Advanced Functions

Terms

Numeric approximation : process of solving a mathematical problem using a numeric algorithm that estimates a solution

• estimate the deriviative of a function at one or more points
• find the roots of a function
• integrate a function

Function handle provides a way to pass a function as a parameter

• special way of making a value that indicates "this is a function & can be used as one"
• uses @

Examples:

sinHandle = @sin;
natLog = @log;
xx = 0.1 : 0.01 : 2*pi;
yy1 = sinHandle(xx);
yy2 = natLog(xx);
plot(xx, yy1, 'b:', xx, yy2, 'r-.')

Anonymous function : a way to define (write) a function handle without defining a separate file for the function

Examples:

quadHandle1 = @(x)x.^2 + 2*x + 3;
quadHandle2 = @(p)(p – 1).^2;
disp(quadHandle1(2) + quadHandle2(-1))

Passing a function to a function

Example:

function result = addNoise(values, noiseFctn)
% Modifies the values by using the given function to add "noise"
result = values + noiseFctn(values);

Using the addNoise function:

xx = -10 : 0.001 : 20;
myFctn = @(x)polyval([0.01, -0.1, -2, 5], x);
yy = myFctn(xx);
noise1 = addNoise(yy, @cos);
noise2 = addNoise(yy, @(x)0.5*sin(5*x));
plot(xx, yy, 'g-', xx, noise1, 'r--', xx, noise2, 'b-.')

Example: estimating the derivative of a function

Recall from calculus:

function dfdx = derivative(fctn, x)
% Estimates the derivative of the given function at the given x value(s)

Using the derivative function:

dyy = derivative(myFctn, xx);
plot(xx, yy, 'g-', xx, dyy, 'k:')

Finding roots using fzero

Find the roots of f(x) = x³ - 3x² + x/2 + 2cos(x)

Define function (in fctn.m)

function val = fctn(x)
val = x.^3 - 3*x.^2 + x/2 + 2*cos(x);
end

Plot

xx = -1 : 0.1 : 3.5;
plot(xx, fctn(xx), 'b', [-1 3.5], [0 0], 'k:', [0 0], [-4 8],  'k:')

Find the roots

root1 =


root2 = 



root3 =

Find where f(x) = 1

Numerical integration with integral

Consider the following intersecting curves

xx = 0 : 0.1 : 5;
yy1 = (xx - 3).^2 + 5;
yy2 = xx.*sin(xx) + 10;
plot(xx, yy1, 'b:',  xx, yy2, 'k')
axis([0 5 0 15])

Area between the curves =

Define function (in f2_minus_f1.m)

function y_diff = f2_minus_f1(x) 
y1 = (x - 3).^2 + 5;
y2 = x.*sin(x) + 10;
y_diff = y2 - y1;

Find where the functions intersect

a =

b =

Numerically integrate to find area between curves

area =

Finding volumes of revolution

General approach: integrate the cross-sectional area of slices perpendicular to the x-axis.

Volume =