Math 1483 - Quiz 1

YOUR NAME HERE

Remember, ALT+7 gives a new text cell, be sure to explain your answers, if needed!

Problem 1

For this problem, we are letting x =(e^0.029), y = (π^0.025), and z = (3^0.026) .  Use these values to answer all parts of this problem.

Part 1

Define x, y, z as the values given in the problem.

Part 2

Use the “NumberForm” command to display each value x, y, and z to 10 places to the right of the decimal.

Part 3

Use inequalities to order the values of x, y, and z from least to greatest. (Even if you know what order they should be in, have Mathematica verify them, that is the point of this problem.)

Part 4

Look up the command “NumberLinePlot” and use it to visually determine the order of three values on a number line.  Be sure to label each point in your command!

Problem 2

Consider the function Q(x) =(x^3)+2(x^2)-12x-2

Part 1

Define Q(x) in Mathematica using proper function notation.

Part 2

Plot Q(x) for -5 <=  x <= 5.

Part 3

Find the x-values for which Q(x) = 0 using the Solve command. Note, since you used the variable x in problem 1, you should clear it before continuing on...

Part 4

Numerically approximate the solutions you found in Part 3 using the NSolve command.

Part 5

Graph both the function Q(x) and the the points where Q(x) =0 on the same graph.  Note:  you will most likely want to use the ListPlot command, and the Show command.  Be sure that your graph displays all the important information.

Problem 3

Consider the function F(x) =((x-1)^(1/3))+(3/((x^2)+1))-1.6

Clear[x]
Clear[y]
Clear[z]

Part 1

Define F(x) in Mathematica using proper function notation.  For the cube root term, look up CubeRoot and use it to help define the first termof the function.

Part 2

Use the Table command to create a list of the function F(x) for the values x=-5, -4, -3,..., 3, 4, 5

Part 3

Graph the function F(x) for -10 < x < 10.

Part 4

Use the NSolve command to find the roots of F(x).  Verify that these roots agree with the graph from the previous part.

Problem 4

Let G(x) = -1+5 x-2 (x^3)+6 (x^4)-3 (x^5)+2 (x^6), and H(x) = -10-7 x+11 (x^2)+14 (x^3)+3 (x^4)-7 (x^5)+3 (x^6).

Clear[x];
Clear[G];
Clear[H];

Part 1

Define the function H(x) and G(x) in Mathematica as given above.

Part 2

Find ALL the values of x for which G(x) = H(x).

Part 3

Determine algebraically the values of x for which G(x) > H(x).  Express your answer in interval notation! (Hint: The Reduce function is your friend)

Part 4

Plot the functions G(x) and H(x) on the same graph, where the graph of  G(x) is magenta in color, and H(x) is cyan in color.  Be sure to plot on a domain which contains all the points of intersection between the two functions.  Also, you will have to set the range so that the intersections are discernable on the graph.

Part  5

Add a plot of the three intersection points of  G(x) and H(x) to the graph from Part 4 to produce a new graph which contains both functions  and now the two points of intersection.  Pick an icon from:

https://reference.wolfram.com/language/guide/ShapesIconsAndRelatedCharacters.html

which is NOT “▽”, and make sure that the points show up on the resulting graph!

First we need to know the actual points to put on the graph.  We know from part 3 that the three real solutions are x=-1, x=1,  and x=3.

Problem 5

Consider the following two lists:

xvals = {2,3,-1,5,6,12,-6,-2,0,1,6,9,10};
yvals = {0,5,2,8,-6,13,-8,0,-2,4,3,7,12};

Part 1

Use the Table command to create a new list xyvals, which is a set of ordered pairs of the form {x,y} where x is from xvals and y is from yvals, in sequence.

Part 2

Look up the TableForm command, and use it to create the following display from the xyvals variable you created in Part 1.  (No cutting and pasting this table into a new cell!)
({{"x", "y"}, {2, 0}, {3, 5}, {-1, 2}, {5, 8}, {6, -6}, {12, 13}, {-6, -8}, {-2, 0}, {0, -2}, {1, 4}, {6, 3}, {9, 7}, {10, 12}})

Part 3

Use the ListPlot command to plot the data points you created in Part 1.  Make sure to pick a nifty plot marker (with a colour of your choice) along with the PlotLegend option to label the data “XY”.

Part 4

Create a new graph of the function y = -0.39617174934225224`+0.8421086268787081` h+0.01101749377758368` (h^2)  and use PlotLegends to label the line as “Curve”.

Part 5

Show the two plots from Parts 3 and 4 on the same graph, make sure all the points and the line are visible on the graph.

Problem 6

Here we will have a function of three variables: K(a,b,c) = (a^2)b+a b c^2 + 2c^3-2b c.

Clear[K]

Part 1

Define the function correctly using function notation.

Part 2

Evaluate K(1,b,1) and K(2,2,c)

Part 3

Solve the equation K(1,1,c) = K(1,3,c)

Part 4

Plot K(1,1,c) and K(1,3,c), be sure to choose a domain and range which clearly illustrates the intersection between the two functions.