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- If (x, y) moves from (x0, y0) to a point (x0 + dx, y0 + dy) nearby, how can you estimate the resulting change in the value of a dif-ferentiable function ƒ(x, y)? Give an example.This is a practice question from my Multivariable Calculus Course. It wants me to use maximization to figure it out, starting with xy + 2xz + 2yz = 192 and using partial derivatives, etc… Thank you.The discriminant ƒxx ƒyy - ƒxy 2 is zero at the origin for each ofthe following functions, so the Second Derivative Test fails there.Determine whether the function has a maximum, a minimum, orneither at the origin by imagining what the surface z = ƒ(x, y)looks like. Describe your reasoning in each case. ƒ(x, y) = x2y2
- The discriminant ƒxx ƒyy - ƒxy 2 is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface z = ƒ(x, y) looks like. Describe your reasoning in each case. a. ƒ(x, y) = x2y2 b. ƒ(x, y) = xy2 c. ƒ(x, y) = x3y3 d. ƒ(x, y) = 1 - x2y2 e. ƒ(x, y) = x3y2 f. ƒ(x, y) =x4y4Find the linearization of the function f(x,y) = ln(x3y6) at the point (x0,y0) = (4, 3). Leave all terms in exact form (do not use decimal approximations).Let f : I → J be a bijective differentiable function froman interval I to an interval J, and let F : I → R be an anti-derivative of f. Find an explicit expression, in terms of f, f-1 and F, for an anti-derivative of f-1: J → I. [Both substitution and integration by parts may come in handy.]
- Show that the function f(z)=sin x cosh y+i cosx sinh y is continious as well as analytic every where.What can you conclude about a real function f(x,y) that has continuous partial derivatives at a point (x,y)?The temperature in degrees centigrade (◦C) at each point (x, y) on a curve 2x + y = 3 is given by T(x, y) = 2x2 + 2x + 5y.Find the lowest temperature on the curve using:(a) Second Derivative Test(b) Lagrange Multipliers
- Show that the function is differentiable by finding values of ε1 and ε2 as designated in the definition of differentiability f(x, y) = x2y , and verify that both ε1 and ε2 approach 0 as (∆x, ∆y)→(0, 0).Demonstrate the use of the method with reflections on the use of numerical methods, find the minimum for the function below: F(x,y)= Ax^2 - Bxy- cy^2= x -y (Xo=4, Yo=4) A=2 B=-2 C=1 Identify the minimum again using the Newton’s method with dynamic . However, use this time numerical derivatives instead of . When using numerical derivatives, only one of the constants is being varied as with partial derivatives. Apply in this case the forward numerical derivative, . Here equals some very small number. For each step , solve first the and optimal using the condition . When taking the derivative of , please remember to consider the inner derivatives for each of the coordinate axes that results as dot product with the main function. In this work it is enough that only the second term in the dot product is analyzed using numerical derivatives. Thus, the function takes the form .Find the firsst derivative of y = arctan [ (3 sin x) / (4 + 5 cos x) ] and the first partial derivative fx, fy, and fz of (x + yw )2 – ez – 2 = 0 and f ( x , y, z ) = 2xyz + tan x + y sin z Make the solution detailed and clear IN A PAPER.