Let f: [a, b] → R5 be defined by f(t) = (4t, 3 t², 2t*, e', 9). List F'(t)

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π

A: The linearization of a function f(x) at x = a is, Now differentiating  given function with respect…

Q: Find the direction at which the function f(x, y, 2) = a In(e-az) increases fastest at (-3,0, 0) O…

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Q: (c(2z+y), %3D ue of c in the joi F of W.

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Q: If f(x, y, z) = excosy + z, show that fxx + fyy + fzz = 0

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Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

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Q: Assume x and y are functions of t. Evaluate dy/dt. 12) x3+y3=9;- dx dt 13) xy + x = 12; dx dt = 4, x…

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Q: Sx, ifx 1 f(x) = Show that f is R- integrable.

A: R- integrable function

Q: Show that F(x, y) = x² + 3y is not uniformly continuous on the whole plane.

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Q: Suppose that z = x/y, where x and y are differentiable functions of u and v such that x = dx/dv 3, у…

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Q: Suppose that z = f(x, y) is differentiable at the point (1024, 64) withf, (1024, 64) = 4 and f,…

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Q: 2. Suppose that f (x, y), g(x, y) are two continuous functions on the xy plane. Is it true that So…

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Q: Which of the following is the gradient vector of the function f(x, y) = 3x ln 3y - 2x²y at point (1,…

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Q: Find the direction at which the function f(x, y, z) = x In(e – az) increases fastest at (-3,0, 0) O…

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Q: Use the definition of differentiability to show that f(x, y) = y² – xy+5x is differentiable…

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Q: Find a potential function for F or determine that F is not conservative.〈F = z, 1, x〉

A: F is conservative when curl F = 0 Given that f=(z, 1, x)=zi+j+xk Now evaluate curl F, curl F = ∇ × F…

Q: Let F(x, y) =x2+xyz+z , find at the point (1,1,1)

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Q: find a potential function for F or determine that F is not conservative. F=(z,x,y)

A: F is conservative when curl F = 0 Given that f=(z, x, y)=zi+xj+yk

Q: 2. For f(z, y) = z-y, prove that f is differentiable at the point (1, 1).

A: Given: f(x, y) = x2 - y2

Q: Let f(x, y) = (x° + ay´, xʻy°, 2x+by), g(u,v,w) = (3u² + bv² + w°, uvw). Find the derivative of…

A: f(x,y)=(x3+ay2,x2y3,2x+by)g(u,v,w)=(3u2+bv2+w3,uvw)To Find: Derivative of f(g(u,v,w)) using the…

Q: Let F(x, y) = G , y 1 )- Calculate the work against F required to move an object from (1, 1) to (3,…

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Q: Show that f(x,y)=exyx is differentiable at point (1,0) using the definition of differentiability.

A: A function is called differentiable, if the derivative of function exists at each and every point in…

Q: Compute the gradient of the function at the given point. f (x, y, z) = In(x² + 7y – 3z²) ; (7,7,7) O…

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Q: Suppose that z = f(a, y) is differentiable at the point (1024, 1024) with, (1024, 1024) = 5…

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Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

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Q: Show that the function f defined on [0, 1] by the conditions f)=2rx when <,r=1, 2, 3,. 1 1=1, 2,…

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Q: I. Show that the function ƒ defined by (1, y) = (1, –1) f(1, v) = { +g (z, y) # (1, –1) + y is not…

A: We will use the sequential characterization of continuity to solve this problem. The detailed…

Q: Let f be a function with continuous first derivatives, defined by f (cos(xy), x2 + y³) xy If we take…

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Q: Let f(r) = Vt3 +1 dt for r2 3. 3

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Q: Find g: R R, continuous in [0. ) and position in (0, ) satisfying g1) %3D )dt

A: Here we have,  g:R→R, continuous in [0,∞) and position in (0,∞) Also given g(1)=1 and…

Q: Find He valve of fe2) when fg functions such that are dilferentrable %3D g(x) >o 1 for all x, while…

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Q: Let f(z)=ex^2 = U(x,y)+iV(x,y) Verify Ux=Vy

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Q: Compute F. dr where F (x, y) : (7xy + x cos x) i+ (-yev + 6x²) j and C is a piecewise closed curve…

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Q: Let f(1,y) = (r+ay°,r²y°,2r + by), g(u, v,w) = (3u² + bv² + w°, uvw). Find the derivative of…

A: Let f(x, y)=x3+ay2, x2y3, 2x+by,  gu, v, w=3u2+bv2+w3, uvw. We need to find the derivative of fgu,…

Q: Let u be a differentiable function of x. Use the fact that ∣u∣ = √u2 to prove that

A: Given that - u = u2          ............(1)To prove - ddxu = u'uuProof - Difefrentiating equation…

Q: Find the lineanzation LlXiy) of f (iy) =x*-5Xy at (2,1). Then use the linearization to approximate f…

A: Here in this question we need to linearize the given function and then approximate the value of…

Q: Choose the gradient of the function f defined by f(x, y, z)= In[1 + (7xy + 13xz + 17yz)²]. O…

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Q: ) Compute f-(, ") and fy(§, T) of f(r, y) = e9 cos(ry)

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: is function oF x $ y xz + y In x = ze-sin (xy) D. Ausumo that z Find zx iF

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Q: Suppose that z = x + iy ∈ C and consider the function f(z) = y^2  + ix^2 Determine where f(z) is…

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Q: Use the limit to find partial derivative of…

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Q: Suppose that f: R2R is a function given by S x²+y² f(x, y) 0 for xy = 0. Show that f(x, y) is…

A: We have to show the given function  f(x, y) is continuous  at ( 0,0)

Q: Let f(r,y, z) = x² + 3x°y + z°, g(u, v) = (u² + v², uv, 3u – 2v). Find the derivative of f(g(u, v))…

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Q: Let f(x,y, z) = In(x²2+ z) + y²e*z – cos(2yz). Find fzyx·

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Q: Suppose f is differentiable at (3, 4), V f(3, 4) = (-V3, 1), /V3 and u = Compute D„f(3, 4). 2

A: ∇f3, 4=-3, 1, u=32, -12

Q: /alue of divergence of function (x2y2)î - (z³y – 4x) ĵ + 4y2 xk at (1, 2, 1) is

A: Let f = (x2y2)i^ - (z3y-4x)j^ + 4y2xk^

Q: 2. Suppose that a function f: R3 → R³ defined by: f (х, у, 2) 3 (2х + Зу — z, sin(тхy?z-1),…

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Q: that F(x, y, z)= Divergence Theoren

A: The given vector field is Fx,y,z=xz2+12.13, sin y, x·y5. The upper hemisphere can be taken as…