Introduction to Programming in MATLAB Assignment

1. Revisiting MATLAB matrices and built-in functions.

a) Factor the following polynomials using MATLAB:

i. x² + 18x + 81

ii. x² + 11x + 24

iii. x² - 10x + 9

iv. x³ - 4x² + 4x - 16

v. 2x² - 16x - 40

b) Multiply the following vectors using "dot" multiplication. What does this type of operation do in contrast to traditional matrix multiplication?

i. [5,6,9]' with [10,10,10]'

ii. [1,1,1]' with [2,5,1]'

iii.

2. Plotting in MATLAB.

a) Consider the following function:

y = f (x) = sin(x) + x² + 6²

Create vectors from -10 to 10 with the following step-sizes:

i. 5

ii. 1

iii. 0.1

iv. 0.001

v. 0.00001

plot the results on the same graph. Add axis labels, title and legend.

b) Create multiple figures, plotting all data from part (a). Create new titles, as necessary. Change the line styles to the following:

i. red, black, green, cyan, magenta - grid on

ii. black diamonds, red circles, blue x's, dashed magenta, dotted green - grid off

iii. black asterisks, red squares, dotted red, blue dashes, magenta squares - grid on

c) Consider the following functions:

Y₁ = 6sin x_i + cos 2x₁

y₂ = 12sin 2x₁ + cos 4x₁

define x₁ from 0 to 2π with the following number of elements:

i. 10 elements

ii. 100 elements

iii. 1000 elements

iv. 10000 elements

for each (i) - (iv) above, evaluate y₁ and y₂. Create a single figure window showing the progression of resolution as elements are added to x₁. Each plot should have a legend, appropriate title and axis label and include the results from both y₁ and y₂. (hint: use subplots).

d) Consider the following two functions:

f₁ (x) = 3x³ + 3x + 87

f₂ (x) = (x/2) + 3 + √64x

evaluate these functions from x = 0 to 100 in a step size that gives acceptable resolution (use your judgment!). Plot the results on the same figure. What is one problem with the figure generated? Find another MATLAB function that will fix this plotting issue and regenerate your plot. Add appropriate axis labels, legends, titles, etc.

e) Consider the data set given by the x and y vectors below:

y= [3, 15, 26, 35, 41, 52, 60, 77, 84, 90, 95, 100]

x= [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60]

Plot this data using a semi-log scale, creating a plot that has a natural logararithmic y-axis. (hint: semilogy()).

3. Working with MATLAB function files.

a) Write a function file, that accepts values of x as inputs and outputs the value of y when the input is passed through the following function:

y = e^√56x+3.21 + 4x + 0.2x²

evaluate the function at a range of x values from 0 to 1 with 143 elements.

b) Consider the functions below, both of which are dependent on a single input, x. Create a function file that will return the value of both calculations when provided a value (or series of values) of x. Evaluate your functions for x values of 1,2,3 and 4 as well as the vector from 0 to 10 with 1000 elements. Plot your results, adding figure labeling, legends, etc. as necessary. You should present two plots, each with two lines.

Y₁ = x + 5

Y₂ = e^3x + sin (x)

c) Modify your function from 3(b) so that, after evaluating the value of yi for a given value of x, an intermediate calculation is performed which divides x by 10 before it is used in evaluating y2. Evaluate your function at the same points as 3(b) and compare the results.

d) Create a function file to evaluate the value of sin x accurate to 9 Taylor series terms.

Recall, the Taylor series for sin x is given by:

sin x ≈ x - x³/3! + x⁵/5! - x⁷/7! + x⁹/9! - x¹¹/11! + x¹³/13! - x¹⁵/15! + x¹⁷/17!

* NOTE: There are multiple ways to do this problem. Think it through before you code!

Evaluate your function at various points on the domain [0, 2π]. (at least 3)

e) Create a function file to evaluate the multivariable function f(x,y,z) below.

f (x, y, z) = (x²y/X² +y²) + Z³x

evaluate the function at the following points: (1,1,1), (1,0,1), (e³,1,5).

f) Consider an object, whose position along the x-axis is given by:

x = -½g t²

create a function that has time as an input and returns the position (along x) of the object, the x-axis velocity at time "t" and the acceleration of the object. (hint: think about how many outputs the function must have and how to relate the velocity and acceleration to the given function). Plot the above quantities from t = 0 to 10 with 100 elements. You should use subplot where, from top to bottom the plots should be in the order: position, velocity, acceleration. All plots should be labeled appropriately.

Create anonymous functions in your script file to define and evaluate the following functions:

i. f (x) = x² + 4x + 6 for x = 1,5,10

ii. f (x, y) = xy + x² y - y² - 1) for (1,4), (-1,-4), (4,1)

iii. f (x, y) = sin x + cos 2y for (0,0), (√2/2, √2/2)

iv. f (x, y) = e^xy + In 4x - log₁₀ 5xy for (1,1), (6,4), (2,10)