Eulers Formula states that e^ix = cos(x) + isin(x). when prooving this formula with the given maclaurin series expansions of e^ix, cos(x), and sin(x). Explain why one is able to rearrange terms in this equality without changing the sum of the series.

Q: Let K be a Galois extension of F of degree pn where p is a prime. Show that there exist Galois…

A:

Q: f/² sin (3) 4x COS ; (**) sin (*) dx (Consider initially the first 2 terms)

A:

Q: Complete the table by finding the balance A when $18,000 is invested at rate r for t years,…

A:

Q: PROBLEM: Derive the z-transform for f(t) = sin oot u(t). z¹sin(@T) ANSWER: F(z) : = 1-2 z ¹ cos(@T)…

A:

Q: 2. Show that the series is conditionally convergent ∞ n=2 cos(nx) √n ln n

A:

Q: How many different initials can someone have if a person has at least two, but no more than five,…

A:

Q: Prove that if n is an integer and n³ + 5 is odd, then n is even using a) A proof by contraposition.…

A:

Q: Free Response Show all your work to support your final answer and 1. Let R be the region in the…

A:

Q: Given vectors (A = a₁ + a₂) and (B = 2a,- 2ay - a₂), when A is defined by Q (1,1,1) find (i) AXB…

A:

Q: a. 2 uxx = Иxy - 4uy azu J2u azu ди b. + +4 +3+u = 0 ?х у дуг ?х ?x2 ди c. k = 2 ,k < 0 ду 02u ?хг

A: Classification of the Partial Differential equations

Q: Numerical Analysis (1) Let f(x)= ex2 + 4x³ . Find the third Taylor polynomial of f about xo = 2.…

A: Given function is fx=ex−2+4x3. We have to find the third Taylor polynomial of the given function f…

Q: (2x + 3) csc²() dx + f 4 tan () dx = 2(2x + 3) cot (#) f(2x + 3) sec² () dx - 4 tan ) dx = 2(2x + 3)…

A:

Q: Solve the following Initial Value Problem using Laplace transfo -9t y" - 12y + 11y = e ;y (0) = 1,…

A: The given equation is y''-12y'+11y=e-9t Taking Laplace transform on both sides Ly''-12y'+11y=Le-9t…

Q: 3.   p=317, q=811 (a) Consider the multiplicative group F∗p and find a generator (primitive root)…

A:

Q: 10. 1²+2²+...+ n² = n(n+1)(2n + 1) 6 integer n ≥ 1. s 12 for every

A:

Q: Find the unit tangent vector of the given curve. r(t) = (7t cost-7 sin t)j + (7t sin t + 7 cos t) k…

A: The solution is given as

Q: A faire dice is tossed twice -11 LI Pace

A: Given, a fair dice is tossed twice.

Q: III. Suppose that a function f(x) satisfies the following properties, together with the table of…

A:

Q: I. Evaluate the following limits. Using L'Hôpital's Rule is not allowed. 3. lim x-3 (x-3) tan(9-x²)…

A: To Evaluate: limx→3(x-3)tan(9-x2)sin2(2x-6) without using L'Hopital's Rule

Q: KeyConcept Law of Cosines In AABC, if sides with lengths a, b, and care opposite angles with…

A: Consider the given triangle △ABC. Here, a=9, b=7 and c=12. We have to find the angle A, B and C.…

Q: Write f(z) = Im(z 32) + zRe(z²) - 5z in the form f(z) = u(x, y) + iv(x, y).

A:

Q: Example 10.28. Solve the differential equations dz dy = 1 + x2, dx -xy for x = 0.3, dx using fourth…

A: Here, We need to use the fourth-order Runge-Kuta method to solve give differential equations.

Q: log: (x+3) 2-1-1-cosh(cos-¹x) 4. lim

A:

Q: 5. Consider that in 1915 that the population in the United States was 100 million. Suppose the…

A:

Q: Q1): The concentration of salt is given as a function of time by: dx -= 37.5-3.5x at the initial…

A: We have dxdt=37.5-3.5x At t=0, x=50 g/L So, t0,x0=0,50 Given step size is h=1 Euler's method is:…

Q: Let x = [1,1], y = [1,1] € R² and let f: R² ⇒ R² with f(z) = 2₁.x + 22.y for any z = [21, 22] € R².…

A:

Q: Let S₁ be the surface generated by x = √1-(y-1)² revolved about the y-axis. Let S₂: y = x² + 2.…

A: This is the surface of revolution S1(x axis (green),y axis(red),z axis(blue)).

Q: Let fn, f be real functions on S2. Show ∞ ∞ ∞ {w: fn(w) + f(w)} = UNU {w: | fn(w) - f (w)| ≥ } }.…

A:

Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

A: Let X be a metric space with metric d. Let , A ⊂ X be a non - empty. We need to show that , dx , A =…

Q: Which one is a normal vector to the line which is perpendicular to the 8x - 3y = 9. OA (8,-3) OB.…

A:

Q: (ix) = (hang R. y=x and y=x²

A:

Q: 3 42 2 -1 3 Let B = Compute det(B) by hand by reducing B to upper -1 31 triangular form using row…

A:

Q: I. Evaluate the following limits. Using L'H 1+ 3/ 1. lim x-1 2x²+x-1

A: The given expression is limx→-11+1x32x2+x-1 limx→-1x3+1x32x2+x-1 use the formula a3+b3=a+ba2+b2-ab…

Q: Which of the following sets of vectors in R³ are linearly dependent? Note. Mark all your choices.…

A: We want to find linear dependent vectors.

Q: -1 5 -2 Consider the matrix A= 2 a) Obtain the eigenvectors (v₁, V2, V3) of A. b) Write a matrix P…

A:

Q: I L is a line in 2-space or 3-space that passes through the points A an 3, then the distance from a…

A:

Q: 3x - 1 is 11 13 79 continuous in and f'(x) exist :) then find th 14 11 13 7 9 at it lies in (13

A:

Q: Create a table showing the differences of Euclidean, Elliptic, and Hyperbolic geometry according to…

A: Euclidean, Elliptic and Hyperbolic geometry are the different classes of three dimensional constant…

Q: dice In the experiment of throwing two fair let (A) be the event that the is odd, (B) Be the event…

A: Two fair dice are thrown. Let A be the event that the first die is odd and B be the event that the…

Q: Solve the following set of four linear equations using Gauss-Seidel iteration method; Continue…

A:

Q: Answer the following problems using the specified method up to four (4) iterations only. Round off…

A:

Q: Suppose that u, v, and ware vectors such that Evaluate the given expression. u-v-2w, 4u+vi* (u, v) -…

A: The axioms of vector operations are as follows: &lt;u+v, v+w&gt;= &lt;u,v+w&gt;+&lt;v,v+w&gt; &lt;u,…

Q: Consider the function f with the following properties: f is continuous everywhere except at x = 2 •…

A:

Q: Consider a system of linear equations A [:] b where -2 b [3] and A-1 RA Find the value of y in the…

A:

Q: (i) Draw the graph whose adjacency matrix is given in Fig. 2.25. (ii) Draw the graph whose incidence…

A: We have to Draw the graph whose adjacency matrix is given in Fig. 2.25. and Draw the graph whose…

Q: Let n P₁ P2 = ppp be the prime factorisation of an integer n > 1. 1: ti A 11. C.

A:

Q: Res (z(z+1)) a) 0 b) 1 c) -1 d) 2 e) i z=00 ||

A: We have to evaluate Res1−zzz+1at ∞. Here, fz=1−zzz+1 Here, poles of fz is z=0 and z=-1. Now,…

Q: Find the solution of the equation P №²₂ a²z อ + cos y ax² ay ² which tends to zero as x→∞ and has…

A:

Q: V. Given: el32-6-2, if x ≤ 2 13 f(x)= [2x - 5], if 2 3 x²-9, 1. Determine if f is continuous or…

A:

Q: (1 + inxy) dx + (1 +5) dy = 0

A: