Calculo 2 De dos variables_9na Edición - Ron Larson

En los ejercicios 1 a 12, hallar la derivada direccional de la función
en en dirección de v.
En los ejercicios 13 a 16, hallar la derivada direccional de la función
en dirección de
En los ejercicios 17 a 20, hallar la derivada direccional de la función
en
en dirección de
En los ejercicios 21 a 26, hallar el gradiente de la función en el
punto dado.
En los ejercicios 27 a 30, utilizar el gradiente para hallar la
derivada direccional de la función en
en la dirección de
En los ejercicios 31 a 40, hallar el gradiente de la función y el
valor máximo de la derivada direccional en el punto dado.
En los ejercicios 41 a 46, utilizar la función
41. Dibujar la gráfica de en el primer octante y marcar el punto
(3, 2, 1) sobre la superficie.
42. Hallar donde usando cada
valor dado de q.
a) b)
c) d)
43. Hallar donde usando cada vector v dado.
a)
b)
c) es el vector que va de a
d) es el vector que va de a
44. Hallar
45. Hallar el valor máximo de la derivada direccional en (3, 2).
46. Hallar un vector unitario de u ortogonal a y calcular
Analizar el significado geométrico del resultado.
D u f3, 2.
f3, 2
fx, y.
4, 5.
3, 2
v
2, 6.
1, 2
v
v 3i 4j
v i j
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
hx, y y cos x y
2, 4
hx, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe yx ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
gx, y xe y ,
3
fx, y sen 2x y ,
6
fx, y
y
x y , 4
fx, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1, v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2, v
1
2 i 3j
f x, y
xy,
P 4, 3 , v
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
f x, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
D u f3, 2,
6
4
3
2
3
4
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
hx, y y cos x y
2, 4
hx, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe yx ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
gx, y xe y ,
3
fx, y sen 2x y ,
6
fx, y
y
x y , 4
fx, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1, v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2, v
1
2 i 3j
f x, y
xy,
P 4, 3 , v
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
f x, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
D u f 3, 2,
f
fx, y 3 x 3 y 2 .
Q.
P
Q.
P
u cos
i sin
j.
P
13.6 Ejercicios
942 CAPÍTULO 13 Funciones de varias variables
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
hx, y y cos x y
2, 4
hx, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe yx ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
gx, y xe y ,
3
fx, y sen 2x y ,
6
fx, y
y
x y , 4
fx, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1 , v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2 , v
1
f x, y xy, P 4, 3 , v 2 i 3j
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
f x, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
hx, y y cos x y
2, 4
hx, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe yx ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
g x, y xe y ,
3
f x, y sen 2x y ,
6
f x, y
y
x y , 4
f x, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1, v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2, v
1
2 i 3j
f x, y
xy,
P 4, 3 , v
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
f x, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
hx, y y cos x y
2, 4
hx, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe yx ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
gx, y xe y ,
3
fx, y sen 2x y ,
6
fx, y
y
x y , 4
fx, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1, v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2, v
1
2 i 3j
f x, y
xy,
P 4, 3 , v
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
fx, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
hx, y y cos x y
2, 4
hx, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe y x ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
gx, y xe y ,
3
fx, y sen 2x y ,
6
fx, y
y
x y , 4
fx, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1, v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2, v
1
2 i 3j
f x, y
xy,
P 4, 3 , v
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
f x, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
hx, y y cos x y
2, 4
hx, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe yx ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
gx, y xe y ,
3
fx, y sen 2x y ,
6
fx, y
y
x y , 4
fx, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1, v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2, v
1
2 i 3j
f x, y
xy,
P 4, 3 , v
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
fx, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–12, find the directional derivative of the function
at in the direction of v.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–16, find the directional derivative of the
function in the direction of the unit vector
13.
14.
15.
16.
In Exercises 17–20, find the directional derivative of the
function at
in the direction of
17.
18.
19.
20.
In Exercises 21–26, find the gradient of the function at the given
point.
21.
22.
23.
24.
25.
26.
In Exercises 27–30, use the gradient to find the directional
derivative of the function at
in the direction of
27.
28.
29.
30.
In Exercises 31–40, find the gradient of the function and the
maximum value of the directional derivative at the given point.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41– 46, consider the function
41. Sketch the graph of in the first octant and plot the point
on the surface.
42. Find where each given
value of
(a)
(b)
(c)
(d)
43. Find where using each given vector
(a)
(b)
(c) is the vector from to
(d) is the vector from to
44. Find
45. Find the maximum value of the directional derivative at
46. Find a unit vector orthogonal to and calculate
Discuss the geometric meaning of the result.
D u f 3, 2 .
f 3, 2
u
3, 2 .
fx, y .
4, 5 .
3, 2
v
2, 6 .
1, 2
v
v 3i 4j
v i j
v.
u
v
v ,
D u f 3, 2 ,
6
4
3
2
3
4
.
u cos i sen j,
D u f 3, 2 ,
3, 2, 1
f
fx, y 3
x
3
y
2 .
2, 0, 4
f x, y, z xe yz 2, 1, 1
w xy 2 z 2 0, 0, 0
w
1
1 x 2 y 2 z 2 1, 4, 2
f x, y, z x 2 y 2 z 2 1, 2
g x, y ln 3 x 2 y 2 0, 5
g x, y ye x 0, 3
h x, y y cos x y
2, 4
h x, y
x tan y
0, 1
f x, y
x
y
y 1
1, 0
f x, y x 2 2xy
Punto
Función
P , 0 , Q 2 ,
f x, y sen 2x cos y,
P 0, 0 , Q 2, 1
f x, y e y sen x,
P 1, 4 , Q 3, 6
f x, y 3x 2 y 2 4,
P 1, 2 , Q 2, 3
g x, y x 2 y 2 1,
Q.
P
4, 3, 1
w x tan y z ,
1, 1, 2
w 3x 2 5y 2 2z 2 ,
3, 4
z cos x 2 y 2 ,
2, 3
z ln x 2 y ,
2, 0
g x, y 2xe yx ,
2, 1
f x, y 3x 5y 2 1,
P 1, 0, 0 , Q 4, 3, 1
h x, y, z ln x y z ,
P 2, 4, 0 , Q 0, 0, 0
g x, y, z xye z ,
P 0, , Q 2 , 0
f x, y cos x y ,
P 1, 1 , Q 4, 5
f x, y x 2 3y 2 ,
Q.
P
2
3
gx, y xe y ,
3
fx, y sen 2x y ,
6
fx, y
y
x y , 4
fx, y x 2 y 2 ,
u cos i + sen j.
P 4, 1, 1 , v 1, 2, 1
h x, y, z
x arctan yz,
P 2, 1, 1 , v 2, 1, 2
h(x, y, z
xyz,
P 1, 2, 1, v 2i j k
f x, y, z xy yz xz,
P 1, 1, 1 , v
3
3
i j k
f x, y, z x 2 y 2 z 2 ,
P 0, 0 , v i j
h x, y e x2 y 2 ,
P 3, 4 , v 3i 4j
g x, y x 2 y 2 ,
P 1, 0 , v
j
g x, y
arccos xy,
P 1, 2 , v
i
h x, y e x sen y,
P 1, 1 , v
j
f x, y
x
y , P 0, 2, v
1
2 i 3j
f x, y
xy,
P 4, 3 , v
2
2
i
j
f x, y x 3 y 3 ,
P 1, 2 , v
3
5 i 4
5 j
f x, y 3x 4xy 9y,
P
942 Chapter 13 Functions of Several Variables
13.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
sen