A Fast Guide to Maple and Partial Derivatives \302\251 Mike May, S.J., Saint Louis University, 2006 - maymk@slu.edu restart;

To use Maple to find partial derivatives we first need to be able to define functions. We start with the simple function f taking (x,y) to x^2 + y^3. This can be done in a number of ways. 1) Use the insert menu to insert an execution group after the cursor (command-J), insert Maple input (command-M) getting a vertical cursor and red type, and define the function with the syntax "f := (x,y) -> x^2+y^3;" We then evaluate the function for a particular (x,y). f := (x,y) -> x^2+y^3-x*y;

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plot(f(x,x^2),x=-2..2);

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h := x^2+y^3-x*y;

LCgqJEkieEc2IiIiIyIiIiokSSJ5R0YlIiIkRicqJkYkRidGKUYnISIi

eval(h,x=2); subs(x=2,h);

LCgiIiUiIiIqJEkieUc2IiIiJEYkRiYhIiM=

LCgiIiUiIiIqJEkieUc2IiIiJEYkRiYhIiM=

h(1,2);

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f(2,z);

LCgiIiUiIiIqJEkiekc2IiIiJEYkRiYhIiM=

2) Use the same method in 2-D math mode. (On a new line insert 2-D Math, command-R.) Note that Maple turns the dash-greater than combination into an arrow. QyQ+SSJnRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIjIiIiKiQ5JSIiJEYxRiVGJUYlRjE=

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print(); # input placeholder

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g(t,t);

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JSFH 3) Use the 2-D math mode and the Expression palette to the left. Click on the expression f:=(a,b)->z, then navigate through the entries with tabs to fill in the expression. 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print(); # input placeholder

Error, invalid parameters for inline function 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print(); # input placeholder

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QyQtSSJoRzYiNiRJImtHRiVJIm1HRiUiIiI=

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print(); # input placeholder

LCYqJEkia0c2IiIiIyIiIiokSSJtR0YlIiIkRic=

For the exercises, instead of defining a function named f, you will define a function named func, and you will use func throughout the exercises. Exercise: 1) Using each of the three methods described above, define the function func(x,y)=sin(x^2+y^3) and evaluate at (x,y)=(sqrt(Pi)/4,0). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

When we did cross sections of a graph before, we connected a function of two variables with a family of functions in one variables. This can be done by creating new functions. It can also be done by using a constant to fill in one of the variables. f := (x,y) -> x^2+y^3-x*y; x0 := 1: y0 := 2: f1 := x -> f(x,y0); f2 := y -> f(x0,y); plot([f1(t), f2(t)], t=-2..2, color=[red, green], legend = ["f1(t)", "f2(t)"]);

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

We can also plot a family of slices f := (x,y) -> x^2+y^3-x*y; plot([f(x,-1), f(x,0.5), f(x,1)],x=-3..3, color=[red,green,blue], legend=["f(x,-1)", "f(x,0.5)", "f(x,1)"]);

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Exercise: 2) Plot the slice of the function func(x,y), which you defined in exercise 1 above, with y =2. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The partial derivative is computed by taking the derivative of a function of two variables, treating one of the variables as a constant. In Maple, this is done with the diff command. It should be noted that the diff command has an "inert" version, Diff. Applying both commands we see that while Maple uses the same command for the derivative and the partial derivative, it uses a different symbol for the partial derivative. f := (x,y) -> x^2+y^3-x*y; f(x,y); Dfx := diff(f(x,y),x); Df1x := diff(f(x,y0),x); Df1xa := diff(f1(x),x); Diff(f(x,y),x); Diff(f1(x),x); Diff(f(x,y0),x);

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Diff(f(x,y),x)=diff(f(x,y),x);

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We can do the same computations in 2D math mode or by using the Expression palette. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= QyQ+SSREZnlHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlRi4iIiI=

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print(); # input placeholder

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Exercise: 4) Evaluate both func and its partial derivative with respect to y at (2,3). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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