1. Starting with the equation a tanh y show that, for real r, y and |rl < 1, y-tanh-1 2. Compute the following derivative dtanh (sin() 3. Assume θ is a real number. Then use Euler's formula eie-cos θ + isin θ to show that coth(i0)-icot(e) 4. Use the definitions to obtain an equation for cosh(3x) in terms of cosh(x) and sinh(x) and their various products (e.g., cosh*(z), cosh(x) sinh3(x) etc.). Do not use the double-angle formula such as cosh(u+...
Please do both of them. Thank you.
Find 3 cosh(2 x) 3 sinh(5 x)] 3 5 2+4 15 a) 225 3s 15 b) О 2-4 2-25 6 c) O 3 s 24 2-25 6 3 s d) O 2+25 3 5 15 e) O 2-4 (2-25) fONone of the above. Question 6 Find e2 sinh (5 x) 4 e cosh(5x) 2 а) О 2 (5 7) 2 (s3) 5 6 54 1 b) О 2 (s3) 2 (s 7) 54...
Find the derivative of the following: f(x)=( Sinh(Sin-(x2)))3 оа. Select one: - 6X(Sinh(Sin-(x^))) Cosh(Sin-1(xº) √1-8" 3(Sinh(Sin-'(x2)))?Cosh(Sin-'(x2)) O b. 71-X 3(Sinh(Sin '(x2))) Cosh(Sin ?(X)) O c. 71-X 6XCosh(Sin-'(x?)) O d. V1-14
Recall three facts about hyperbolic cosine and hyperbolic sine functions: dr cosh(x) = sinh(x), di sinh(x) = cosh(x), cosh? (x) – sinh? (x) = 1. Compute the Wronskian of cosh(2) and sinh(2). Using your question 1's answer, cosh() and sinh(x) are linearly independent on R True False
1. Ssinhx – cosh xd sinh x - cosh x 2 coth'(x) + 1 coth2(X) - 1 dx 3. S cosh² x dx 4. S sinhᵒ x dx ctanhx- sinh x sinh 2x
Problem 7. (20 points) We consider the function tanh(z) sinh(z) tanh(z)=cosh(z) where , For any integer 0, we denote by Qe the positively oriented square whose edges lie along the lines z-t(k+1) π and y = ± (k+)π -(km 4p. (a) Show that for any z z + iy e C, |cosh(z)12-sinh2(x) + cos2(y). 2p (b) Recall that tanh is analytic at the origin and that tanh () 1 - tanh2(). Compute the tanh(z) limit l := lim (Problem 7...
2 -e Recall that cosh(x) er te 2 and sinh(2) Any general solution of y'' – y=0, can be written as y(x) = ci cosh(x) + C2 sinh(x), for arbitrary constants C1, C2. O True O False
' ' 3. The hyperbolic trigonometric functions are defined by sinh(t) , The following identities might be helpful in the following problem. You should convince yourself that they are true, but you are not required to write up proofs/derivations cosh2 1-sinh2に1, cosh2 1 + sinh2にcosh(21), cosh 1 -sinh 1-cosh 1 sinh 1, dl dl (a) For a real number t, define sinh-1に1n(t + Vt2 + 1) Show that sinh(t) is the unique eaber u such that sinhu t (b) Use...
Please solve all three. Thank you very
much
5. (a) Let a be a constant (we can write “a ER” to mean “a is a real number”). Verify that y(x) = ci cos(ax) + C2 sin(ax) is a solution for y" = -a’y, where C1,C2 ER. (b) Consider the hyperbolic trigonometric functions defined by cosh(x) = et tex 2 ex – e- sinh(x) = * d Show that I cosh(x) = sinh(x) and sinh(x) = cosh(x). (e) Verify that y(x)...
Problem 5. Prove that parametric equations: x a-cosh(s) (a > 0) or back half(a < 0) of hyperboloid of one sheet: Χ t), y b-sinh(s) cos (t) zc-sinh(s) sin( t), (x,y,z) lies on the front half L" a2 b2 c2 Problem 6 What graph of this Compute the arc length : rit)- < sin t, cos t, 2Vt', when 0<t < function: a) Compute the arc length : re)-3cos(9) and 0 < θ < π/2 b) Problem 7. Find parametric...