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Calculus 3 Review 2

This topic is a collection of problems and concepts that might help someone understand their working knowledge of Calculus 3.

Show all work and justify each step.

(1) Find parametric equations for the tangent line to the graph of

at the highest point on the graph.

    Solution. We want where and since

we find that Further,


and so

Also, and therefore, the parametric equations for the tangent line at the highest point are

(2) Find the length of the graph of the vector function

from to

    Solution. We determine and so

Therefore, which is the desired length.

(3) If show that

    Solution. Let and so then

Therefore,


(4) Suppose the system

can be solved for and in terms of and so that and Use implicit differentiation to find the partial derivatives and

    Solution. We use implicit differentiation on

to find

We have,

and so

Also,

and so

So we can solve the system

calculus 3 review 2 _gr_44.gif]

to obtain

(5) Find the direction from

in which the function

increases most rapidly and compute the magnitude of the greatest rate of increase.

Solution. We compute the gradient:

and at

we have,

Therefore,

is the magnitude of the greatest rate of increase and occurs in the direction of

(6) Find the directional derivative of

in the direction toward the origin.

Solution. We use the unit vector

and find

to compute

(7) Find the absolute extrema of

on the closed bounded set

defined by the disk

Solution. We find

and

and so

only at

To check the boundary

we let

and

for

Then

and so

Thus

when

which yields the points

respectively. Therefore,

is a minimum and

is the maximum value and

does not yield extrema values.

(8) A wire of length

is cut into three pieces that are bent to form a circle, a square, and an equilateral triangle. How should the cuts be made to minimize the sum of the total area?

Solution. Let

be the length of each side of the square,

be the length of each side of the triangle, and

the radius of the circle. We wish to optimize

subject to

and

Since

we have

with

Then

when simultaneously,

We obtain

Since

and

(9) Minimize

subject to

and

Solution. We set

and

and we use the Lagrange multipliers to set up the system

calculus 3 review 2 _gr_108.gif]

which is

to find

which leads to

as the minimize subject to

and

(10) Find the maximum and minimum values of

on the surface

Solution. We set

and we use the Lagrange multiplier

and solve the system

calculus 3 review 2 _gr_119.gif]

which is

to find

Therefore,

calculus 3 review 2 _gr_122.gif]

is the minimum and

calculus 3 review 2 _gr_123.gif]

is the maximum value.

(11) Use iterated integration to compute

calculus 3 review 2 _gr_125.gif]

where

Solution. By Fubini's theorem,

calculus 3 review 2 _gr_127.gif]

(12) Find the volume of the solid bounded below by the rectangle

in the

-plane and above by the graph of

Solution. The volume is given by the following double integral and can be compute using Fubini's theorem.

calculus 3 review 2 _gr_132.gif]

(13) Sketch the region of integration and write an equivalent integral with the order of integration reversed given

Solution. We graph the region:

calculus 3 review 2 _gr_139.gif]

The iterated integral

is with the region as a vertically simple region. Also,

and so

(14) Find the volume of the solid bounded above by the paraboloid

and below by the plane

Solution. The projected region in the

-plane is

which is symmetric about both the

-axis and the

-axis. Thus,

Cite this as:
Calculus 3 Review 2
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/calculus-3-review-2.html

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