Calculus Answers

Nathan is designing a box to keep his pet newt in. To make the box, he is going to start with a solid rectangle and cut squares with sides x cm in length from each corner. The dimensions of the solid rectangle are 51 cm by 45 cm.

a) Determine an equation for a function that models the volume of the box
b) Determine what dimensions can be used to create a box with a volume equal to 7750cm^3. Round final answers to one decimal place
c) Explain why not all of the solutions to the equation could be possible lengths of the squares that Nathan is going to cut out of the rectangle?

Show your work clearly for full marks

Find the absolute maximum and absolute minimum of the function f(x,y)=2x^3+y^4 on the region {(x,y)|x^2+y^2≤4}.

If the absolute max or min occurs at multiple points list them all, separated by commas.

Absolute minimum value: ___

attained at ___.

Absolute maximum value: ___

attained at ___.

The function f(x,y)=xy(1−6x−5y) has 4 critical points. List them and select the type of critical point.

Points should be entered as ordered pairs and listed in increasing lexicographic order. By that we mean that (x,y) comes before (z,w) if x<z or if x=z and y<w.

First point ___of type ____

Second point___ of type ___

Third point ___ of type ___

Fourth point ___ of type___

Consider the function
f(x,y)=(2x−x^2)(8y−y^2).
Find and classify all critical points of the function.
First list out all the first and second partial deriatives.
There are several critical points to be listed. List them lexicograhically, that is in ascending order by x-coordinates, and for equal x-coordinates in ascending order by y-coordinates (e.g., (1,1), (2, -1), (2, 3) is a correct order)

A student was asked to find the equation of the tangent plane to the surface z=x^4−y^5 at the point (x,y)=(4,1). The student's answer was z=255+4x^3 (x−4)−(5y^4)(y−1).

(a) At a glance, how do you know this is wrong. What mistakes did the student make? Select all that apply.

A. The answer is not a linear function.
B. The partial derivatives were not evaluated a the point.
C. The 255 should not be in the answer.
D. The (x - 4) and (y - 1) should be x and y.
E. All of the above

(b) Find the correct equation for the tangent plane.
z=

Suppose that f(x,y) is a smooth function and that its partial derivatives have the values, fx(2,8)=−4 and fy(2,8)=−5. Given that f(2,8)=6, use this information to estimate the value of f(3,9). Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation.
Estimate of (integer value) f(2,9):

Estimate of (integer value) f(3,8):

Estimate of (integer value) f(3,9):

If f(x,y,z)=xe^(2y)sin(9z), then the gradient is

∇f(x,y,z)=

Find the directional derivative of f(x,y,z)=z^3−x^(2)y at the point (-3, -4, 1) in the direction of the vector v=⟨4,−3,1⟩.

Suppose f(x,y)=1x^2+3xy−2y^2, P=(3,2), and u=(−5/13,12/13).

A. Compute the gradient of f.
∇f=___i+___j
Note: Your answers should be expressions of x and y; e.g. "3x - 4y"

B. Evaluate the gradient at the point P.
(∇f)(3,2)=___i+___j
Note: Your answers should be numbers

C. Compute the directional derivative of f at P in the direction u .
(Duf)(P)=

Note: Your answer should be a number

Find the absolute minimum of the function f(x, y) = 3 +xy-x-2yon the closed triangular region with vertices (1,0), (5,0), and (1,4).