Section 11.1 Section 11.2

Jim Lambers
MAT 280
Summer Session 2012-13
Homework Set 1
This homework set is due on Friday, June 21. Except where otherwise noted, each problem is worth
5 points.
Section 11.1
1. Find the domain and range of the function f (x, y, z) =
p
xyz 2 .
2. Find the domain and range of the function f (x, y) = ln |x2 + y 2 − 4|.
3. Let f (x, y) = y + x2 . Sketch the level curves f (x, y) = k, for k = 0, 1, 2, 3, in the xy-plane.
Use these level curves to sketch the graph of f in 3-D space.
4. Describe the level curves of the function f (x, y) = ln |x2 + 4y 2 |. Use this description to sketch
the graph of f in 3-D space.
5. Sketch two sections of the function f (x, y) = |x| + |y| in 3-D space, and at least two level
curves in the xy-plane. Use this information to sketch the graph of f in 3-D space.
6. Repeat the previous problem with f (x, y) = |x + y|.
Section 11.2
1. Compute
xy 2
,
(x,y)→(0,0) x4 + y 3
lim
if it exists.
2. Compute
xy 2
,
(x,y)→(0,0) x3 + y 3
lim
if it exists.
3. Compute
xy 2
,
(x,y)→(0,0) x2 + y 2
lim
if it exists.
1
4. Suppose that f (x, y) is a rational function, meaning that it has the form
f (x, y) =
p(x, y)
,
q(x, y)
where p(x, y) and q(x, y) are both polynomials. Furthermore, suppose that p(0, 0) = q(0, 0) =
0. Explain how you can use the exponents of the terms in p(x, y) and q(x, y) to make an
educated guess as to whether f (x, y) has a limit as (x, y) → (0, 0), or, if it does not, why it
does not.
p
5. Show that the function f (x, y, z) = x2 + y 2 + z 2 is continuous on all of R3 .
6. Recall that the triple product of the vectors u, v and w in R3 is defined by the following
function f : R9 → R:
f (u, v, w) = u · (v × w),
where × denotes the cross product, and · denotes the dot product. Show that this function is
continuous on all of R9 .
7. Show that the vector-valued function f : R2 → R3 , defined by
D
v
u
v
u
v
uE
f (u, v) = 1 + cos
cos u, 1 + cos
sin u, sin
,
2
2
2
2
2
2
is continuous on all of R2 .
√
8. Determine where the function f (x, y) = tan( xy) is continuous.
9. Determine where the function defined by
(
f (x, y) =
(1+x)2 −(1−y)2
x+y
2 + 2x
x 6= −y
x = −y
is continuous.
10. Suppose that f : R2 → R is continuous on all of R2 . Show that |f | is also continuous on all
of R2 . Hint: consider the cases of f (x, y) > 0, f (x, y) < 0, and f (x, y) = 0 separately.
11. Compute the following limits, or explain why they do not exist. If the limit does exist, it
must be proven using the definition of a limit and/or limit laws.
(a)
x2 y + y 2
(x,y)→(0,0) 2x2 + y 2
lim
(b)
x4 + 2x3 y + x2 y 2
x2 + y 2
(x,y)→(0,0)
lim
2
Section 11.3
1. For each of the following functions, compute all first partial derivatives.
(a)
f (x, y) = sec x tan y + ln | sec x + tan y|
(b)
v
u
f (u, v) = 1 + cos
cos u
2
2
(c)
√
f (x, y, z) = 2
x2 +y 2 +z 2
2. For each of the following functions, determine whether the function is increasing as a function
of x, and as a function of y, when x = −1 and y = 1.
(a)
f (x, y) = xy + y 2
(b)
f (x, y) =
sin(πx) cos(πy)
p
x2 + y 2
(c)
f (x, y) =
ex + ey
ex − ey
3. (10 points) For each of the functions from Problem 2, compute all second partial derivatives.
4. Given a function z = f (x, y), what can the second partial derivatives tell us about the graph
of a function?
5. Use implicit differentiation to compute wx , wy and wz , where w = f (x, y, z) is implicitly
described by the equation
sin(wx) cos(wy) = ln(w2 + z 2 ).
6. (10 points) Compute all second partial derivatives of the following functions. Be sure to use
Clairaut’s Theorem wherever applicable.
(a)
f (x, y) = cos2 x sin2 y
(b)
f (x, y, t) = e−(x
3
2 +y 2 )/t
7. Given the equation
xyz = ln |x2 + y 2 + z 2 |,
compute the first partial derivatives of z with respect to x and y.
Section 11.4
1. Compute the equation of the tangent plane of f (x, y) = ex cos y + e−x sin y at (x0 , y0 ) =
(−1, π/2). Then, use the linearization of f at this point to approximate the value of f (−1.1, 1.6).
How accurate is this approximation?
2. Compute the equation of the tangent space of f (x, y, z) = 12 ln |(x − 1)2 + (y + 2)2 + z 2 | at
(x0 , y0 , z0 ) = (1, 1, 1). Then, use the linearization of f at this point to approximate the value
of f (1.01, 0.99, 1.05). How accurate is this approximation?
3. Let f (x, y) = x2 + y 2 . Use the definition of differentiability to show that this function is
differentiable at (x0 , y0 ) = (1, 1).
4. Suppose that the coordinates of two points (x1 , y1 ) = (2, −3) and (x2 , y2 ) = (7, −5) are
obtained by measurements, for which the maximum error in each is 0.01. Estimate the
maximum error in the distance between the two points.
5. Consider the function f (x, y) = ex+y . Let (x0 , y0 ) = (1, 1).
(a) Compute the equation of the tangent plane of f at (x0 , y0 ).
(b) Compute the linearization of f at (x0 , y0 ).
(c) Use the linearization of f at (x0 , y0 ) to estimate f (0.9, 1.1).
6. Consider a window pane with the shape shown in the figure below. Suppose that according
to measurements, h = 30 inches. and w = 10 inches. Assuming these measurements can be
off by at most 0.1 inches, estimate the maximum possible error in the calculation of the area
of the pane.
Section 11.5
1. For each of the following pairs of functions f and g, use the Chain Rule to compute all of the
first partial derivatives of f ◦ g with respect to the independent variables of g.
(a)
f (x, y, z) =
p
x2 + y 2 + z 2 ,
4
g(t) = het cos t, et sin t, tet i
Figure 1: Window pane for problem 6, Section 11.4
(b)
f (x, y) = ln
p
x2 + y 2 ,
g(u, v) = hu cos v, u sin vi
(c)
f (x, y) = hx3 + 3y 2 , y 3 − 3x2 i,
g(t) = hcos t, sin ti
(d)
f (u, v) =
D
1+
v
u
v
u
v
uE
cos
cos u, 1 + cos
sin u, sin
,
2
2
2
2
2
2
g(x, y) = hx2 y, xy 2 i
2. Compute the indicated derivatives or partial derivatives of the functions that are implicitly
defined by the following equations. Then obtain the equation of the tangent line (or tangent
plane) to the curve (surface) at the indicated point.
5
√
(a) F (x, y) = (x2 + y 2 )2 − 2(x2 − y 2 ) = 0, dy/dx, (x0 , y0 ) = ( 3/2, 1/2)
p
(b) F (x, y, z) = x2 + y 2 + z 2 − 8 x2 + y 2 + 15 = 0, ∂z/∂x, ∂z/∂y, (x0 , y0 , z0 ) = (4, 0, 1)
3. Given
w = f (x, y) = ex
2 +y 2
and
g(u, v) = hx(u, v), y(u, v)i = hu cos v, u sin vi,
compute the first partial derivatives of w with respect to u and v (that is, the first partial
derivatives of f ◦ g).
Section 11.6
1. Compute the directional derivatives of each function at the indicated point, in the direction
of the given vector.
√
√
√
(a) f (x, y, z) = x2 yz 3 + x3 y 2 z, (x0 , y0 , z0 ) = (1, −1, 2), u = h3/ 50, 4/ 50, 5/ 50i
(b) f (x, y) = 4x2 + 9y 2 , (x0 , y0 ) = (3, 2), u is the unit vector in the xy-plane that makes the
angle θ = π/6 with the positive x-axis
2. For each of the following functions, compute the direction along which the function increases
most rapidly from the given point.
(a) f (x, y) = x2 + y 2 , (x0 , y0 ) = (1, 4)
(b) f (x, y, z) = ez cos x sin y, (x0 , y0 , z0 ) = (π/6, π/6, 1)
3. For each of the following implicitly defined surfaces, compute the equations of the tangent
plane and normal line at the indicated point.
(a) F (x, y, z) = x2 + y 2 + z 2 − 9 = 0, (x0 , y0 , z0 ) = (2, −1, 2)
p
p
√
(b) F (x, y, z) = x2 + y 2 + z 2 + 16 − 8 x2 + y 2 − s2 = 0, (x0 , y0 , z0 ) = (1, 4, 32 − 8 17)
4. Let f (x, y, z) = xe−y + ye−z + ze−x and let (x0 , y0 , z0 ) = (0, 1, −1).
(a) Given u =
√1 h1, −2, 3i,
14
compute Du f (x0 , y0 , z0 ).
(b) In which direction u is Du f (x0 , y0 , z0 ) maximized, and what is the value of Du (x0 , y0 , z0 )
for this direction?
5. Consider the surface that is implicitly defined by the equation
xy + xz = 4 tan−1 (yz).
Compute the equation of the tangent plane and normal line of the surface at (x0 , y0 , z0 ) =
(π/2, 1, 1).
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Section 11.7
1. Find the local maximum and minimum values, as well as saddle points, of the following
functions.
(a) f (x, y) = x3 + y 3 − 3xy
(b) g(x, y) = x2 + y + 1/(x2 y)
2. Find the absolute maximum and minimum values of the following functions on the indicated
domains.
(a) f (x, y) = x3 + y 3 + 3x2 − 6y 2 , |x| ≤ 1, |y| ≤ 1
(b) f (x, y) = x2 − 2y 2 − 4x + 5y, on the triangle with vertices (−2, 0), (2, 0) and (0, 4)
3. Find the local maximum and minimum values and saddle points of the function
f (x, y) = x3 + 3xy 2 + 4x2 + y 2 .
4. Find the local maximum and minimum values and saddle points of the function
f (x, y) = x3 − 6xy + 8y 3 .
Section 11.8
1. (10 points) Find the maximum and minimum values of the function f (x, y, z) = xyz, subject
to the constraints x2 + y 2 + z 2 = 3.
2. (10 points) Find the points on the intersection of the cone z 2 = 4x2 + 4y 2 and the plane
2x + 4z = 5 that are closest, and furthest away, from the origin.
3. (10 points) Find the absolute maximum and minimum values of
f (x, y) = 8x2 + 4xy + 5y 2
on the region
x2 + y 2 ≤ 9.
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