your linear approximation at x=-1?

Exercise 2

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

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.

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:

Get all real-valued second-order critical numbers for f(x) as decimal values, i.e. solve f''(x)=0. These are points about which concavity can change, the inflection points.

Inflection points:

Make a plot of the graph of f(x) to verify your calculations for the y-intercept, the local max/min, and the inflection points.

Exercise 3

Let g(x)=(e^(2x)) and h(x)=3(e^x)-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.

Get decimal numbers for the x-values of the intersection points.
	Intersection points: x≈

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

Exercise 4

Let g(x)=(x^3)-6x and h(x)=(x^2)-3x-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 regions bounded by these two graphs.

Get exact numbers for the x-values of the intersection points.
	Intersection points: x≈

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

Exercise 5

Let p(x)=(x^3) and q(x)=-(x^2)+3.

Plot the graphs of g(x) and of h(x)  together on the same axes for various ranges of x-values in the first quadrant until you have one that clearly shows the regions  bounded by these two graphs for x≥0.

Get the x-values of that intersection point.
	Intersection point: x≈

Find the volume for the solid obtained by rotating about the x-axis the region bounded by y=p(x) and y=q(x) for x≥0.
	Volume: