If use Taylor's method of order three to approximate an initial value problem with f(t, y) = et-y, we need h h² T(3³) (t,y) = f(t,y) + ½ f'(t,y) + / ƒ"(t, y), 3! Write out the explicit expression of T(³) (t, y) (no ƒ, ƒ', or f" appears in the expression, should only involve t, h and y). (Do not carry out the actual calculations)

Q: 3. Consider f (x, y) = a +6V-10. Compute the following: (a) (b)

A: Given,          fx,y=x4+6y-10

Q: If y = f(x), then f'(x), y', [Select] dy da and f(x) all represent the of fat x.

A: If , then to determine the meaning of .

Q: Suppose that f (1) = 1, f'(2) = 5, f'(1) = 4, g(1) = 2, g'(1) = 7. Find a. (fo g)'(1) b. (9((x)))…

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Q: f(x) = (13 + 6t² + 1) dt

A: SOLUTION-

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

A: Chain Rule

Q: Let h(t) = 5t³.2 – 3t-3.2. Compute the following. h'(t) = h' (2) = h" (t) = h" (2) =

A: The following formulas of differentiation has been used to solve this questiona. f'(x)=ddxf(x)b.…

Q: Compute r''(t) and r'''(t) for the following function. r(t) = (9t2 + 3,t+ 1,1) = (9r?. Find r'(t).…

A: Find the first, second, and third derivative of the following vector-valued function. r(t) = <9t2…

Q: TLP iadade mnts & smt Findf F'(t) =sec(t) (seclE)+ tan (ED), -st < f- FCt) = -<ts f (--7 2.

A: We simplify f'(t) by expanding  To find f(t) we integrate f'(t) C=integrating constant

Q: Find y as a function of if y(0) = 18, y'(0) = 28, y'(0) = 21. y(x) = y"-9y" + 20y = 24e,

A:

Q: 5 3. Let F(x) /t2 17 dt. Find F'(x)

A: Given:

Q: f'(x) = f'(4) - L

A:

Q: 13. Find ƒ. (1)ƒ'(x)=x² + 3√x (2) f(t)=2cost+sec²t,-/2 <t</2,ƒ(π/3)=4; (3)ƒ*(x) = 6x+sinx.

A:

Q: Knowing fli) = 3, f (1) = 27. find d (7) at

A:

Q: f(T) is a twice differentiable function with f(0) = 5, f(1) = 4, f'(1) = 2, Then Calo

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Q: Find an expression for f (t, p) and evaluate that expression when t = 0. Find an expression for f(t,…

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Q: Compute the derivative (r₁(1) r2(1)) if r₁(t) = (t¹0, 1³, tº) 73 10t r2(t) = (et, e³t, elor) (Use…

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Q: Find r'(t), r"(t), r'(t) · r"(t), and r'(t) x r"(t). - r(t) = 3t²i – 4tj + ²t³k 2 (a) r' (t) (b)…

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Q: 2. Use the theorem in Sec. 23 to show that f'(z) and its derivative f"(z) exist everywhere, and find…

A: Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: et f(r, y) (-(4r + y)) Then %3D ardy

A: The given differential equation is: f(x,y)=-(4x+y)7

Q: Find y as a function of a if y"" - 12y" + 35y' = 24e", y(0) = 27, y'(0) = 11, y'(0) = 30. y(x) =

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Q: Find f. f(t): = 12t f"(t) = 3 9 √t' f(4) = 32, - 22t + 6 X f'(4) = 14

A: f''(t)=9tgiven  f(4)=32f'(4)=14∫xndx=xn+1n+1+c

Q: 14. The function g is defined as g(x) = 3x + | (1+?) dt. What is the value of g'(2)? %3D (A) 20 (B)…

A: The given function is- gx=3x+∫1x21+t2dt Then evaluating the integral-…

Q: Let r(t) = (- Find 7' (t) r' (t) = ( (..) -5 - 2t -3t+3' 5t² - 4 - 3t² -6³-5

A:

Q: If r(t) =  , find  r'(t), T(1), r''(t), and  r'(t) × r ''(t).

A:

Q: F:[- 2, 2]→ R, F(x) = (- 3Xf + 1) di

A: Given: Fx=∫-2x2t-3t2+1dt For finding derivative of given function, we will use fundamental theorem…

Q: 8. Find f(4) if f(0) >0 and [f(x)]² = [(f(t))² + (f'(t))²]dt + 4.

A: We do this using Lebniz rule for differentiation under integral sign. The detailed solution is…

Q: 15 [ = f(x) = H t²dt then

A: f(x)=∫x15t2dt

Q: Show that f(x)g(x) = (x – a)" Cmdn' -m- n'=0 m=0

A:

Q: 2) f(1)=47² -ť², (0<t<2x)

A:

Q: Find r'(t), r"(t), r'(t) · r"(t), and r'(t) x r"(t). r(t) = 2t²1 - 4tj + (a) r(t) (b) r"(t) (c)…

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Q: If f(x) = then ƒ"(x)= = x S² (t³ + 4t 0 (t³ + 4t² + 6) dt SI

A: given,  f(x)=∫0x(t3+4t2+6)dtwe know→(i)  ∫xndx=xn+1n+1+c(ii)  d dx(xn)=nxn-1(iii)  d dx(constant)=0

Q: Find r'(t), r"(t), r'(t) r"(t), and r'(t) x r"(t). 1 t³ k 2 r(t) = 3t²1 – 3tj + 2 (a) r(t) (b) r"(t)…

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Q: Jack drives east away from Hilltop bakery at 60 mph while Jill drives toward the bakery at 40 mph…

A: Given that,

Q: find f'(t) and f"(t).

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Q: Calculate r' (t), T(t), and T(1) where r(t) = (2t³, 4t³). r' (t) = T(t) = T(1) = = 6t², 12t²

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Q: If F (s) = L {f(t)} and f" (t) +3f' (t) + f(t) = 28 (t), where f' (0-) = f(0) = 0, ti f" (t) + 3ƒ'…

A: