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8. 2 sinº x-3 sin x-1=0 11. 4 sin’ x + 7 sin x = 6 9. tanx-1=0 12. ( tan x-)(tan x-3) = 0
1. cos 4 x-sinº x = cos 2x 6 6 2. sin x + COS x = 1-3sin ?x cos” x 3. cos 2x = 1-tanx 1+tanx 4. 2sinx cosx = cos(x-y) – cos (x+y)
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0 Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
3. Let W (x, t) = (coswt)(a cos nx+b sin nx) and (x, t) = (exp -kn-t)(a cos nx+ bsin nx). Here n is a positive integer, 2,t are real variables, and a, b,w, k are real constants with k positive. a. Evaluate W(x,0), H (2,0) and əW/ət(x,0) for all c. b. Show OH/ət = k(32H/8x2) for all x,t. c. Find some positive constant c so that w2w/at2 = c(32W/8x?) for all x, t.
It is known that Fourier series of f(x)=x is 2° 2(-1)" + "sin(nx) (n 1 on interval [-T, T). Use this to find the value of the infinite sum 1 - + 1 1 5 7 3
Solve for å by showing all the details. a) tan” x — 4 sinº x = 0 where 0 < x <a b) sec(sin-1 Væ2 – 1) c) In(x) – In(3x2 + 2) = ln(K)
QUESTION 3 Evaluate the integral by using multiple substitutions. SV1 1 + sin2 (x-7) sin (x-7) cos (x-7) dx o 3 (1+ sinº x) (1 + sin? x)3/2 + c 3 O AV1 + sin?(x - 7) +C og (1 + sin? (x - 7)) 3/2 + c O (1 + cos2 (x - 7) 3/2 + c
3. [6 marks] Solve the IVP: cos x + y sin x = sinº x, y(0) = 2.
It is known that the Fourier series of f(x)=x is 2,6 21– 1)* * 'sin(nx) on [-1,1). n 1 1 1 Use this to find the value of the infinite series 1 - + + .... 3 5 7
Question 3 Suppose that f(x) nx2 1 + nx?' for all x ER. What is fo) for all nEN.