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![1- Verify the orthogonality properties of sines and cosines on the interval [-L, L]
So,
m #n
[si
sin(mmx/L) sin(nлx/L) dx =
LL,
m = n
m #n
cos(max/L) cos(nnx/L) dx = {
m = n
Lo
[²³
sin(mлx/L) cos(nnx/L) dx = 0 all m, n = 1,2,3,...](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0436312-80b7-4238-97bc-7d69c35381d6%2F1886f9a5-691b-42a9-92af-873f652184ad%2Fsmdvn9x_processed.jpeg&w=3840&q=75)
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- Use Taylor’s formula to find a quadratic approximation of ƒ(x, y) = cos x cos y at the origin. Estimate the error in the approximation if | x| ≤ 0.1 and | y | ≤ 0.1.Integrate the trig identity 2 cos j x cos kx = cos(j + k )x + cos(j -k )x to show that cos j x is orthogonal to cos kx, provided j -/- k. What is the result when j = k?Q1: show that1) d/dz [cosz] = -sin z , 2) * 1+tan2 z = sec2 z
- Show that cos jx is orthogonal to cos kx in C [ -π,π] for j ≠k,j, k >=1laplace of g(x) = sin21/3x cos3 2/5xSuppose that the position of one particle at time t is given by x1 = 3 sin(t), y1 = 2 cos(t), 0 ≤ t ≤ 2? and the position of a second particle is given by x2 = −3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2?. (a) Graph the paths of both particles. How many points of intersection are there? __ points of intersection (b) Are any of these points of intersection collision points? That is, are the particles ever at the same place at the same time? If so, find the collision points. (Enter your answers as a comma-separated list of ordered pairs of the form (x, y). If there are no collision points, enter DNE.) (x, y) =__ (c) Describe what happens if the path of the second particle is given by x2 = 3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2?. The circle is centered at (x, y) = __ There are _ intersection point(s), and there are _ collision point(s).