Exercises

Visualization

Generate 100 points using the formula: t↦(cos(2t)cos(t), cos(2t)sin(t)) for t∈[0,2π].

Plot the points.

Use the AspectRatio option to set the same scale for both axes.

By using the options of ListPlot make sure, that there are no axes drawn, the background is black and the colour of the points is white.

Algebra

Partial fraction decomposition

((x+1)/((x^2)(x-1)))

Complex numbers

((15-5i)/(3+4i))=?

Calculate the eigenvalues of

M=(({{a, b}, {-b, a}}))

Let

M(θ)=(({{cos(θ), sin(θ)}, {-sin(θ), cos(θ)}})).

Prove that (((M(θ)))^2)=M(2θ).

Solving equations

Solve the quadratic equation (x^2)-x-1=0.

Find all real solutions of the following cubic equation: (x^3)-3(x^2)+4x-2=0.

Find all real solutions of 2sin(x)=tan(x).

Find the solution of ln(x+2)=tan(x), x∈(-(π/2),(π/2)). Visualize the result by plotting the functions x↦ln(x+2), x↦tan(x) and their intersection.

Calculus

Find the values of α>0 such that α·x>ln(x) for all x>0.

Make an interactive plot of the functions: x↦α x and x↦ln(x) by manipulating the parameter α.

Consider the case when the line α·x is tangent to the graph of ln(x).

At which point do they touch?

For which value of α do they touch?

Fitting

data=Module[{model},
model=NDSolveValue[{x''[t]+0.5x'[t]+2x[t]==0,x[0]==1,x'[0]==0},x[t],{t,0,15}];
Table[{t,model+RandomVariate[NormalDistribution[0,0.05]]},{t,0,15,0.2}]
];

Visualize the given data.

Make an interactive visualization of the data along with following function: t↦A (e^(-(t/τ)))cos(ω t+ϕ), where A, τ, ω>0 and 0≤ϕ<2π are parameters. Then make sure you understand the effects of the parameters, and try to fit the given model to the data by hand.

Using NonlinearModelFit fit the model to the data and visualize the result. Print out the ANOVATable.