Question 9. (a) Complete the statement for the Fundamental Theorem of Calculus Part II: Suppose f is a function integrable on the interval [a, b]. If F is any antiderivative of f then [² f(x) dx = (b) (c) Use the theorem above to compute e-t - 4 sin(4t) dt. -T Use the theorem above to compute fdx. 2

Q: Consider the function f(r) whose second derivative is f(z)- 8z+10 sin(r). If/@)- 4 and f'0)-4, what…

A:

Q: (b) Use induction and the Product Rule to prove d. -I" = nr"-1 for all n E No.

A: First principle of mathematical Induction : Suppose Pn is any statement dependent on n  where n∈ℕ .…

Q: Show that the functions f: R → R defined below are continuous. (a) f(=) =2-z; (b) f(z) = 2 - 3z. (c)…

A: a). fx=2-x Let c be any real number. Now, fc=2-c & limx→cfx=limx→c2-x=2-c Clearly, limx→cfx=fc…

Q: Question 2 (a) Find all the analytic functions for which Justify all of your steps. (b) Prove that…

A:

Q: Let fi(x) E Fun(R, R) be the constant function 1, this is, fi(x) = 1 for all x ER. Recall that the…

A:

Q: Let f be continuous on the interval [0, b], where f(x) + f(b − x) ≠ 0 on [0, b].

A: Let I=∫0bfxfx+fb-xdx   I=∫0bfxfx+fb-xdx                     ....... (1) use ∫0afxdx=∫0afa-xdx…

Q: Exercise 1.1: For the function f(x) = sin(x) in the domain (0, π), (a) Sketch the function f(x) in…

A: The given function  in the interval .We have to sketch the graph, find the values of  and , and find…

Q: From algebraic form of function ez, prove that  |ez| = ex

A:

Q: Determine values of a, b that belong to Real Numbers so that attached function be continuous in all…

A: Step 1: Step 2: Step 3: Step 4:

Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx<0x20≤x<12−x1≤x<2x−3x≥2

Q: Let f be a differentiable function such that f f(x) sec² x dx = f(x) tan x-f3x² tan x dx . Which of…

A: Consider

Q: P c=0 and I= 1. d (a) Use the definition of the derivative to prove IP d. -I" = nr"-1 for all n E…

A: Given , Function =xn To Prove :ddxxn= nxn-1  using Quotient rule

Q: The real part of an analytic function f(z) where z = x + iy is given by ey cos (x). The imaginary…

A:

Q: tinuous, prove that T/2 T/2 f(cos x) dx = f(sin x) dx (b) Use part (a) to evaluate /2 cos?x dx and…

A:

Q: Please solve this problem with a good explanation. It is a linear Algebra  question Thank you.

A: (a) The given f is in V if it satisfies

Q: 2 dt t² +2t +3 18. Evaluate 8 dz z² +8z+ 25 20. Evaluate o

A:

Q: 3X 812x

A: To solve the following problem

Q: a,b,c

A: For (a): We know that                       . So, the domain of sech (x) is (-∞, ∞). And the range…