1. Use the definition to find f'(e) if it exists *, if z< 3 S(x) = c= 3 -3r, if r 23

Q: Prove that 1 + p'2 p4 where primes denote differentiation with respect to 's'. 1. x"2 + y"2 + z"2 %D

A:

Q: 34) If f is an even function that is continuous on [-1,3] and if f f(t)dt f₁f(t)dt = 8, then f…

A:

Q: 9. 3 If f is continuous and f (u) du = 3, then / (4xf(x²) + f(3x)) dx is equal to

A:

Q: + [ƒ(2)₁ Suppose that fand gare continuous functions such that 1 [ g(x)dz -3, find f(x)dx= 2 and +…

A:

Q: Suppose f(x) is the function such that for a > 0 we have f(x) = (e" – 1) In(x) and for x < 0 we have…

A:

Q: 2.if f (x) = x2 + 3, with domain R then f(x) has f(x) %3D O T F

A: Here the given function is f(x)= x2+3 with domain R We need to find it has inverse or not.

Q: Suppose that h(x) has a continuous derivative h'(x), h(0) = -2, h(T/4) = 2 and tan(x)h'(x) dx = 4.…

A:

Q: 5. Show that the function h(r, y) defined by if + y 1 is discontinuous.

A:

Q: If f is continuous on [a, b] and "If(x)| dx = 0, what can you conclude about f ?

A: Solution:Given If f is continuous on a,b and ∫abf(x) dx = 0To conclude about f ?

Q: Find the devivative of f cx) = by Hhe de filzifion.

A: Here we differentiate the given function by it's definition.

Q: 2. If f is differentiable at a, then Duf(a) = Vf(a) · u. Moreover, if Vf(a) + 0, it gives the…

A:

Q: of of and Find for the following function. f(x.y) = e xy In (2y) of dy

A:

Q: Suppose that the first derivative of f(x) = is approximated by the following formula %3D 3x+4 -2f(x…

A:

Q: If A(x) = f(t)dt is increasing on [0,10], then f(x) must be increasing on [O,10]. True False

A: Given

Q: Demonstrate that for the integral of p(x)/q(x) where p(x) is of degree 2 and q(x) of degree 3 it is…

A:

Q: If functions ƒ(x) and g(x) are continuous for

A:

Q: B) Show that fx) = sin for x * 0. for x= 0. X = is not continuous.

A:

Q: Зx2 — 1 Q7- Let f(x){ cx + d Vx + 8 if x 1 Determine e and d so that f is continuous everywhere (as…

A:

Q: I'm stuck on this question and I don't know how I should be approaching this. What should I do?

A:

Q: 4 Suppose that f(1) 3, f(4) = 5, f '(1) = 4, f '(4) = 2, and f " is continuous. Find the value of xf…

A:

Q: If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I…

A: We have to find f(c)

Q: 1 Let f(x) = 8 + In (t) dt, for x2 1. Find f'(2).

A:

Q: 4 Suppose that f(1) = 9, f(4) = -7, f'(1) = 6, f'(4) = -1, and f" is continuous. Find the value of…

A:

Q: 7. Suppose that f: R R is differentiable at c and that f(c) = 0. Show that g(x):= |f(x)| is…

A:

Q: If we use Simpson's rule for a function f(x) = X +1 with n=4 , then the approximate of S f(x)dx is

A:

Q: 3 B Consider the following function, z= x -2xyty't yn Lind the alues of ag Ddz,z,when redgyel dy 6.

A: Given : z = x2 -2xy + y3 + yw

Q: 1. Find grad f for the given function below. Ax, y, z) = 5xy + 3yz – 4zx

A:

Q: The limit of f(x) = sin(x) as x approaches t is O T/x n/2

A:

Q: Assume that f'(c) = 0 and that f'(x) changes from + to - at the critical point x = c. This means…

A:

Q: Suppose that f(x, t) is the probability of getting x suc-cesses during a time interval of length t…

A:

Q: 1)Differentiate the following f(x) = x³In (3x² – 2x + 7) e sin (x) g(x) = %3D x² – 1 h(x) = log,(x +…

A: We’ll answer the 1st question since we answer only one question at a time. Please submit a new…

Q: 3. Suppose that {f(t)} = F(s) with F a differentiable function and that L{f'(t)} Exists. Then…

A:

Q: Suppose that f (x) has a continuous derivative, сп/2 f(-1/2) = 0, f(T/2) = 1/4 and f (x) dx = 3.…

A: The method of substitution is a very significant method in evaluating definite integral. The…

Q: b) Show that the function f (x) = { * x² sin ! for x E (0, 1] for x = 0 is absolutely continuous on…

A:

Q: 11. Suppose f(x) dr = 4, f(x) dx = 5, and | f(x) dx = –1. Then f(x) dx =

A: From given data: ∫03fxdx=4 .....1∫23fxdx=-1 .....2 From (1) and (2): f3-f0=4  .......3f3-f2=-1…

Q: If the function f(x) is defined on the 2 intervallo, TT) and f'(x) = -cos X (sinx) than f(x) has…

A: Given that fx is a function defined on an interval 0,π whose derivative f'x=-cosx1+sinx2 To find the…

Q: Suppose that f(x) is continuous on [0, 9] and Fx) is an anti-derivative of f. If (0)=9 and F(9)=3,…

A:

Q: (Read the dėlihition Caleluny.) |x2, if x l; is differentiable at 1. 2x-1, ifx > I; 4. Prove that…

A:

Q: 1. Evaluate the definite integral S ,2 C (9x² + 5)dx by the limit definition.

A:

Q: 1 dt, for x > 4. In(t) Let F(x) 4

A:

Q: '18 6. If f is continuous and f(x) dx = 16, find f(2x) dx. %3D

A:

Q: The tangent line to y=f(x) at (7,0) passes through the point (6,−7). Compute the following. a.)…

A: Topic:- calculus

Q: (2) Is the function f(x) : R → R, sin a 1 ifa # πh,k & Z, ifa = Tk, k ε Z f (x) Borel measurable?

A: Consider the given function, fx=1sinxifx≠πk,k∈Z1ifx=πk,k∈Z   Thus, the given function have infinite…

Q: h(x) = (5x-1 d) m(x) = (4x² – 9x} %3D

A: Product rule of differentiation: u·v' = u·v' + v·u'   ddxxn = n·xn-1 ddxf(x)n = n·f(x)n-1·f'(x)

Q: If f(x) = x' = 4x² + 10 determine the contavity when x=1. Hint: First find the second derivative O[®…

A:

Q: The Gamma Function is defined by [(x) = fx-¹e-tdt, positive integer then prove that for na T(n) = (n…

A: The given Gamma function is Γx=∫0∞tx-1e-tdt. Prove that for n a positive integer then Γn=n-1!.   As…