1) Find all of the second partial derivatives of f(x,y)=3(x^4)y-2xy+5x(y^3).

2) Find an equation of the tangent plane to z=32-3(x^2)-4(y^2) at the point (2,1,16).

Ant on a metal plate. The temperature at a point ( x, y) on a metal plate is T(x, y) = 4x² - 4xy + y² . An ant on the plate walks around the circle of radius 10 centered at the origin.

a) What are the highest and lowest temperatures encountered by the ant?

b) Suppose the ant has changed its trajectory and is walking around the circle of radius 5. Is the highest temperature encountered by the ant greater or less compared to the one in part a)?

Find the curvature of the parametrized curve ~r(t) =< 2t 2 , 4 + t, −t 2 >.

Graph the polynomial function​f(x)=8x⁵+ 52x⁴+80x³ and determine the end behavior of the graph of the function.

Explain the concept of optimization. Explain the general procedure used to do an optimization problem. Use an example and include steps

For the function f(x,y) = 4xy - x^3 - 2y^2 find and label any relative extrema or saddle points. Use the D test to classify. Give your answers in (x,y,z) form. Use factions, not decimals.

Use spherical coordinates to evaluate the triple integral ∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where E is the region bounded by the spheres x^2 + y^2 + z^2 = 4 and x^2 + y^2 + z^2 = 9

A mass weighing 8 pounds stretches a spring 4 feet. The medium through the mass moves offers a damping force numerically equal to sqrt(2) times the instantaneous velocity. Find the equation of motion if mass is released from equilibrium position with a downward velocity of 5 ft/s. Find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant? The acceleration of gravity is g = 32 ft/s²

The plane

y + z = 7

intersects the cylinder

x² + y² = 41

in an ellipse. Find parametric equations for the tangent line to this ellipse at the point

(4, 5, 2).

(Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)

Refer to f(x)=1/x^2

1)Give an upper bound for the error you get when using the fourth degree Taylor polynomial centered at c = 2 to approximate f(2.1)

2)What is the actual error of your approximation? (Use a calculator.)

(x+5y)^21 expand using Pascal's triangle

The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 4 m and w = h = 1 m, and ℓ and w are increasing at a rate of 8 m/s while h is decreasing at a rate of 2 m/s. At that instant find the rates at which the following quantities are changing.

(a) The volume.

(b) The surface area.

(c) The length of a diagonal. (Round your answer to two decimal places.)

(b)For the following IVP, find the solution for x>0,

.                           

   y'''=1+3x2-4x3+9√x .                                     

               y1=8          

               y'1=9

               y''1=10              

(c) Test for exactness and solve the IVP,

(x+5exy)dx=-exdy

                            y0=-2                                         

Use the formulas for arc length and surface area and the equation x^2+y^2=R^2

(a)... to derive the formula for the surface area of a sphere with radius R

(b)... to derive the formula for the circumference of a circle with radius R

A company manufactures and sells x television sets per month. The monthly cost equation is?(?)=??,???+???and the monthly demand−price equation is ?=???−???Use thisinformation to answerquestions below.

1.a.What is the maximum monthly revenue?

b.What price should the company charge for each television set?

2.a.How many television sets should be produced to realize the maximum monthly profit?

b.What is the maximum monthly profit