MATH 021 Lab 1 (Part 2)

Getting Started: Exercises

By Helen P. Read, University of Vermont. Last revised August 2013.

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Show your work (input and output) in this file, and record your answers in the AnswerBoxes provided.

Exercise 1

(a)

Define f(x)=1176+350 x+38 (x^2)+5 (x^3)-(x^4) and plot it for x in the interval [-15,15].

How many x-intercepts does f(x) have?

Find all of the x-intercepts exactly, and then convert the exact values into decimal approximations to 8 significant digits. Use Solve to find the exact values. Then use N[] to find the decimal approximations once you have the exact values.

(b)

The point on the graph of f(x) with the largest y-value is called the absolute maximum. Make another plot of f(x), and adjust the domain of the plot to get a close-up view of the absolute maximum.

(c)

Make a table of ordered pairs (x,f(x)) around the absolute maximum, in increments of (1/10).

As a hypothetical example, if the maximum is located somewhere between x=1 and x=2, you would make a table from x=1 to x=2, in increments of (1/10). (The point of this is to use the table to estimate the location of the absolute maximum.)

Display the table as a Grid, with the x-coordinates given to 2 significant digits, and the y-coordinates to 5 significant digits. That is, use N[ , 2]  and N[ ,5] respectively. Use exact numbers (integers or fractions, NOT decimals) for the start number, end number, and increment when you set up the table (it will not work correctly if you use decimals for any of those numbers).

Use your table to estimate the (x,y) coordinates of the absolute maximum.

Later in the semester we will learn how to find the exact coordinates of maximum and minimum points on a graph.

Exercise 2

(a)

Define a fuction for g(x)=cos(x). Plot g(x) for x in the interval [-2π,2π].

(b)

The following Manipulate panel will allow you to plot y=a cos(x) for various values of a. Evaluate the cell, then experiment with different values by dragging the slider to the right and left and checking/unchecking the negative box.

Manipulate[Plot[a z Cos[x],{x,-2π,2π},PlotLabel->a z Cos[x],PlotRange->{-5,5},Ticks->{Range[-2π,2π,π/2],Automatic}],{{a,1,"size of a"},.05,5,.05,Appearance->"Labeled"},{{z,1,"negative"},{1,-1},Checkbox}]

Explain how multiplying cosx by a constant affects the graph. What happens if the constant is large in size? What if it is small? What if it is negative?

(c)

The following Manipulate panel will allow you to plot y=cos(b x) for various values of b. After evaluating it, use the slider to experiment with different values of b.

Manipulate[Plot[Cos[b x],{x,-2π,2π},PlotLabel->Cos[b x],PlotRange->{-2,2},Ticks->{Range[-2π,2π,π/2],Automatic}],{{b,1},.25,8,.25,Appearance->"Labeled"}]

Explain how multiplying x by a constant affects the graph of the cosine function. What happens if the constant is large? What if it is small?

(d)

Use the following Manipulate panel to plot y=cos(x)+c for various values of c.

Manipulate[Plot[Cos[x]+c,{x,-2π,2π},PlotLabel->Cos[x]+c,PlotRange->{-5,5},Ticks->{Range[-2π,2π,π/2],Automatic}],{{c,0},-6,6,.1,Appearance->"Labeled"}]

How does adding or subtracting a constant affect the graph of the cosine function?

(e)

Define a function h(x)=a cos(b x)+c with specific constants a, b, and c chosen so that h(x) cycles three times as fast as cos(x), has an amplitude that is 6 times as high as cos(x), is flipped upside down compared with cos(x) and shifted 3 units up. Plot h(x) together with g(x)=cos(x), for x in the interval [-2π,2π].

Note: You should have already defined g(x)=cos(x) up in part (a); you do not need to define it over again, but if you quit and come back, make sure you re-evaluate it with Shift-Enter before plotting it together with h(x). (Mathematica never remembers anything if you quit and come back.)

What is the range of the function h(x) that you defined? Give your answer in interval notation.