(d) f"(t) – 4f(t) = sin 2t where f(0) = 1 and f'(0) = -2 Taking Laplace transforms of both sides of the equati L{f"(t)} – L{4f(t)} = L{sin 2t} | %3D so that [s F(s) – sf (0) – f'(0)] – 4F(s) =- s2 + 22 That is (s2 – 4)F(s) – s,1 – (-2) = %3D s2 + 22 S-2 it. 2 so F(s) = (s2 – 4)(s² + 2²) – 4 1 1 1 16 s+ 2 15 1 %3D 16 s - 2 8 s2 +22

Q: Let 3 4 f(x) = (x° + 5x + 8)* f'(x) = f'(3) =

A: Chain Rule

Q: y" – 7y' + 12y = 4t where y(0) = 1 and y'(0) = 0

A:

Q: 4. Find (f-1)'(1) if f(x)=x' +8x+sin(3.x) а. Find (f-1)'(2) if f(x)= Vx' +x +xr+1 b.

A: To find the derivative of the inverse of a function

Q: Q3) for f(v) = sec2(3v – 1) and g(v) = tan?(3v – 1) show that %3D %3! f'(v) = g'(v).

A:

Q: a.) If f(x)= sin(In(x* )), find f'(x). b.) If g(x) = 5x find g'(x). %3D .2 x +3

A: (a) Consider the provided function, fx=sinlnx3 Find the derivative of the provided function. Apply…

Q: 2. Let F(x) = x s sin (t2) dt. Find F'(x). (a) sin (x²) dx (b) sin (x²) (c) sin (x²) + C (d) 2x sin…

A:

Q: Solve using Laplace transforms. y" - y' + 5y = 58(t- T/6) sin t y(0) = y'(0) = 0 SHOW WORK.

A: To find- Solve the Laplace Transforms. y'' - y' + 5y = 5δt - π6sint         y0 = y'0 = 0     Concept…

Q: y"" - 4y" + 5y' - 2y = 7t-5, y(0) = 0, y'(0) = 0, y'(0) = -2

A:

Q: Let (t) = Find '(t) '(t)=<

A:

Q: Find f. f"(0) = sin(0) + cos(0), f(0) = 3, f'(0) = 4 f(0) =

A: Objective is to find f.

Q: (a) If (In a] = , evaluate dx d [In x²] dx

A: Since you have asked multiple questions in single request so we  will be answering only first…

Q: Q4 B) Find y' for the functions cos(xy) + 3*+cos ly+ In(xy) + log-7 dy

A:

Q: Find r'(t). 13 r(t) = 3 cos (t)i + 5 sin (t)j + k %3D r'(t) %3D

A: Given:

Q: a) Suppose that f(4)=3 and f'(4)=-5. Find g'(4) where g(x)=Vx f(x) b) Let y=cosx-sin.x , show that…

A:

Q: 6. Let f(x,y) = sin(x + y) + arcsin (xy) a. Find the linearization L(x,y) at the point (0, ")

A: We have to find linearization of given function f(x,y): Concept: The linearization of a function…

Q: Let F (4t° + 2) i + sin(2t) j+ e*k Find F' (t), F"(t) and F" (t) (1) F'(t) (2) F"(t) (3) F"(t) =

A:

Q: Find f. f "(t) = 5e + 8 sin t, f(0) = 0, f(x) = 0 f(t) :

A:

Q: Find f. f(0) = f "(0) = sin(0) + cos(0), f(0) = 1, f(0) = 1, f'(0) = 3

A: Given that the expression

Q: Let 7(t) = Find '(t) F (t) =

A:

Q: Suppose r' (t) = (cos(t), et, 2t) and r(0) = (1, 0, 3). Find r(t).

A:

Q: Let F = (3t + 2) i + sin(2t) j+ 3ť²k Find F'(t), F"(t) and F"(t) 1) F'(t) = (2) F" (t) 3) F" (t) =

A: Solve for the equation for the derivative

Q: Q3) for f(t) = sec?(6t – 1) and g(t) = tan?(6t - 1) show that %3D f'(t) = g'(t).

A:

Q: 7. F(t) = cos?- sin? 2 8. F(t) = cos ( bt 9. F(t) = 8 sin- cos %3D

A:

Q: y"+y' = 0; y(0) = 0, y '(0) = 1 cost + sint t + cost + sint A) 1- e-t t-e-t O

A:

Q: v" - 4v" +5v' - 2y = 7t-5. v(0) = 0. v'(0) = 0. v"(0) = -2

A:

Q: The inverse Laplace transform of the function F(s) = = a) y(t) = 2 cost + 3 sint b) y(t) = 2e' cost…

A: