Let f(x, y, z) = 3x?y+ yz3, where x = 3 – t2, y = 2t + t3, and z = V1+ 3t. Calculate when t = 1.

Q: Let f (x, y) = 3x2 + 2xy + 5y. (a) Calculate df/dx (1, 4). (b) Evaluate df/dy at (x, y) = (1, 4).

A: Differentiate the function f with respect to x treating y as a constant to calculate ∂f∂x.

Q: Use the linear approximation to f (x, y) = √x/y at (9, 4) to estimate √9.1/3.9.

A: f(x,y)=xy at (9,4) to estimate 9.13.9

Q: Let f (x, y) = 2x3 + x2y - y2. Compute d2f/dx2, d2f/dy2, and d2f/dxdy at (x, y) = (1, 2).

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Q: 2. Show that the function u = e(xy) + VrY 4(y/x) provides the equation Pu = 0. dy? %3D -

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Q: Let f(z) = 5z2 + 4z. Find u(x,y) if z = 8.17 + 3.6i. %3D Round off your final answer in five decimal…

A: Please find the attachment for the details solution

Q: Let f(x, y) = . Use the Linear Approximation at an appropriate point (a, b) to estimate…

A: Given,f(x,y) = x2y2+1To estimate,f(4.02,0.93) using the LinearApproximation at an appropriate point…

Q: Find V x F for F(x, y, z) = -IZ -yz √₁² + y²² + √²² +2²² =j + √√x² + y²k.

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Q: Find fryz from the functions f(x, y, z) = 4x³y² – e²y* ++ 4y – x16 %3D - |

A: To find the third order partial derivative of the function fxyz f(x,y,z)=4x3y2-ezy4+z3x2+4y-x16

Q: Find the linear approximation of the function f (x, y, z) (3, 2, 6) and use it to approximate the…

A: Given that  f(x,y,z)=√(x^2+y^2+z^2) To find the linear approximation of f(x,y,z) and also find the…

Q: Let f(x, y) = x2 /(y2 + 1). Use the Linear Approximation at an appropriate point (a, b) to estimate…

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Q: Evaluate fSS 6e(²+y²+z²)/² dV over B where B= {(x, y, 2) : x² + y² + z² < 1}. Take T = 3.

A: Since we have    x2 + y2 + z2  expression in the integral and region B is the sphere of radius 1…

Q: Use the Linear Approximation to f(x, y) = x/y at (9, 4) to estimate 9.1/3.9.

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Q: Let F(z, v)-v +y+3z+1-0

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Q: integrate ƒ over the given curve. ƒ(x, y) = x3&gt;y, C: y = x2/2, 0 &lt;=x&lt;= 2.

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Q: zl*dz, where C is x(t) t2, y(t) , t:2 1 Evaluate /=

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

Q: Find a linear approximation to the function f(x,y) at the indicated point. 2x y f(x,y) = at (1,2) 1…

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

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Q: Suppose that a = x(t) and y = y(t) are both functions of t. If y + xy – 3x = 23, and dy/dt = 1 when…

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Q: Find fryz from the functions f (x, y, z) = 4x³y² – e²y+ ++ 4y – x16

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Q: Suppose that f(x, y) : at which {(x, y) | 0 < x < 4, – x < ys va}. 1+ x D

A: Consider a given function:  The objective is to evaluate the following double integral of fx, y…

Q: Evaluate f, xyz dV, where Q = {(x, y, z): 0 < x < 2y, 0 <y< 1, 0 < z < x+y}.

A: According to question,Q = x,y,z : 0 &lt; x &lt; 2y, 0 &lt; y &lt; 1, 0 &lt; z  &lt; x+y

Q: 4. Suppose that z = x³y, x= t², and y = t³, find . dt = 12t11 dt 11t10 b) = 10t° dt d) = 9t® dt

A: This question is related to differentiation, We will solve it using given information.

Q: Find the linear approximation to f(x, y, z) = at the point (2, –1, –2): f(r, y, z) z

A: A Givenfx,y,z=xyz at the point 2,-1,-2linear approximation at the point…

Q: Show that the function y(x, dy/at = a2y/ar2 = dy/ax = %3D %3!

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

Q: 2. Show that the function u = p(xy) + Vry ý(y/x) provides the equation n, = 0. dy? U -

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Q: Q4: Given: d²y =[f(x-t)dt -show that = f (x)· dx? y %3D .2

A: Differentiation of integration

Q: Find a linear approximation to the function f(x,y) at the indicated point. 2x°y f(x,y) = at (5,1) 1…

A: Linear approximation to the function f(x ,y) at (a, b) :       L(x ,y) = f(a ,b) +(x-a) f x (a, b) +…

Q: Find the linear approximation to f(x, y, z) = " at the point (2, -1,1): f(x, y, z) z

A: Given the function

Q: Compute the discriminant D(x, y) of the function. S(x, y) = x' + y* – 6x – 2y + 1

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Q: Find the local linear approximation to f(x, y) = 4x² + y² at P(2,3)

A: We have to find the linear approximation of above function at (2,3) please see the explanation…

Q: Suppose that w = xe"/, where x = t², y = 1 – 6t, and z =1+ 4t. dw Find dt

A: w=xeyz, where x=t2 , y=1-6t and z=1+4t

Q: Find fryz from the functions f (x, y, z) = 4x³y² – e²y* + + 4y – x16

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Q: Let f (x, y) = 2x³ + 4y³ + 19xy Find D = frx · fyy - fey at the point (2,2)

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Q: f(x,y)=x2+y3, where x=t2+t3,y=t3+t9find df/dt at t=1.

A: Topic:- application of derivatives

Q: Show that the function u = p(xy) + VrY v(y/x) provides the uation x2 = 0. dy?

A: First find the partial derivative of the given function with  respect to x :

Q: Suppose that f(x, y) = e 42? - ly + 7z + 2y then find the discriminant of f.

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Q: Use the linear approximation of f(æ, y) = e+y at (0, 0) to estimate f(-0.03, -0.03)

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Q: F(x) = F' (2) , where (t4 - 3t2 )dt %3D d. Vt dt x2 dx

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Q: Assume that w= f(s° +t*) and f'(x) = e*. Find and dt D I3D 次咖一。

A: Finding the expression of w and proceed the differentiation

Q: Find the differential of f(x, y) = vx2 + y³ at the point (1,2). df

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Q: ) Find the direction in which f (x, y) = x²+xyincreases most rapidly at P(1, –1).

A: 1.) given multivariable function is, fx,y=x2+xy and point is P(1,-1). Divergence: Divergence of the…

Q: Use the Chain Rule to Find dz/dt, where z=2xsiny, x=t^5, and y=3t^3.

A: Chain rule  dz/dt=(dz/dx)(dx/dt)+(dz/dy)(dy/dt)

Q: find the potential function for F= (6x+5y)i+(5x+4y)j Select one: а. f(x,y) = 3x² +5xy + 3y? %3D b.…

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Q: Find the image of the line z =3 +iy under f(z) = e

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Q: ) Find dw if w = xe³/², x = t², y = 1 — t, and z = dt 1 + 2t.

A: Topic:- application of derivatives

Q: Let f(x,y) = 5x² + 6y² + 3xy – 5x – 8y + 36 What is the first coordinate of the critical point?

A: Given data:   The given expression of the function is f(x,y)=5x2+6y2+3xy-5x-8y+36.     Equate the…

Q: f(x, y) = /32² + 3y fz(4, 5) = VT23 %3D

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Q: X+Z+3Siny X +2y-2

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Q: dz dt || z = cos(x + 3y), x = 8t³, y = 9/t

A: By the use of chain rule we need to find the required derivative.