MAT331,

Feb 2nd, 2010

How to remove output.

Submit homework (remember to remove output and run again the worksheet to make sure everything works. Then remove it again before saving.) Do not add commets when submitting.

Functions and Composition. Expressions

f:=x->sin(x)+sin(x^2);

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f(Pi);

LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJEkjUGlHRiUiIiM=

f(y+17);

LCYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmSSJ5R0YoIiIiIiM8RixGLC1GJDYjKiRGKiIiI0Ys

g:=x->x^(1/2);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JCMiIiIiIiNGJUYlRiU=

(g @ f)(y);

KiQsJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ5R0YpIiIiLUYlNiMqJEYrIiIjRiwjRixGMA==

Exercise: Find all the values of x which make x^4-4 equal to 2.

Add all these values.

mySol:=[solve(x^4-4=2,x)];

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Find a decimal approximation of each the values of x which make x^4-4 equal to 2

evalf(mySol);

NyYkIishZSUzbDohIipeI0YjJCErIWUlM2w6RiVeI0Yn

evalf(mySol[3]);

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Add all the above approximations.

sum(mySol[i] ,i=1..4 );

IiIh

Find the solutions of the system of equations 3x+y=24, 2x-y=9. Illustrate

solve([ 3*x+y=24, 2*x-y=9],{x,y });

PCQvSSJ4RzYiIyIjTCIiJi9JInlHRiUjIiNARig=

line1:=solve( 3*x+y=24,y); line2:=solve( 2*x-y=9,y);

LCZJInhHNiIhIiQiI0MiIiI=

LCZJInhHNiIiIiMhIioiIiI=

plot([line1, line2], x=-2..10, color=[blue, yellow],thickness=3, scaling=constrained);

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

Storing and expressing answers

Find the sum of the first ten positive integers. Simplify your answer.

sum(mat331, mat331 =1..10);

IiNi

Find the sum of the first n positive integers. Simplify your answer.

simplify(sum(mat331, mat331 =1..n));

LCYqJEkibkc2IiIiIyMiIiJGJkYkRic=

Find the sum of the first n squares positive integers. Simplify your answer.

simplify(sum(mat331^2, mat331 =1..n));

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JSFH

Consider the expression 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

a. Rewrite this expression as a function of the variable x

b. Rewrite this expression as a function of the variables x and a.

c. Evaluate the expression for x=3.

d. Evaluate the expression for x=z^{1/2}

e. Assign this expression to the variable VAR and take the square root of VAR.

h:=x->25*x^3*ln(a*x)/ln(10);

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h(y);

LCQqKEkieUc2IiIiJC1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJkkiYUdGJSIiIkYkRi9GLy1GKDYjIiM1ISIiIiNE

k:=(x,a)->25*x^3*ln(a*x)/ln(10);

Zio2JEkieEc2IkkiYUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKDkkIiIkLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomOSUiIiJGLEY2RjYtRi82IyIjNSEiIiIjREYlRiVGJQ==

k(1,4);

LCQqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiMiIiItRiU2IyIjNSEiIiIjXQ==

h(3);

LCQqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkiYUdGKSIiJCIiIi1GJTYjIiM1ISIiIiR2Jw==

k(3,a);

LCQqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkiYUdGKSIiJCIiIi1GJTYjIiM1ISIiIiR2Jw==

h(z^(1/2));

LCQqKEkiekc2IiMiIiQiIiMtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJImFHRiUiIiJGJCNGMUYoRjEtRio2IyIjNSEiIiIjRA==

h(sqrt(z));

LCQqKEkiekc2IiMiIiQiIiMtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJImFHRiUiIiJGJCNGMUYoRjEtRio2IyIjNSEiIiIjRA==

VAR:=h(x);

LCQqKEkieEc2IiIiJC1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJkkiYUdGJSIiIkYkRi9GLy1GKDYjIiM1ISIiIiNE

sqrt(VAR);

LCQqJCooSSJ4RzYiIiIkLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IyomSSJhR0YmIiIiRiVGMEYwLUYpNiMiIzUhIiIjRjAiIiMiIiY=

Lists and sets.

myList:=[1,24,5,1];

NyYiIiIiI0MiIiZGIw==

myList[2];

IiND

mySet:={1,24,5,1};

PCUiIiIiIiYiI0M=

Plot the graph of your favorite polynomial of degree 6. How do you think Maple "decides" the vertical and horizontal scales?

plot(x^6+3*x,thickness=3);

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Plot options

Parametric plots

plot([t^2*cos(t),sin(t),t=0..100 ], thickness=3, color=blue );

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Plot a set of points. Exercise: Define a sequence of points of the form (i,i^2+3) , for i between -3 and 10 and plot these points.

JSFH

3D plots

Problems 4 and 5.

?prime

The function g is defined as follows:

g:=k->sum(ithprime(ii) , ii=1..k);

Zio2I0kia0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQtSSlpdGhwcmltZUdGKzYjSSNpaUdGJS9GMjsiIiI5JEYlRiVGJQ==

g(3);

IiM1

for i from 1 to 10 do g(i); od;

IiIj

IiIm

IiM1

IiM8

IiNH

IiNU

IiNl

IiN4

IiQrIg==

IiRIIg==

We will solve by trial and error.

evalb(g(10000)> 100000);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

Since 10000 gives me a value of g larger than the desired value 100000, I'll with a smaller input

evalb(g(1000)> 100000);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

evalb(g(100)> 100000);

SSZmYWxzZUclKnByb3RlY3RlZEc=

From the two lines above we know that the value we are looking for is between 100 and 1000

evalb(g(500)> 100000);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

Thereofre, he value we are looking for is between 100 and 500

If a curve fits, wear it (Chapter 2 of the online book)

EXERCISE: Suppose we measure the height of a baby, and we get the following results:

When the baby is 1 day old, the height is 20.1 in.

When the baby is 15 days old, the height is 20.6 in.

When the baby is 20 days old, the height is 21.2 in.

When the baby is 29 days old, the height is 21.9 in.

JSFH

1. Make a graph of the points determined by the above data

2. What is the height of the baby at 25 days old?

3. Find a polynomial passing through all the points.

4. Find the line passing through all the points.

EXERCISE: Given the points, [1,-1/2], [2,1], [2.4,1] and [3,1/3] find a piecewise linear function with domain the interval [1,3] passing through all of them.

EXERCISE: Is it possible to find a linear function passing through [1,-1/2], [2,1] and [3,1/3]? Why?

EXERCISE: Find the linear function passing closest to the points [1,-1/2], [2,1] and [3,1/3]

EXERCISE: Find the quadratic function passing closest to the points [1,-1/2], [2,1] and [3,1/3]

EXERCISE: Find the quadratic function passing closest to the points [1,2], [2.4,1.5] , [3,1] and [-0.4,3]

EXERCISE: Is it possible to find a linear function passing through the following points? Justify your answer.

EXERCISE: Find the linear function passing closest to the points [1,-1/2], [2,1], [2.4,1] and [3,1/3]

JSFH

pts:=[ [1,-1/2], [2,1], [2.4,1] , [3,1/3] ];

QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi

Note: The graph below works only if all the vertical blue segments have length larger than 0.

L:=x->x+3;

display(plot(pts, x = 0 .. 5, style = point, symbol = box, color = green), plot([L(x)], x = 0 .. 5, color = red), seq(plot([pts[i][1], t, t = pts[i][2] .. L(pts[i][1])], thickness = 3, color = blue), i = 1 .. 3));

EXERCISE: Consider the points

newpts := [seq([i, 1.5*i+1], i = -1 .. 5), [8, 1.5*8+.5], seq([i, 1.5*i+1], i = 9 .. 11)];

a. Find a polynomial passing through those points.

b. Fit the points with a line using the least squares method.

c. Fit the points with a quadratic function using the least squares method.

d. Find a piecewise linear function passing through all the points.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=