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Q1:(3) Define g(z) - Inr+i0 <0<T. where z=rei and r>0 and (i) Sketch the domain D of the function g(z) and show that g(z) is analytic on this domain and find g'(z). (ii) Show that the composite function G(z) = g(2²+1) is analytic on the half-plane H = {z=z+iy : x>0}, with derivative 2z 2² +1 G(z)= (4) (i) Define what it means for a subset of C to be open. (ii) Define what it means for a subset of C to be a domain. (iii) Suppose f(2)= u(x, y) +iu(x, y), z=1+iy, x,y ER, for z in a domain D. Prove that if f is analytic on this domain D then f'(2)=0 for all z in D, and hence f is constant on D.See Answer
Q2: Solve the IVP. 4 y^{\prime \prime}+12 y^{\prime}+9 y=0 \quad y(0)=1 \quad y^{\prime}(0)=2See Answer
Q3: \text { What is the Fourier transform of the function } 4 \cos (5 t) \text { on the interval }[-\pi, \pi] ? \text { a. } \cos (2 t)+\cos (4 t) b. none of the others c. 2 cos (t) +7 sin(2t) d. 4 cos(5t) 7 sin(2t) + 4 cos(5t)e.See Answer
Q4: \text { or the continuous odd function } f(t) \text { if } \int_{0}^{a} f(t) d t=4, a>0 \text { , then } \int_{-a}^{a} f(t) d t=? a. 4 b. all of the others c. 2 d. 8 e. OSee Answer
Q5: \text { What is the Fourier transform of the function } 2 \sin (3 t) \text { on the interval }[-\pi, \pi] \text { ? } \text { a. } \sin (t)+\sin (2 t) \text { b. } 3 \sin (2 t)+4 \sin (3 t) \text { c. } 2 \sin (3 t) \text { d. } 4 \sin (t)+2 \sin (2 t) e. none of the othersSee Answer
Q6: Using (13) sketch or graph a figure (similar to Fig. 291 in Sec. 12.3) of the deflection u(x, t) of a vibrating string(length L = 1, ends fixed, c = 1) starting with initial velocity 0 and initial deflection (k small, say, k = 0.01). \text { 7. } f(x)=k \sin 2 \pi x See Answer
Q7: Find all trigonometric functions of the acute angle 0 in the right triangle below: See Answer
Q8: \text { What is the fundamental period of the function } f(t)=\cos (3 t) ? \text { a. } \frac{2 \pi}{3} b. none of the others \text { c. } \frac{\pi}{2} \text { d. } 4 \pi \text { e. } 8 \piSee Answer
Q9: . Consider the eigenvalue problem y^{\prime \prime}+\lambda y=0 ; \quad y^{\prime}(0)=0, \quad y(\pi)=0 .See Answer
Q10: A nonlinear dynamic system is described by the following three state variable equations: \dot{q}_{1}=q_{2}-q_{3} \dot{q}_{2}=u+q_{2}+\frac{1}{2}\left|q_{1}\right| q_{1} \dot{q}_{3}=3 q_{1}+6 q_{3} Assume that 73 = 2. What is the value of 71?See Answer
Q11: Here are yesterday's high temperatures (in Fahrenheit) in 11 U.S. cities. 47, 49, 62, 65, 68, 68, 71, 72, 75, 78, 82 Notice that the temperatures are ordered from least to greatest.Give the five-number summary and the interquartile range for the data set. Five-number summary Minimum:_____ Lower quartile: _______ Median:________ Upper quartile: ______ Maximum:_______ Interquartile range:_______ See Answer
Q12: 5. Use appropriate tests to determine the convergence or divergence of the above series \text { (a) } \sum_{n=1}^{\infty} \frac{n^{2}+2 n+3}{4 n^{5}+n^{2}+3} \text { (b) } \sum^{\infty} \frac{\sqrt{n+c}}{\sqrt{n^{3}}+1} \text { where } c>3 \text { is a constant } \text { (c) } \sum_{n=1}^{\infty} \frac{n^{2}+3 n+1}{3 n^{2}+4 n-1} \text { (d) } \sum_{n=1}^{\infty} \frac{4+2 \sin (n)}{3 n^{c}+4} \text { where } c>1 \text { is a constant } \text { (e) } \sum^{\infty} \frac{4+(-1)^{n}}{3 n^{c}+4} \text { where } 0<c<1 \text { is a constant } \text { (f) } \sum_{n=2}^{\infty} \frac{1}{n \ln n} \text { (g) } \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{c}} \text { where } c>1 \text { is a constant } \text { (h) } \sum_{n=1}^{\infty} \frac{(-1)^{n} n^{2}}{3^{n}+2} \text { (i) } \sum_{n=1}^{\infty} \frac{c^{n}}{n^{4}+3} \text { where } c>1 \text { is a constant } \text { (j) } \sum_{n=1}^{\infty} \frac{n^{2}+2}{(-1)^{n} \cdot n !} (k) \sum_{n=1}^{\infty} \frac{n !}{n^{2}} \text { (1) } \sum_{n=1}^{\infty}\left(-\frac{1}{3}+\frac{1}{n}\right)^{n} \text { (m) } \sum_{n=1}^{\infty}\left(c+\frac{1}{n}\right)^{n} \text { where } c>1 \text { is a constant }See Answer
Q13: Given following differential equation: ÿ + 10y + 16y = 0 It has initial conditions y(0) = 1 and y(0) = 2 Solve it by the trial solution method. True or false, the solution is: y(t)=-\frac{2}{3} e^{-2 t}+\frac{5}{3} e^{-8 t}See Answer
Q14: . We want to solve the partial differential equation (PDE) \frac{\partial^{2} U(x, y)}{\partial x^{2}}+\frac{\partial^{2} U(x, y)}{\partial y^{2}}=3 x+2 y subject to Dirichlet boundary conditions with g1(x, y) = Y, 92(x, y) = e®+y, and g3(x, y)figure below; h = k = 1/4).= x (see a. Discretise the partial differential equation using a centred difference scheme with error O(h2)in each dimension. Explain what a stencil is and present it for the given problem.[8] b. Use the stencil derived in (a) to present the system of equations to solve. Find the solutions for the problem at P1, P2 and P3.[9] c. Explain how to compute the numerical solution if the condition on g1 (left hand side of the triangle) changes to a Neumann boundary condition.[8]See Answer
Q15: O Make a histogram of 7500 integer values you generated. Use bins 10 units wide.Describe the shape of this distribution and calculate the sample mean and SD from the data you generated. Are these values similar to the theoretical values you found in (3)?See Answer
Q16: Lula is trying to find and classify the discontinuities of some functions, but she has some misconceptions.In this problem, your job is to explain what’s wrong with Lula’s reasoning. "As x → 3, the denominator x – 3 tends to 0; there's nothing that I can cancel in the numerator and denominator, so g must have a vertical asymptote at x = 3." - What is incorrect about Lula's reasoning? Is Lula's conclusion that g has a vertical asymptote at x = 3 correct? Give a concrete example to show Lula that her reasoning is incorrect. (In other words, give a different function g for which the same reasoning would lead Lula to the wrong conclusion.) next, lula looks at g(x) cos/x-3 she recognizes that there is a discontinuity at x=3 and she saysSee Answer
Q17: The average value of the function f(x) = x5 on the interval [ 0'4] is equal to 1/4 f40x5dxSee Answer
Q18: State the Central Limit Theorem and explain what it means in your own words.See Answer
Q19: What is the volume of the solid obtained by revolving about the x-axis the region bounded by the lines y = 4, x = 0, x = 3, and the x-axis? А. 24л В. 12л С. 48л D. 96лSee Answer
Q20: Use =RANDBETWEEN(1, 100) to create 250 samples of size n = 30 each by choosing generating random integer (whole) numbers between 1 and 100.See Answer

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