Educational Materials

Introduction to Mathematica for High School Math (for Students and Teachers) #9

Ruth Dover

Author

Ruth Dover

Title

Introduction to Mathematica for High School Math (for Students and Teachers) #9

Description

Introduction to Mathematica for High School Math

Tutorial 9: Solving

R. Dover, IMSA

Basics

Solve

is a good place to start. Check out the following. Note the double

sign:

Solve[

-x-60,x]

Solve[

-x+60,x]

Solve[2

Sin[x]

-Sin[x]-10,x]

(Impressive, huh?):

Solve[

+xy+3

4,x]

Edit the preceding line to solve for

rather than

Another Curiosity

Try these literal equations:

Solve[a

+bx+c0,x]

This is about as we would expect, but we must also remember the restrictions.

Reduce

can help this issue, but the solution does become more complicated. The

is the logical "and," while

refers to the logical "or."

In[]:=

Reduce[a

+bx+c0,x]

Some Work… and Some Do Not

Solve[

-6x+20,x]

Interesting. Though it is a cubic equation, there were not nice roots. Basically, the Wolfram Language™ is telling you that you really do not want to see the exact solutions. It gives approximations and notation to show you that it changed things behind the scenes. Hovering over the outputs shows that it is the solution to that cubic. Oh…

Instead, try

NSolve

NSolve[

-6x+20,x]

NSolve[

+3x6,x]

Try some others.

But this will not take care of everything:

NSolve[x-Cos[x]0,x]

Another Method

Solve and NSolve work on algebraic equations. The command

FindRoot

works on other stuff. It is a variation of Newton's method, and it requires an initial input that is fairly close to the actual root being sought. Consider the equation

x=cos(x)

. Plot the two functions defined by the two sides of the equality to see what is reasonable:

Plot[{x,Cos[x]},{x,-2,2}]

Given this graph, start with an initial guess of

x=1

FindRoot[x==Cos[x],{x,1}]

As another example, consider the cubic

f(x)=

−7

and let

g(x)=x+2sin(3x−1)

. First, we will define the functions and plot them together to get a clue as to where they intersect:

f[x_]:=

-7;g[x_]:=x+2Sin[3x-1]

Plot[{f[x],g[x]},{x,-5.4,2}]

There is a point of intersection near

x=−5

, so we will enter that first:

FindRoot[f[x]g[x],{x,-5}]

Find the other two solutions (use Insert ▶ Input from Above and then edit the initial guess in the command):

Clear[f,g]

Checking

First, we will define a function

f[x_]:=

+12

-305

-1056x+3220

We will call our solution set

solns

solns=Solve[f[x]0,x]

Check these values by replacing them into the function:

f[x]/.solns

This shows one of the reasons that Mathematica® presents the solution set in this format: to make substitution quite easy. Then clear both

and

solns

Clear[f,solns]

Systems

Here, we will solve a system of two equations,

y=2x–3

and

-5x+2xy+

. First, we will graph the system to check for approximate solutions. We will use a new method to do this. We will create and name each plot separately (but withhold the actual plot) and then use

Show

to put them together:

gr1=Plot[2x-3,{x,-2,4}];

gr2=ContourPlot[

-5x+2xy+

6,{x,-2,5},{y,-5,5}];

Show[gr1,gr2]

To solve, we need a list of equations and a list of variables for which we wish to solve:

NSolve[{

-5x+2xy+

6,y2x-3},{x,y}]

Exercises

Use NSolve with the function

k(x)=

−6

+3x+8

. When you are done, clear your function.

Next, we will review several commands with one extended exercise.

Consider two polynomials:

f(x)=

−6

+2x+5

and

g(x)=

−3x+1

. First, define both functions.

Now, plot both functions together, finding an appropriate domain. Add options to distinguish the graphs more clearly.

Now try to find the intersections of the equations using NSolve.

Substitute this list into either

to find the corresponding

values. Do the ordered pairs seem to agree with your graphical approximations?

Cite this as: Ruth Dover, "Introduction to Mathematica for High School Math (for Students and Teachers) #9" from the Notebook Archive (2021), https://notebookarchive.org/2021-09-6h2b887