(4b) 6. (20%) Find the Laplace Transform F(s) = of f(t) if f(t) = 1. Hence find (f(t)) F(s) = L (f(t)) = L (1) = ] e¯*f (t)dt = √ e¯™ª.1 dt = √ e™ªdt 0 d if (e') = e² dt { e'dt = e' (6a) (6b) (6c)

Q: Let f(e) = In[r*(z + 6}*(z° + 6)"] f'(x) =

A: Given: Now,         f'(x)=lnx4x+62x2+69'        =1x4x+62x2+69x4x+62x2+69'…

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

A: Chain Rule

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

A:

Q: Let g(x) - [₁6'd -3 g'(x) = -0 6t²dt. Find g'(x).

A:

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: Given 9 (x) = S* t² cos(t)dt use FTC Part 1 to compute g' (x).

A:

Q: (11) Let G(x) = | a In(1+t²)dt. (a) Find G(1). (b) Find G'(1). (c) Find G"(1).

A: a) G(1)=0 b) G'(1)=piln(1+x^2)|x=1 G'(1)=pi ln(2) c) G''(x)=pi(1/(1+x^2))2x G''(1)=2pi/2 G"(1)=pi

Q: psin(rx) (b) Let g(æ) 1 and f(x) g(t) dt. Find f'(2). x4 + 1

A: We have to use derivative under the functional limit formulae first then put x=2.

Q: Let F(x) be the function defined as follows: F(x)= (1+ In(t))*dt What is the value of F'(e) ?

A:

Q: Find G(1), G'(0), and G'(π/4), where G(x) =

A: Here we have;  G(x)=∫1xtant dt

Q: tan x If h(x) Vt + vt dt, then find ht(x). 0,

A: We apply here Leibniz rule

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A: Second fundamental theorem : Suppose f is continuous function in an open interval I .Let a∈I and a…

Q: Find the area of each circle, given the radius r or the diameter d. Leave your answer in tems of t.

A: See the detailed solution below.

Q: Find dy/dx if y= cos-1 In(x+1)

A:

Q: Let f(x, y) = x² + y² and r(t) = (cos 1, sin t). (a) Find f(r(1)) without making any calculations.…

A: a) Equation of the circle with the centre at the origin that is (0,0) is x2+y2=a2 then every point…

Q: 35. A particle travels along a curve y = f(x) as in Figure 15. Let L(t) be the particle's distance…

A:

Q: 1. Let A(x) = | e -tdt. (a) Sketch a graph of e-t and use your graph to illustrate what A(x)…

A: Given:Here A(x)=∫1xe-tdtTo find:a)sketch the graph of e-tb)find the value of A(2)c)find…

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A:

Q: A tank initially contains 60 litres of pure water. Brine containing 2 gm of salt per litre is…

A: Using the given data.

Q: The graph of f is given to the right. Let g(x) = f(t) dt. (a) Find g(3). (b) Find g'(-2). (c) Find…

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

Q: Suppose f(x) = [ 6t² /1+t° dt .

A:

Q: In a rectangle, the width is one-third the length. The total perimeter is 40 inches. Find the length…

A: Solution: Given:    Width is one third of the length of rectangle     and the perimeter of rectangle…

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

A:

Q: Exercise 1. A student is asked to prove If m is even and n is odd, then m² + n² is odd. 2 The…

A: Let us answer above questions

Q: 1. a)∫9/x t^4 cos(t)dt f'(x)= b)∫x^2/1 ((t^2+3)/sin t)dt f'(x)=

A:

Q: Compute F'(x) for each of the following functions: (a) F(x)= | ecos(t) dt; 1 (b) F(x) = | Vĩ+tª dt;…

A:

Q: (11) Let G(x) = 1 In(1+t²)dt. (a) Find G(1).

A: Solution Given G(x)=∫1xπ ln(1+t2)dt.We need to find the G(1)