If f(0) = 8 and L(f(t)) = F(s), then L[(t - 1)e"-)u(t - 1)f" (t- 1)] = 8e- e"(s - 1)2F' (s - 1) - 2e*(s - 1)F(s - 1) 8e - e"(s + 1)2F'(s + 1) - 2e"(s + 1)F(s + 1) 8-e (s-1) F'(s - 1) - 2(s - 1)F(s - 1) Sel-1) - els-1) (s - 1)2F'(s - 1) - 2el-" (s – 1)F(s - 1) 8e - e"(s - 2) F'(s + 1) - 2e*(s - 2)F(s- 2)

Q: 1. Let T = [-2 1 0 5], 5 1 1 0 A = [¦ ¦] ·--[¦ ¦ ). -- [¦ ¦ ¦]. - -[] [16], B = = [1 2] C = D = 0 3…

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Q: 57. If f(x) = (A) (0,00) fen²x + (n (ex) (n² (ex) dx; f(1) = 0 and g(x) = 1 -f(x). Then the domain…

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Q: +(n) = 3 (²³-9x² +33m - 18 enfor 1) Ⓒincreasing in (-0,0) (Ⓒineressing M- L decreasing in (2,8) in…

A: The function is given by:-Since, the natural log is present in the function.Domain of the function…

Q: [2x*dx (Ans. 3 2 x' +c ); (xi) J3x°dx - 3 In\x)| 3 |3x'dx (Ans. (Se* –x? +2)dx , (x + 0) (x) x' +c);…

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Q: Let f(x) = x7 Then f'(x) is 5æ5 + 3x³ – 3æ – 4. За — - - f'(3) is f' '(x) is and f''(3) is

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Q: If f(x) = (x² +2) e², what is f'(x)?

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Q: Q4. Let flW=( Gué-4u-7) (4ax4 -8x#+15) Find tle regueetad closiuadtiul (10)

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Q: (d) dx (e) S,(4x + 2)5 + (6 – 4x) dx (f) S,' 4x3 – 3x* dx

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Q: AF f(1) = 6 glD=5 h(1) = 3 f'(5)=-7 g'(5) = -11 (5) = 15 f" (-11) -2 - g" (-11) = 3 f(1) = 4 g(1) =…

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Q: If f(t)=(t^2+4t+3)(5t^2+6). find f('t) and f'(3)

A: Given: f(t)=(t^2+4t+3)(5t^2+6) find f('t) and f'(3) = ?

Q: Given f()= 2x + 1 and ()= r*+ 2x-1, fnd (f- (1).Then valute the difence when x=2.

A: Sum and difference of the two functions

Q: f(t) = (t + 4t + 8)(7t¯² + 6t).

A: Given function is: ft=t2+4t+87t-2+6t-3 Then we get,…

Q: f(x, y) = x²/(1 + y²) (a) f(-5, 0) (b) f(14, 1) (-) (c) | (d) f(-8, y)

A: Put value of x and y in given equation.

Q: f (n) + g(n) = 0(max(f(n), g(n)))

A: Big Theta notation (Θ) :  f(n) is said to be Θ(g(n)) if there exist positive constants C1 and C2…

Q: f(x) then: f'(x) = f'(1) -

A: We are asked to find the derivatives of the function.

Q: If f(x) (1)? = 2 x² + 3x x+1 " then what is

A: Given: f(x)=x2+3xx+1

Q: 4+ (-1.9, 2.8) f(x) = x - 3x - x 5 -4 -3-2 -1 1 2 3 4 -3- -4 (a) (b) gure 3.21 (a) One section of…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: If f(x, y) = (5x- y²) (a) f(1,4) find the following.

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Q: If f'(x) = 3x + 2x and f(2) = 3, then f(1) = %3D (А) —10 (B) -7 (C) 10 - (D) 13

A: We have given f'(x) and f(2) =3 .  We have to find the value of f(1)

Q: = (t² + 5t + 5) (5t² + 4), find ƒ'(t). If f(t) f’(t) = Find ƒʼ(3). ƒ'(3) =

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Q: 3 +5 - dx =

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Q: 5-s s² + 4s-5 (A) Find L¹(F(s)}, knowing that: F(s)=- DA

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Q: r³ (x + 2) 6. Find f'(6) for f(x) = V (x + 3) Væ2 – 27 11 (a) 18 11 (b) 216 11

A: Some common derivative rules are, Also,

Q: F(x) = 3Jx#6 + K Given find the value of k if f(-3) =19 f'-2) =- 3 ·

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Q: If we let f(x) = 1-4x/1+4x What are.... f’(5) = f”(5) = f”’(5) =

A: we have to find derivative at that point

Q: lo UcE - Uxx=0 4(x,0) = f(x) 4= (x2,0) = g(x)

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Q: the derivative () of = f(x) = 2

A: Please refer to the image below

Q: (-y dx + xdy) = (a) d(xy) (b) d(y/x) (c) d{ In(xy) } (d) -d(x/y)

A: By quotient rule , duv = v du-u dvv2

Q: Find the derivative of f(z) = (x³ + 5)(z³ – 5) %3D (8z")(5x*) (8z +5)(z – 5) + (z³ + 5)(5x* – 5) (8z…

A: We need to use the product rule.

Q: If F(x) = f(g(x)),

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Q: f"(z) = 7z – 1 and f'(-2) =-1 and f(-2) = -4. Find f'(z) = and find f(4) = !!

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Q: If f(x) = (2² + 5x + 3)“, then s'(#) - [ f'(x) -

A: To find the derivative of the given function, we need to use chain rule of differentiation and some…

Q: If f(4) = 3 and f'(x) 2 3 for 4 s xs 9, how small can f(9) possibly be? (9) 2

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.