Lab 1: Student Assignment

Week 1, January 11-15

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[...]}})

Exercises to submit

Exercise 1

Plot the graph of ((3(x^3)-5x)/((x^2)+x+1)) for -3≤x≤3.

From the graph estimate the values of x for which this graph has a local maximum point. Try to get 1 decimal place of accuracy.
	Local maximum points: x≈

From the graph estimate the values of x for which this graph has a local minimum point. Try to get 1 decimal place of accuracy.
	Local minimum points: x≈

Exercise 2

Let g(x)=((4x)/(1+(x^2))) and h(x)=x+1.

Plot the graphs of g(x) and of h(x)  together on the same axes for -3≤x≤3.

From your plot estimate the values of x where these two graphs intersect. Try to get 1 decimal place of accuracy.
	Intersection points: x≈

Exercise 3

Let f(x)=sin(x).

Determine an equation for the tangent line to f(x)=sin(x) at x=0, y=m x + b.
	y=

Plot graphs of f(x) and your line m x+b on the same graph for -2≤x≤2.

Exercise 4

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(2x).

Determine the linear approximation for f(x) at x=0.
	tan(2x)≈m x+b=

Using your values for m and b, plot the absolute value of the difference of this function and its linear approximation
	|tan(2x)-(m x+b)|
(Note: Use Abs[...] for the absolute value)
By varying the values of x for which you plot this difference, estimate the x-values for which this difference is no more than 0.25.
	≤x≤

Exercise 5

Consider the function f(x)=((x^((x/5)))/(1+(x^2)))

Plot it for various ranges of x-values until you get one that shows all the important aspects of the graph.

From the graph, what is the domain of this function?
	Domain:

From the graph, what are the intervals of decrease?
	Intervals of decrease:

From the graph, what are the intervals of increase?
	Intervals of increase: