10. Using Laplace transform, solve the initial-value problem: y' + 2y = f(t), y(0) = 0, where f(t)= = 2 for 0 < t < 1, (t - 2)2 for 1< t < 2, 0 for t> 2.

Q: i) Find the tangent and normal lines to the curve r*y° +sin(xy)+ln(1+y)+sin(2x) = sin(2) at the…

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Q: From the previous page: the function f(x) = e*(x² + x – 1) has derivatives f'(x) = e*(x² + 3x),…

A: Consider the following function: fx=exx2+x-1f'x=exx2+3x Consider the part (e): Linear approximation…

Q: sin(t) dt , 1-4x 1+ t2 4 If f(x) then f' (x) =

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Q: (1) Let g(x) :| f(t) dt. What is g'(x)? a

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Q: Q4: (1) y = 3logst (2) y = t sin¹(t) + √1-t² 4 (3) y = (sech t) (1 - In(sech t)) (4) y = (t² + 2t)…

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Q: Consider the function | dt, defined in [-2, +). Find the local max/min of f(x).

A: Local maxima and minima  Use lebnitz theorem for differentiation

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…

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Q: Find the critical Point(s) of the function

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Q: If L{f(t)} = F(S) then show that L{f(t)} = SL{f(t)}-f(0) (sint) hence find L (dt

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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.

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Q: 3x2 Given that F(x) = In t dt, what is F'(x)? 3x 4 + t2 ||

A: Given, Fx = ∫3x3x2lnt4+t2dt  We have to find F'x.

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: secx 1 Given g(x) : z dt , find g'(x). t2

A: gx = ∫7sexcx1t2dt           = ∫7sexcx t-2 dt        = t-2+1-2+7secx      =t-17secx-1 = secx-1-7-1-1…

Q: Just use the definition L{f (t)} = ° e-st f (t) dt with 2 integrals, one for 0 1

A: Laplace transferom

Q: ) y = 3logst ) y deriv = (sech t) (1 - In(sech t

A: A function's derivative is the function whose value at x is f′. (x). The graph of a function's…

Q: Find the absolute maximum and minimum of f(t) = -2cos2t + sin4t on  the interval [0,pi/4]

A: The given function is, f(t)=-2cos2t+sin4t and the given interval is, 0,π4. Obtain the value of the…

Q: g(x) = 2x−7+4 2x-π+4 o TT/2 sin² x g(t) dt

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Q: The graph of the function y = 3 sinh 4x has a relative maximum at a point O (0.0) O (0, 3) O (3.0)…

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Q: 2 (a) Consider the functions f(x) = 1– a* and F(x) =| f(t) dt. i. Write a simplified form of F(x) =…

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Q: (b) Prove that sint -dt > 0 t+1 for all a > 0.

A: Given:- ∫0x sintt+1dt &gt;0

Q: 4) Knowing that f = f(t), using the expressions of L[f'] and L[f"], derive an expression for d'f…

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Q: Let f(x) = х+ 1 (a) Evaluate f(0), f(3), and f(a + 3). f(0) : V3 f(3) 4 f(а + 3). (b) Find the…

A: Topic -functions qnd average rate of change of function over certain interval

Q: Given f(x)=4x3, 0<=x<=1 Find moment generating function of X

A: The moment generating function of any function, f(x) is calculated asMXt=∫-∞∞e-txfxdx