Let (t) < v2-1. , et In(t+ 1)>, a) Find the domain of F(t) b) Find " (t)

Q: b) Let f be a differentiable function of one variable, and let z = f(3x² – 5y²). Show that az əz…

A: Given  z = 3x2 - 5y2

Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

A: Given that,  fx, y=sin15x2+3y2 The quadratic approximation of the function fx, y at the point x0, y0…

Q: : Apply Rolle's theorem for the function f(z) = 7x² – 28z+ 21 over the interval [1,3]. er.

A: The given function is, f(x)=7x2-28x+21 over the interval [1,3]. Observe that, the given function…

Q: → [0, ∞) defined by f (x) = Vx + 2. Prove that f is 17. Consider the function f: (-2, 0) surjective.

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Q: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

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Q: Verify that fxy = fyx for the following funtion.  f(x,y) = ex + y + 1 fxy = ? fyx = ?

A: Given,            fx,y=ex+y+1

Q: Let f(x)= V x³ at the interval of [0,2] ; then , use the mean valune theorem to find c.

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Q: what is the quadratic polynomial for the function f(x,y) sin(xy) + cos(x/y) at (pie/4, 1)

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Q: Show that the function f(x, y) x4 + y? has no limit as (x, y) approaches (0,0).

A: Note: Since we can solve at most one problem at a time, we have solved the first one. Kindly…

Q: Define the function f(x, v) = 4x³-x³y-4y5 3x5-3y5 at the origin so that it becomes continuous there.

A: A function is said to be continuous at (a,b) if the the value of function at (a,b) is equal to…

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) = 1 + t3 dt

A: To find F'x Fx=∫0xt21+t3dt

Q: Find linearization L(x) of f (x) = Vx at a =4 and use to approximate 7.8.

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

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Q: Show that the function is differentiable by finding values of ε1 and ε2 as designated in the…

A: Given, the function f(x, y) = 5x – 10y + y3, we have to show that the function is differential by…

Q: Find the linearization L(x) of f (x) = tan x at a =

A:

Q: (5) Find the most general function f(t) so that the curve F(t) =, shall lie in a plane.

A: The most general function for f(t) is a constant function, f(t) = c:   &lt;a cost, a sint, c&gt;

Q: Show that the function is differentiable by finding values of ε1 and ε2 as designated in the…

A: Given  f(x,y)=x2y

Q: lim, - f{x)= 0, then f(x)dx diverges Select one: O True False

A: When the limit exist or equal to 0 then the series is convergent otherwise divergent.

Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f(x) = 2x 2 +…

A: Given:                fx=2x2+10x,     a=3 Find: f3+h and f3 f3+h=23+h2+103+h…

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Let A(x) f(t)dt, with f(x) shown in the graph below. -3- ** * T?

A: As per the fundamental theorem of calculus, given a function A(x)=∫0xf(t) dt, the derivative of this…

Q: Show that f(x) = 2a + 3b is continuous, where a and b are constants

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Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A:

Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

Q: Find the linearization of the function at a . f(x) = sin x    , a = pie /6

A: The given function fx=sinx, a=π6

Q: Integrate f(x) from x = 0 to 10 using Three-Point Gauss-Legendre Quadrature. f(x) = 5000+24563x -…

A: Given f(x)=5000+24563x-7418x2+768x3-31x4+0.14x5 Let I=∫010f(x)dx Here a=0 and b=10

Q: Let f(x, y) = ln (3x + y2). Show that fxy = fyx

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Q: For f(z) = 2z', find ip using the definition,

A: The limit definition of a derivative of a function, f(x) at a point, x=a is given by .

Q: Let f(x, y) = − sin2 (2x − y) Evaluate fx and fy at (π / 4, π / 3)

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Q: Find the global maximum and global minimum of the function f(x)=sinx−x in the interval [−π,π].

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Q: Coo show that the f(x,y)=X+4xy-3y gdegrad is homogenaow of

A: Concept used: The function is homogeneous of degree n if fkx,ky=knfx,y. Given: fx,y=x3+xy2-3y3…

Q: Find the best quadratic approximation to f on the interval [ -1, 1].                     f(x) = lxl

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Q: GrRve the domnin f n(x)= -9-4x|usq iaeoul.ntaton

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Q: Q2 (A) At what points f(x, y, z) in which the function f(x, y, z) = continuous? is

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Q: lim, - „f(x) = 0, then fx)dx diverges Select one: O True False

A: The given statement is

Q: Use the Second Fundamental Theorem of Calculus to find F '(x) F(x) |Vt4 + 1 dt -7

A: 2nd Fundamental Theorem : For F(x)=∫axf(t) dt, if f is continues on [a, x], then F'(x)=f(x).

Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function -5x f(1) = | V1+ tª…

A: To differentiate the function by using first fundamental theorem of calculus f(x)=integration (x to…

Q: Consider the function f: (-2,0) → [0, 0) defined by f (x) = Vx + 2. Prove that f is suriective.

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Q: A function y(t) satisfies th

A: Given, the differential equation                       dydt=-y4-3y3+4y2

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: It is given that,Fx=∫-2xt2-2tdt We have to use the Second Fundamental Theorem of Calculus to find…

Q: Show that f (x, y) = has no limit as (x, y)→(0,0) Vx2+y2

A: The objective is to show that f(x,y) has no limit as (x,y) approaches to (0,0).

Q: 2. Use the Intermediate Value Theorem to prove that the equation e* = 4 - x³is solvable on the…

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Q: Estimate the global maximum value for the function F on the interval [-2,1] if F(x)= || dt . t2 +1

A: The given function is: F(x)=∫-2x tt2+1 dt

Q: Let f(z) = 1 Find all the points where f(z) %3D Z + 4 is analytic.

A:

Q: 4. Let F : R² → R² defined by F(x,y) = (2x +3y, –x +y). Is F a linear? Why?

A: Given

Q: Determine local extremum f(x,g) = xy(4-x-). %3D

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Q: what is the limit of f(t) as t approaches 3. F(t)= ((1/t)-1/3)/t-3?

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Q: Suppose that a function f(x, y) satisfies fxx fyy – (fxy)² > 0 at a point (a, b). How is the sign of…

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