Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Solutions to a series of problems related to partial derivatives and differentiability, including finding domains, ranges, limits, and tangent planes, as well as applying the chain rule and the method of Lagrange multipliers. The problems also involve optimization, homogeneous functions, and the wave equation.
Typology: Assignments
1 / 14
This page cannot be seen from the preview
Don't miss anything!
- 12장 답안입니다. 자세한 답안이 아닌 학습 시 정답을 확인하기 위한 간략한 답안입니다. 시험에서는 자세한
§ 12.1. Functions of Several variables.
.
(a) evaluate ^
(b) Find a domain of (^) .
(c) Find a range of .
(d) Sketch the graph of the function .
.
§ 12.2. Limits and Continuity.
(a) lim →
.
(b) lim →
.
(c) lim →
sin
(sol1) ≤
sin
≤ sin^
since
≤ . And sin^
→ as → .
By the Squeeze theorem, (^) lim
→
sin
.
(sol2) We are enough to show that ∀ ∃ such that if ≺ ^
then
We have
≤
^
.
So, if we choose and let ≺ ^
, then ^
.
(d) (^) lim
→
cos
.
(e) (^) lim
→
lines. ⅰ) On the -axis, for
≠ , so → as → ^ along the -axis. ⅱ) Approaching ^ along the curve
gives
for ≠ , so along this path →
as → .
Since ⅰ) ≠ ⅱ), the limit does not exist. □
(f) lim
→
.
Let be given. Take so that for all , ≤ .
§ 12.3. Partial Derivatives.
(a)
(b)
cos
sin
cos
(c)
(a) ^
, (^) ^
(b)
(^)
(^)
.
≠
(^) lim →
arcsin , find .
^ is -independent. (^)
, (^)
, (^)
. By Clairaut’s
Theorem, (^)
.
is a solution of the
(It is three-dimensional Laplace equation.)
, (^)
, (^)
so that (^) (^) (^) .
if ≠
if
(a) Find (^) ^ and (^) ^ when ≠ .
,
(b) Find (^) ^ and (^) .
→
, and similarly,
(c) Show that and
→
, and similarly,
(d) Does the result of part (c) contradict Clairaut's Theorem?
Now as → ^ along the -axis,
(^) →. While as → ^ along the -axis, (^) → . Thus (^) is not continuous at
and Clairaut's Theorem is not applied. So there is no contradiction.
(^) is a solution of the differential equations
and
.
and
, we have
. Also, since
and
, we have (^)
so that (^)
.
distance at the point in (a) the direction and (b) the direction.
(b) similar to (a)
intersects the plane in an ellipse. Find the parametric
equations for the tangent line to this ellipse at the point .
at . [Hint : Show that (^) lim
→
→
(^) lim ∆ ∆ →
Since is differentiable at ,
^ ∆ ^ ∆ ^ ∆ (^) ∆ (^) ∆ (^) ∆ (^) ∆^ where (^) (^) → as
∆∆ → , Thus (^) ∆ (^) ∆
(^) Taking limit
to both sides as ∆∆ → gives (^) lim
∆ (^) ∆ →
∆ ∆ . Thus is continuous
at .
cm if the tin is cm thick. (Consider the can have top and bottom)
if ≠
if
Show that (^) ^ and (^) ^ both exist, but is not differentiable at .
^ doesn't exist (consider the limit along the x-axis and
). So, is discontinuous at and thus not differentiable at .
§ 12.5. The Chain Rule.
(a) arcsin
(b)
(c)
cos
sin ,
sin
and , find
.
dimensions are and , and and are increasing at a rate of while is
decreasing ar a rate of . 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
§ 12.6. Directional Derivatives and the Gradient Vector.
(a)
sin
sin
(b) tan^
(a) Find the gradient of .
(a) sin
(b) ^
At , ∇
ball, which we take to be the origin. The temperature at the point is
∘ .
(b) Show that at any point in the ball the direction of greatest increase in temperature is given by a
vector that points toward the origin.
(^) , and since (^) is the position vector of
,
where and are measured in meters you are standing at a point with coordinates .
The positive -axis points east and the positive -axis points north.
(a) If you walk due south, will you start to ascend or descend? At what rate?
(b) If you walk northwest will you start to ascend or descend? At what rate?
vertical meter per horizontal meter
(c) In which direction is the slope largest? What is the rate of ascent in that direction? At what
angle above horizontal does the path in that direction begin?
surface intersect in a common point.
At , the tangent plane is
Since
All tangent planes intersect in a common point .
.
and the ellipsoid
at the point .
§ 12.8. Lagrange Multiplies
given constant.
(a) (^)
. Hence, no maximum value, minimum value : (^)
(b)
is maximum value,
is minimum value.
(c) (^) (^) (^) (^) (^) (^) (^)
(^)
(^)
is maximum and
is minimum.
coordinate planes and one vertex in the plane .
perimeter is a square.
Then , ∇ ∇ ⇒ ^
⇒ and the rectangle with
maximum area is a square with side length
temperature on the ellipsoid.
is the highest temperature.
given that (^)
are positive
numbers and
, where is a constant.
, has maximum value
(b) Deduce from part (a) that if (^) (^) ⋯ (^) are positive numbers, then
. Under what circumstances are these two means equal?
,
≤
is maximum and ±
is minimum.
