3. One version of the Fundamental Theorem of Calculus tells us that if F is any antiderivative of the function f, then f (x) dx F(b) – F(a). Use this to evaluate the following definite integrals. »4 (a) / e + Vx dx (x+ 1)² dx (b) х ~16 (c) dx Vr3 (4) / 4 dx (e) (t² + 3)(t – 1) dt (f) Зет 2 sin x dx

Q: Get the derivative of all the functions. 1. ?(?) = 9?^5 + 2?^4 + 3?^3 + 7?^2 + 1 2. ?(?) = 5?^2−2…

A: To  find the derivative with respect to x

Q: Evaluate the definite integral ¹* (2²—7) ²10x¹ da

A:

Q: Let f(z) be an analytic function defined on the complex plane, where z=x+iy and ii is the imaginary…

A: The derivative of a function in complex analysis is defined similarly to that in real analysis.For a…

Q: f (0) = 1, f(1) = 4, f(2) = 3, f(3) = 5, f(4) = -7. | Use the substitution rule for definite…

A:

Q: The method of u-substitution is the integral analog of the chain rule of derivatives and the method…

A: Let say we have two functions f(x) & g(x)Now combine the function f(g(x))Hence, the derivative…

Q: f(z) = (x+iy)^3 is analytic or not

A:

Q: Use the clerivative rules to tind the deriVative of y. y=(x² + y> x²? (x-1)s y= (4x-1)3 2x+)

A:

Q: 3 Let f(x) = 2 + In(bx), where b is a positive constant. Find f'(x).

A:

Q: f(x,y)=xe*/2+275x²y³, then at (1,1) is equal to əx²əy³

A:

Q: Prove the Calculus identity sin-(x) + sin-(/1 – x²) 2 x € [0,1], where sin-, denotes the usual real…

A:

Q: y = x* + 2 sin(x) + xº + 24*.

A:

Q: Let f(y) = ⌊y⌋. Find a) f^-1({0}). b) f^−1({−1, 0, 1}). c) f^−1({y | 0 <y< 1}).

A: The given function f(y) is floor function:   The floor function is defined as: fy: The greatest…

Q: Cz sin r 5. If f : R → R is continuous, find F"(x) for F(x) = f(t) dt. %3D

A: F'(x) of the given function

Q: Suppose that a function h satisfies −x^2 ≤ h(x) ≤ x^2 for all x. Prove that h is differentiable at x…

A:

Q: For the f(x) = 1/1-X^2 and g(x)= x^2/1-X^2 A) shows that these functions have the derivative B) what…

A: We need to find relation between f and g.

Q: Is the scalar valued function f(x)=(x^2-1)/(x+1) continuous?

A: This is a problem related to limit and continuity of the function. Please follow the procedure given…

Q: For each of the following functions, find (∂z)/(∂x) and (∂z)/(∂y) (d) If z =…

A: Given: z=ax+by2+eax+by+sinax+by To find: ∂z∂x and ∂z∂y To show: b∂z∂x=a∂z∂y

Q: Let f(x) = 8 sin r + 3 cos r f'(x)- f(똥) =|

A: The given function is f(x)=8sinx+3cosxTo find f'(x) and f'5π6

Q: Suppose f is a function for which f' and f" continuous on 1, 3. You are given that f(1) = 4, f'(1) =…

A: Apply the integration by parts method, ∫uvdx=u∫vdx-∫dudx∫vdxdx So,…

Q: What is the residue of the function f(z) = (z^2 + 1) / (z^3 - 2z) at the point z = i?

A: We have to calculate the residue at

Q: Let f(x) f'(x) = 2 cos x + 7 tan x 2 sin(x) + 7 sec²(x) 1' (5) = 6

A: