MATLABFunctionTutorial.pdf

MATLAB for Function Tutorial in Module 8.2

Introduction to Computational Science: Modeling and Simulation for the Sciences

Angela B. Shiflet and George W. ShifletWofford College

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Download the associated MATLAB file FunctionTutorial.m .

Introduction

In this chapter, we deal with models that are driven by the data. In such a situation, we

have data measurements and wish to obtain a function that roughly goes through a plot of

the data points capturing the trend of the data, or fitting the data . Subsequently, we can

use the function to find estimates at places where data does not exist or to perform further

computations. Moreover, determination of an appropriate fitting function can sometimes

deepen our understanding of the reasons for the pattern of the data.

In this module, we consider several important functions, some which we have

already used. By being familiar with basic functions and function transformations, the

modeler can sometimes more readily fit a function to the data.

Linear Function

The concept of a linear function was essential in our discussions of the derivative and

simulation techniques, such as Euler's Method. Here, we review some of the

characteristics of functions whose graphs are lines.

The command in Quick Review Question 1 plots the graph of the linear function y

= 2 t + 1. This line has y -intercept 1, because y = 1 when t = 0. Thus, the graph crosses

MATLAB Function Tutorial

the y -axis when t = 0. With data measurements where t represents time, the y -intercept

indicates the initial data value. The slope of this particular line is 2, which is the

coefficient of t . Consequently, when we go over 1 unit to the right, the graph rises by 2

units.

Definitions

A linear function , whose graph is a straight line, has the following

form:

y = mx + b

The y -intercept , which is b , is the value of y when x = 0, or the place where the line

crosses the y -axis. The slope , m , is the change in y over the change in x . Thus, if

the line goes through points ( x 1 , y 1 ) and ( x 2 , y 2 ), the slope is as follows:

m =

Δ x = y 2 − y 1

x 2 − x 1

Quick Review Question 1

a. Execute the following MATLAB command, which plots the above function,

f ( t ) = 2 t + 1, from t = -3 to 3:

t = -3:0.1:3;

plot(t, 2*t + 1)

b. By replacing xxxxxx with the appropriate equation, complete the command to

plot f along with the equation of the line with the same slope as f but with y -

intercept 3. Distinguish between the graphs of f and the new function, such as

by color, line thickness, or dashing.

plot(t, 2*t + 1, t, xxxxxx, '--')

Δ y

MATLAB Function Tutorial

c. Copy the command from Part b, and change the second function to have a y -

intercept of -3.

d. Describe the effect that changing the y -intercept has on the graph of the line.

e. Copy the command from Part b, and change the second function to have the

same y -intercept as f but slope 3.

f. Copy the command from Part b, and change the second function to have the

same y -intercept as f but slope -3.

g. Describe the effect that changing the slope has on the graph of the line.

Quadratic Function

In Module 2.3 on "Rate of Change" and Module 2.4 on "Fundamental Concepts of

Integral Calculus," we considered a ball thrown upward off a bridge 11 m high with an

initial velocity of 15 m/sec. The function of height of the ball with respect to time is the

following quadratic function:

s ( t ) = -4.9 t 2 + 15 t + 11

The general form of a quadratic function is as follows:

f ( x ) = a 2 x 2 + a 1 x + a 0

where a 2 , a 1 , and a 0 are real numbers. The graph of the ball's height s ( t ) in Figure 2.3.1 of

the "Rate of Change" module is a parabola that is concave down. The next two Quick

Review Questions develop some of the characteristics of quadratic functions.

Definition A quadratic function has the following form:

f ( x ) = a 2 x 2 + a 1 x + a 0

where a 2 , a 1 , and a 0 are real numbers. Its graph is a parabola .

MATLAB Function Tutorial

Quick Review Question 2

a. Execute the following MATLAB command, which plots the above function,

s ( t ) = -4.9 t 2 + 15 t + 11, from t = -1 to 4 (NOTE that we use .^ for

exponentiation of the sequence for t .):

t = -2:0.1:4;

plot(t, -4.9*t.^2 + 15*t + 11)

b. Give the command to plot s ( t ) and another function with the same shape that

crosses the y -axis at 2. Have the graph of the new function be dashed.

c. Using calculus, determine the time t at which the ball reaches its highest point.

Verify your answer by referring to the graph.

d. What effect does changing the sign of the coefficient of t 2 have on the graph?

Quick Review Question 3 In this question, we consider various transformations on a

function. Use .^ for exponentiation of the sequence for t .

a. Plot t 2 , t 2 + 3, and t 2 - 3 on the same graph, using dashing for the second curve

and dotting for the third.

b. Describe the effect of adding a positive number to a function.

c. Describe the effect of subtracting a positive number from a function.

d. Plot t 2 , ( t + 3) 2 , and ( t - 3) 2 on the same graph, using dashing for the second

curve and dotting for the third.

e. Describe the effect of adding a positive number to the independent variable in

a function.

f. Describe the effect of subtracting a positive number from the independent

variable in a function.

g. Plot t 2 and - t 2 on the same graph, using dashing for the second curve.

MATLAB Function Tutorial

h. Describe the effect of multiplying a function by -1.

i. Plot t 2 , 5 t 2 , and 0.2 t 2 on the same graph, using dashing for the second curve

and dotting for the third.

j. Describe the effect of multiplying the function by number greater than 1.

k. Describe the effect of multiplying the function by positive number less than 1.

Polynomial Function

Linear and quadratic functions are polynomial functions of degree 1 and 2, respectively.

The general form of a polynomial function of degree n is as follows:

f ( x ) = a n x n + … + a 1 x + a 0

where a n , …, a 1 , and a 0 are real numbers and n is a nonnegative integer. The graph of

such a function with degree greater than 1 consists of alternating hills and valleys. The

quadratic function of degree 2 has one hill or valley. In general, a polynomial of degree n

has at most n - 1 hills and valleys.

Definition A polynomial function of degree n has the following form:

f ( x ) = a n x n + … + a 1 x + a 0

where a n , …, a 1 , and a 0 are real numbers and n is a nonnegative integer.

Quick Review Question 4

a. Execute the following MATLAB command that plots the polynomial function

p ( t ) = t 3 - 4 t 2 - t + 4 from t = -2 to 5 with .^ for exponentiation of the sequence

for t :

t = -2:0.1:5;

plot(t, t.^3 - 4*t.^2 - t + 4)

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