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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: 2. Let varphi be the following sentence in a first-order language with one one-place function symbol f: \varphi=((\forall x \exists y(x=f y)) \wedge(\forall x \forall y(f x=f y \rightarrow x=y))) (i) Give a short explanation of what it means for a structure to satisfy the sentence varphi above. (ii) Show that there are types of countable structures which satisfy p.uncountably many different isomorphism (iii) For each n > 0, let psin be the sentence \forall x(\neg(x=\underbrace{f \cdots f}_{n \text { times }} x)) Show that there are only countably many different isomorphism types of countable structures which satisfy both varphi and psin for all n > 0.See Answer
Q3: Show how to use the "slide method" to determine the GCF and LCM of 3250 and 4000.See Answer
Q4: Solve the IVP. 4 y^{\prime \prime}+12 y^{\prime}+9 y=0 \quad y(0)=1 \quad y^{\prime}(0)=2See Answer
Q5: (a) Determine which of the following maps are linear (giving reasons for your answers). f: R_{3}-\rightarrow R_{2} \quad\left[x_{1, X 2, X 3}\right\} 7-\rightarrow[x 2+1, x 3-1\} f: \mathrm{R} 3-\rightarrow \mathrm{R} \quad[x 1, X 2, X 3\} 7-\rightarrow X 1 \times 2 \times 3 f: \mathrm{R} 3-\rightarrow \mathrm{R} 3 \quad[X 1, X 2, X 3\} 7-\rightarrow[X 1+X 2+2 X 3,0, X 2\}See Answer
Q6: \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
Q7: \text { The function } \sin (2 x) \text { is an even function on the interval }[-\pi, \pi] \text { . } True FalseSee Answer
Q8: For which value of a > 0 the functions f(t) =t and f(t) 1 areorthogonalon the interval [-a, a] ? a. 1 b. 3 c. all of the others d. 4 e. 2See Answer
Q9: a) Confirm that the function f(t) = 3t is an odd fuction (show all your work). b) Find the Fourier sine series of the function f(t) = 3ton the interval [-7, 7].See Answer
Q10: \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
Q11: 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
Q12: The functions f(t) = 1 and f(t)t are orthogonal on the interval [-1,1]. O True O FalseSee Answer
Q13: \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
Q14: Find all trigonometric functions of the acute angle 0 in the right triangle below: See Answer
Q15: \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
Q16: . Consider the eigenvalue problem y^{\prime \prime}+\lambda y=0 ; \quad y^{\prime}(0)=0, \quad y(\pi)=0 .See Answer
Q17: O Find the values of x for which the following functions have stationary values and using your results, sketch a graph of each function.bed tomoidelngog oIT y=e^{2 x}-2 x y=4 e^{x}-2 x-3See Answer
Q18: 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
Q19: 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
Q20: 1. CAS PROJECT. Effect of Damping. Consider a vibrating system of your choice modeled by y^{\prime \prime}+c y^{\prime}+k y=\delta(t) (b) What happens if e is kept constant and k iscontinuously increased, starting from 0? (c) Extend your results to a system with two8-functions on the right, acting at different times.See Answer

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