(a) Evaluate the integral [21,22,23,21] 3(z − 1)²dz, z₁ = 1, %2 = 1 + i, z3 = −1 + i. (b) Show that when f(z) = x³ + (1 - y)³, it is legitimate to write f'(z) = 3x² only when z = i.

Q: What is the total change of f(x) if f'(x)= sin(x) from [0, πT ]?

A: For a given f'(x)=f(x) where f'(x) is the antiderivative of the f(x) and in the interval x=a to x=b.

Q: Is g = {(x, y) ∈ [−5, 5] × R : x^2 + y^2 = 25} a function from [−5, 5] to R.

A:

Q: Suppose f(x, y) = x" + y*. Find fa(1, 1), f,(1, 1), and fz (x, y)

A: Given :f(x,y)=xy+yx

Q: :Presentation of data include Mathematical presentation Tabular presentation. Graphical…

A: Let us solve the given data based problem in the next step.

Q: ex: show that (H.W) (xJ,(x)' = -x Jv+1

A:

Q: Suppose v(t) gives a particle's velocity at time t , in miles per hour. Write an integral expression…

A: Let the particle velocity v(t) at time t , (in miles per hour)

Q: 6x2+5 If f(x) = In x4/5 find f' (x).

A:

Q: 4x−³+2 3 8. S ² X - 2 - X dx

A: Given :∫4x-3+2x-2-xdx

Q: Calculate Vf when f 3x z-y7+ 4x'y+ 2x - 3y-5 at the point (1, 1, 0).

A:

Q: Let 7'(t) = (3+ 4t, cos(t), 6e*) and F(0) = (7, 2, 5). Determine 7(t). †(t) = (, , )

A:

Q: dy Prove that dx of x2 +2x is f'(x)=2x + 2

A:

Q: Let F(t) = (4t³ - 1)j + (5t+2)k. Find F'(t). 7"(t) O 288t 288t³j 05+288t³ O 36t³ +5 O None of these

A:

Q: Show that y" + ay' = b, wherein a and b are constants, can be reduced to y' =+C₁e-ax using 2…

A:

Q: Let f(t) = 4t* – 4t + 5e'. Then f'(t) =

A: find the derivative

Q: Find the function F(x) such that F' (x) = 8x and F(0) = 1 O 1+4x^2 O 4x^2

A: topic- Functions

Q: A 2xe ²x² B Xex² Ⓒ2e+² D 2xex² Let f(x)= ex², find f'(x)

A: given  f(x)=ex2d dx(ex)=exd dx(xn)=nxn-1

Q: -1 Ax) = 7xsin (x) f'(x) =

A: SOLUTION-

Q: 4. Let f(x, y)=2x*y' . Find f,(x,y)

A:

Q: 3dx For f x-Vx = O x= u^(1/2) x= u^2 x= u None of the choices

A: Given integration is  ∫3 dxx-x.

Q: Let f(x, y) = (- (3z + y))°. Then

A:

Q: 12 y' (12x +5) (4y* +3)° b. If y* +3y-4x' = 5x+1, then prove that y"(4y +3) = 24.x--

A:

Q: Situations that can be modelled by the exponential function.

A:

Q: b y' = 2 + ₁e-ax a

A: details steps are given in step 2

Q: Show that f(x1, x2) = 2 - (x1 - 1)^2 - e^x2^2 is concave.

A: the given function is fx1,x2=2-x1-12-ex22 To find the function is concave we have to show all its…

Q: 8. Find all the roots of f(x) =x* - 4x' + 6x² – 4x+5 if 2-iis one root.

A: Topic- roots of polynomial

Q: Suppose that f(x + h) – f(x) = 7hx? – 8hx + 5h²x – 3h? – 7h³. Find f'(x).

A: The quotient difference formula is given as, f'x=limh→0fx+h-fxh it is given that…

Q: 8. Find all the roots of f(x) = x' - 4x' +6x² – 4x+5 if 2– i is one root.

A: We will find all the remaining roots as following.

Q: Зх-11 19. dx (2х+1)(х-3)

A: According to our guidelines, we can answer only the first question, the rest can be reposted.

Q: x) [in(3x²-36)]². Then f'(x) = A. 12x In (3x²-36) 3x²-36 O A. O B. O C. O D. O E. 2 In (3x²-36)…

A:

Q: Differentiate: y= x*+1 x +1 + xinx (A (x*+1) x + 1 + Inx x+1 + Inx D (x*+1)(x + 1 + Inx)

A:

Q: Suppose f(r) = (In z)e. What is f'(x)?

A: d/dx (f(x)g(x))= f(x)g'(x)+g(x)f'(x)

Q: Let f(z) = 4z +1 f(2) =

A: Given  f(x)= 4x+1

Q: What is the total change of f(x), if f'(x) = 2x - 3, over the interval [1, 3]? -3 O-2 O 2 03

A: