Differential Equations Exercise 1 : Classification — Linear vs Nonlinear Which of the following are linear first- and second-order DEs with constant coefficients? $y’ + x^2 y = 2x$ $5y’’ - 2y’ - 4x = 3y$ $y^4 + 2y’’ + 3y’ = 0$ $\sin x, y’’ - y = 0$ $y’’ - x^5 = 2$ $2y’’ - y’ + \frac{3}{2}y = 0$ Exercise 2 : Homogeneity and Order Which of the following are homogeneous and non-homogeneous DEs and what is the order in each case? ...
Differential Calculus Exercise 1 : Sequences and Limits of Sequences Calculate the limiting value of the following sequences for $n \to \infty$. $a_n = \frac{\sqrt{n}}{n}$ $a_n = \frac{5 + n}{2n}$ $a_n = \left(-\frac{1}{4}\right)^n - 1$ $a_n = \frac{2}{n} + 1$ $a_n = \frac{n^3 + 1}{2n^3 + n^2 + n}$ $a_n = 2 + 2^{-n}$ $a_n = \frac{n^2 - 1}{(n + 1)^2} + 5$ Exercise 2 : Limits of Functions Calculate the following limits: ...
Applications of Integration Exercise 1 : Area Bounded by Parabola Calculate the area bounded by the positive branch of the parabola $y^2 = 25x$, the $x$-axis and the ordinates where $x = 0$ and $x = 36$. Exercise 2 : Area Between a Symmetric Curve and the x-axis Calculate the area bounded by the positive branch of the curve $y^2 = (7 - x)(5 + x)$, the $x$-axis and the ordinates where $x = -5$ and $x = 1$. ...
Theory of Errors Exercise 1 : Types of Errors In the following examples, state whether the error is systematic (constant $S$) or random $R$. A 100-metre race is held in a school during a sports day. The judges start their stopwatches when the sound of the starting pistol reaches them. What type of error arises in this case? The timing of the start and end of the 100-metre race is subject to individual fluctuations, e.g. reaction time. What kind of error is it? The zero point of a voltmeter has been wrongly set. The measurements are therefore subject to an error. What kind of error is it? The resistance of a copper coil is obtained by measuring the current flowing through when a voltage is applied to it. As the coil warms up the resistance increases. What kind of error will ensue? Exercise 2 : Sample Statistics Nine different rock samples are taken from a crater on the moon whose densities are determined with the following results: $3.6, 3.3, 3.2, 3.0, 3.2, 3.1, 3.0, 3.1, 3.3 \frac{g}{cm^3}$. Calculate the mean value and standard deviation of the parent population. The velocity of a body travelling along a straight line is measured 10 times. The results are $1.30, 1.27, 1.25, 1.26, 1.29, 1.31, 1.23, 1.33, 1.24 \frac{m}{s}$. Calculate the mean value and the standard deviation. Exercise 3 : Uniform Distribution Moments A continuous random variable has the following density function: ...
Probability Distributions Exercise 1 : Random Variable Two dice are thrown. Calculate the mean value of the random variable ‘sum of the number of spots’. Exercise 2 : Linear Continuous Distribution A random variable has the probability distribution: $$ f(x) = \begin{cases} \frac{x}{2}, & 0 \leq x \leq 2 \\ 0, & \text{otherwise} \end{cases} $$ Compute the mean value of the random variable. Exercise 3 : Binomial Probability 60% of the students who start to study for an engineering degree complete their studies and obtain a degree. What is the probability that in a group of 10 arbitrarily chosen students in the first term of study 8 will obtain a degree? ...
Probability Calculus Exercise 1 : Menu Selection A menu contains five dishes from which two can be chosen freely. Give the sample space. Exercise 2 : Card Probability A set of cards consists of 16 red and 16 black cards. What is the probability of drawing a black card out of the pile? Exercise 3 : Die Divisibility What is the probability that when tossing a die the number that appears will be divisible by 3? ...
Fourier Integrals and Fourier Transforms Exercise 1 : Fourier Series For the following periodic function: $$ f(t) = \begin{cases} -1 & \text{for } -\frac{T}{2} \leq t < 0 \\ 1 & \text{for } 0 < t \leq \frac{T}{2} \end{cases} $$ Calculate the Fourier series for $$ f(t) = \begin{cases} 0 & \text{for } -\frac{T}{2} \leq t < -\frac{t_0}{2} \\ -1 & \text{for } -\frac{t_0}{2} \leq t < 0 \\ 1 & \text{for } 0 < t \leq \frac{t_0}{2} \\ 0 & \text{for } \frac{t_0}{2} < t \leq \frac{T}{2} \end{cases} $$Exercise 2 : Limiting Spectrum (T \to \infty) Now perform the limiting process for $T \to \infty$ and obtain the amplitude spectrum. ...
Fourier Series; Harmonic Analysis Exercise 1 : Symmetric rectangular pulse Obtain the Fourier series of the 2\pi-periodic function $$ f(x)=\begin{cases} 0, & -\pi\le x< -\tfrac{\pi}{2},\\ 1, & -\tfrac{\pi}{2}\le x< \tfrac{\pi}{2},\\ 0, & \tfrac{\pi}{2}\le x\le \pi, \end{cases} $$State whether the periodic extension is even or odd, compute the nonzero Fourier coefficients, and write the series in sine/cosine form. Exercise 2 : Odd square wave For the 2\pi-periodic function $$ f(x)=\begin{cases} 1, & -\pi\le x<0,\\ -1, & 0\le x\le\pi, \end{cases} $$Compute the Fourier series, indicate whether only sine terms appear, and derive the explicit formula for the coefficients. ...
Integral Calculus Exercise 1 : Primitives Find the primitives of the following functions and the value of the constant: $f(x) = 3x$ given $F(1) = 2$ $f(x) = 2x + 3$ given $F(1) = 0$ Exercise 2 : Definite Integrals (Cosine) Evaluate the following definite integrals: $\int_{0}^{\pi/2} 3 \cos x , dx$ $\int_{-\pi/2}^{\pi/2} 3 \cos x , dx$ $\int_{0}^{\pi} 3 \cos x , dx$ Exercise 3 : Absolute Areas Obtain the absolute values of the areas corresponding to the following integrals: ...
Theory of Vector Fields Exercise 1 : Basic Operations on Vector Fields Given the vector fields: $\mathbf{F}_1 = x^2 \hat{\mathbf{z}}$ $\mathbf{F}_2 = x \hat{\mathbf{x}} + y \hat{\mathbf{y}} + z \hat{\mathbf{z}}$ $\mathbf{F}_3 = yz\hat{x} + zx\hat{y} + xy\hat{z}$ Calculate $\nabla \cdot \mathbf{F}_i$ and $\nabla \times \mathbf{F}_i$ for $i=1,2,3$. Which fields are conservative? Find scalar potentials where possible. Which fields are solenoidal? Find vector potentials where possible. Show that $\mathbf{F}_3$ can be expressed both as a gradient and a curl. Exercise 2 : Helmholtz Theorem Implications For a vector field $\mathbf{F}$ in 3D, prove the following implications: ...