8/18/2019 01 Directional Derivatives and Gradient
1/130
Directional Derivatives and the Gradient Vector
Institute of Mathematics
University of the PhilippinesDiliman
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
2/130
Recall: Derivatives
Let y = f (x).
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
3/130
Recall: Derivatives
Let y = f (x).
dy
dx = f (x) = lim
h→0f (x + h) − f (x)
h
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
4/130
Recall: Derivatives
Let y = f (x).
dy
dx = f (x) = lim
h→0f (x + h) − f (x)
h
f (x) represents the rate of change of y = f (x) with respectto x
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
5/130
Recall: Partial Derivatives
Let z = f (x, y).
∂z
∂x = f x(x, y) = lim
h→0f (x + h, y) − f (x, y)
h
∂z
∂y = f y(x, y) = lim
h→0f (x, y + h) − f (x, y)
h
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
6/130
Recall: Partial Derivatives
Let z = f (x, y).
∂z
∂x = f x(x, y) = lim
h→0f (x + h, y) − f (x, y)
h
∂z
∂y = f y(x, y) = lim
h→0f (x, y + h) − f (x, y)
h
f x represents the rate of changeof f with respect to x (indirection parallel to the x-axis)
when y is fixedf y represents the rate of changeof f with respect to y (indirection parallel to the y-axis)when x is fixed
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
7/130
Directional Derivative
Goal:
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
8/130
Directional Derivative
Goal: Study the rate of change of f in an arbitrary direction.
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
9/130
Directional Derivative
Goal: Study the rate of change of f in an arbitrary direction.
Suppose we want to find the rate of change of f at (x0, y0) inthe direction of a unit vector u = a, b.
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
10/130
Directional Derivative
Goal: Study the rate of change of f in an arbitrary direction.
Suppose we want to find the rate of change of f at (x0, y0) inthe direction of a unit vector u = a, b. That is, the slope of the tangent line to C at P .
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
11/130
Directional Derivative
Goal: Study the rate of change of f in an arbitrary direction.
Suppose we want to find the rate of change of f at (x0, y0) inthe direction of a unit vector u = a, b. That is, the slope of the tangent line to C at P .
If Q(x,y,z) is anotherpoint on C and P , Q arethe projections of P andQ on the xy-plane, resp.,then the vector P Q isparallel to u
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
12/130
Directional Derivative
Goal: Study the rate of change of f in an arbitrary direction.
Suppose we want to find the rate of change of f at (x0, y0) inthe direction of a unit vector u = a, b. That is, the slope of the tangent line to C at P .
If Q(x,y,z) is anotherpoint on C and P , Q arethe projections of P andQ on the xy-plane, resp.,then the vector P Q isparallel to u so for somescalar h,
P Q = hu = ha, hb .
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
13/130
Directional Derivative
Therefore,
x − x0 = ha ⇒ x = x0 + hay − y0 = hb ⇒ y = y0 + hb
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
14/130
Directional Derivative
Therefore,
x − x0 = ha ⇒ x = x0 + hay − y0 = hb ⇒ y = y0 + hb
The rate of change of f (with respect to distance) in thedirection of u at the point (x0, y0) is
limh→0∆f
h
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
15/130
Directional Derivative
Therefore,
x − x0 = ha ⇒ x = x0 + hay − y0 = hb ⇒ y = y0 + hb
The rate of change of f (with respect to distance) in thedirection of u at the point (x0, y0) is
limh→0∆f
h = limh→0f (x, y)
− f (x0, y0)
h
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
16/130
Directional Derivative
Therefore,
x − x0 = ha ⇒ x = x0 + hay − y0 = hb ⇒ y = y0 + hb
The rate of change of f (with respect to distance) in thedirection of u at the point (x0, y0) is
limh→0∆f
h = limh→0f (x, y)
− f (x0, y0)
h
= limh→0
f (x0 + ha, y0 + hb) − f (x0, y0)h
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
17/130
Directional Derivative
Definition
Let f be a function of x and y. The directional derivative of f at a point (x, y) along a unit vector u = a, b, denoted byDuf (x, y), is given by
Duf (x, y) := limh→0
f (x + ha, y + hb) − f (x, y)h
,
if this limit exists.
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
18/130
Directional Derivative
Definition
Let f be a function of x and y. The directional derivative of f at a point (x, y) along a unit vector u = a, b, denoted byDuf (x, y), is given by
Duf (x, y) := limh→0
f (x + ha, y + hb) − f (x, y)h
,
if this limit exists.
Remarks:1
If u = ı̂ = 1, 0, thenDı̂f (x, y) = lim
h→0f (x + h, y) − f (x, y)
h
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
19/130
8/18/2019 01 Directional Derivatives and Gradient
20/130
Directional Derivative
Definition
Let f be a function of x and y. The directional derivative of f at a point (x, y) along a unit vector u = a, b, denoted byDuf (x, y), is given by
Duf (x, y) := limh→0
f (x + ha, y + hb) − f (x, y)h
,
if this limit exists.
Remarks:1
If u = ı̂ = 1, 0, thenDı̂f (x, y) = lim
h→0f (x + h, y) − f (x, y)
h = f x(x, y).
2 If u = ˆ = 0, 1, thenDˆ f (x, y) = lim
h→0
f (x, y + h) − f (x, y)
hDirectional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
21/130
Directional Derivative
Definition
Let f be a function of x and y. The directional derivative of f at a point (x, y) along a unit vector u = a, b, denoted byDuf (x, y), is given by
Duf (x, y) := limh→0
f (x + ha, y + hb) − f (x, y)h
,
if this limit exists.
Remarks:1
If u = ˆı = 1
,0, then
Dı̂f (x, y) = limh→0
f (x + h, y) − f (x, y)h
= f x(x, y).
2 If u = ˆ = 0, 1, thenDˆ f (x, y) = lim
h→0
f (x, y + h) − f (x, y)
h
= f y(x, y).
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
22/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 3
2 ,−1
2 at the point (1,−1).
Directional Derivatives & Gradient
l
8/18/2019 01 Directional Derivatives and Gradient
23/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 3
2 ,−1
2 at the point (1,−1).
Duf (1,−1)
Directional Derivatives & Gradient
E l 1
8/18/2019 01 Directional Derivatives and Gradient
24/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 3
2 ,−1
2 at the point (1,−1).
Duf (1,−1) = limh→0
f (1 +√
32 h,−1 − h
2) − f (1,−1)
h
Directional Derivatives & Gradient
E l 1
8/18/2019 01 Directional Derivatives and Gradient
25/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 32 ,−1
2 at the point (1,−1).
Duf (1,−1) = limh→0
f (1 +√
32 h,−1 − h
2) − f (1,−1)
h
= limh→0
1
h
(1 + √ 32 h
2
+ 3
−1 − h
2
2− 4
Directional Derivatives & Gradient
E l 1
8/18/2019 01 Directional Derivatives and Gradient
26/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 32 ,−1
2 at the point (1,−1).
Duf (1,−1) = limh→0
f (1 +√
32 h,−1 − h
2) − f (1,−1)
h
= limh→0
1
h
(1 + √ 32 h
2
+ 3
−1 − h
2
2− 4
= lim
h→0
1
h1 +
√ 3h +
3
4
h2 + 31 + h + h2
4−
4
Directional Derivatives & Gradient
E l 1
8/18/2019 01 Directional Derivatives and Gradient
27/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 32 ,−1
2 at the point (1,−1).
Duf (1,−1) = limh→0
f (1 +√
32 h,−1 − h
2) − f (1,−1)
h
= limh→0
1
h
(1 + √ 32 h
2
+ 3
−1 − h
2
2− 4
= lim
h→0
1
h1 +
√ 3h +
3
4
h2 + 31 + h + h2
4−
4= lim
h→01
h
√ 3h +
3
2h2 + 3h
Directional Derivatives & Gradient
E l 1
8/18/2019 01 Directional Derivatives and Gradient
28/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 32 ,−1
2 at the point (1,−1).
Duf (1,−1) = limh→0
f (1 +√
32 h,−1 − h
2) − f (1,−1)
h
= limh→0
1
h
(1 + √ 32 h
2
+ 3
−1 − h
2
2− 4
= lim
h→0
1
h1 +
√ 3h +
3
4
h2 + 31 + h + h2
4−
4= lim
h→01
h
√ 3h +
3
2h2 + 3h
= limh→
0√
3 + 3
2h + 3
Directional Derivatives & Gradient
Example 1
8/18/2019 01 Directional Derivatives and Gradient
29/130
Example 1
Example
Use the definition to find the directional derivative of f (x, y) = x2 + 3y2 along u = √ 32 ,−1
2 at the point (1,−1).
Duf (1,−1) = limh→0
f (1 +√
32 h,−1 − h
2) − f (1,−1)
h
= limh→0
1
h
(1 + √ 32 h
2
+ 3
−1 − h
2
2− 4
= lim
h→0
1
h1 +
√ 3h +
3
4
h2 + 31 + h + h2
4−
4= lim
h→01
h
√ 3h +
3
2h2 + 3h
= limh
→0
√ 3 +
3
2h + 3 =
√ 3 + 3
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
30/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
31/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb).
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
32/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
33/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
34/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
= Duf (x0, y0).
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
35/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
= Duf (x0, y0).
Let g(h) = f (x, y) where x = x0 + ha, y = y0 + hb. By Chain Rule,
g(h)
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
36/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
= Duf (x0, y0).
Let g(h) = f (x, y) where x = x0 + ha, y = y0 + hb. By Chain Rule,
g(h) =
∂f
∂x
dx
dh +
∂f
∂y
dy
dh
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
37/130
Directional Derivative
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
= Duf (x0, y0).
Let g(h) = f (x, y) where x = x0 + ha, y = y0 + hb. By Chain Rule,
g(h) =
∂f
∂x
dx
dh +
∂f
∂y
dy
dh = f x(x, y)a + f y(x, y)b
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
38/130
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
= Duf (x0, y0).
Let g(h) = f (x, y) where x = x0 + ha, y = y0 + hb. By Chain Rule,
g(h) =
∂f
∂x
dx
dh +
∂f
∂y
dy
dh = f x(x, y)a + f y(x, y)b
h = 0, x = x0, y = y0 ⇒
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
39/130
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
= Duf (x0, y0).
Let g(h) = f (x, y) where x = x0 + ha, y = y0 + hb. By Chain Rule,
g(h) =
∂f
∂x
dx
dh +
∂f
∂y
dy
dh = f x(x, y)a + f y(x, y)b
h = 0, x = x0, y = y0 ⇒ g(0) = f x(x0, y0)a + f y(x0, y0)b
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
40/130
Theorem
If f is a differentiable function of x and y, then f has a directional derivative at point (x0, y0) in the domain of f , in the direction of any unit vector u = a, b and
Duf (x0, y0) = f x(x0, y0)a + f y(x0, y0)b.
Proof. Define g(h) = f (x0 + ha, y0 + hb). Then
g(0) = lim
h→0
g(h) − g(0)
h = lim
h→0
f (x0 + ha, y0 + hb) − f (x0, y0)
h
= Duf (x0, y0).
Let g(h) = f (x, y) where x = x0 + ha, y = y0 + hb. By Chain Rule,
g(h) =
∂f
∂x
dx
dh +
∂f
∂y
dy
dh = f x(x, y)a + f y(x, y)b
h = 0, x = x0, y = y0 ⇒ g(0) = f x(x0, y0)a + f y(x0, y0)b = Duf (x0, y0)
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
41/130
p
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
42/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y)
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
43/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
44/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2)
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
45/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
46/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y)
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
47/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
48/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2)
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
49/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2) =
−6
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
50/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2) =
−6
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
51/130
Example
Given f (x, y) = x2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2) =
−6
Since PQ = 3,−3 is not a unit vector, take u =
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
52/130
Example
Given f (x, y) = x
2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2) =
−6
Since PQ = 3,−3 is not a unit vector, take u = √
22 ,−
√ 2
2
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
53/130
Example
Given f (x, y) = x
2
− 2xy − y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2) =
−6
Since PQ = 3,−3 is not a unit vector, take u = √
22 ,−
√ 2
2
Duf (1, 2) = f x(1, 2)
√ 2
2
+ f y(1, 2)
−
√ 2
2
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
54/130
Example
Given f
(x, y
) = x2
− 2xy
−y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2) =
−6
Since PQ = 3,−3 is not a unit vector, take u = √
22 ,−
√ 2
2
Duf (1, 2) = f x(1, 2)
√ 2
2
+ f y(1, 2)
−
√ 2
2
= −2√
2
2
− 6
−
√ 2
2
Directional Derivatives & Gradient
Example 2
8/18/2019 01 Directional Derivatives and Gradient
55/130
Example
Given f
(x, y
) = x2
− 2xy
−y2
. Find the directional derivative of f at P (1, 2) in the direction from P to Q(4,−1).
f x(x, y) = 2x − 2y ⇒ f x(1, 2) = −2f y(x, y) =
−2x
− 2y
⇒ f y(1, 2) =
−6
Since PQ = 3,−3 is not a unit vector, take u = √
22 ,−
√ 2
2
Duf (1, 2) = f x(1, 2)
√ 2
2
+ f y(1, 2)
−
√ 2
2
= −2√
2
2
− 6
−
√ 2
2
= 2√
2
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
56/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
57/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .The unit vector in the same direction as the given vector is
u = 3,
−4 3,−4
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
58/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .The unit vector in the same direction as the given vector is
u = 3,
−4 3,−4 =
3,−
4 32 + (−4)2
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
59/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .The unit vector in the same direction as the given vector is
u = 3,
−4 3,−4 =
3,−
4 32 + (−4)2 =
3
5 ,−4
5
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
60/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .The unit vector in the same direction as the given vector is
u = 3,
−4 3,−4 =
3,−
4 32 + (−4)2 =
3
5 ,−4
5
Hence,
Duf (x, y) = f x(x, y)3
5
+ f y(x, y)−45
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
61/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .The unit vector in the same direction as the given vector is
u = 3,
−4 3,−4 =
3,−
4 32 + (−4)2 =
3
5 ,−4
5
Hence,
Duf (x, y) = f x(x, y)3
5
+ f y(x, y)−45
=y2 + y sin(xy)
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
62/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .The unit vector in the same direction as the given vector is
u = 3,
−4 3,−4 =
3,−
4 32 + (−4)2 =
3
5 ,−4
5
Hence,
Duf (x, y) = f x(x, y)3
5
+ f y(x, y)−45
=y2 + y sin(xy)
35
Directional Derivatives & Gradient
Example 3
8/18/2019 01 Directional Derivatives and Gradient
63/130
Example
Determine the directional derivative of f (x, y) = xy2
− cos(xy)at any point, in the direction of the vector 3 ı̂ − 4ˆ .The unit vector in the same direction as the given vector is
u = 3,
−4 3,−4 =
3,−
4 32 + (−4)2 =
3
5 ,−4
5
Hence,
Duf (x, y) = f x(x, y)3
5
+ f y(x, y)−45
=y2 + y sin(xy)
35
+ (2xy + x sin(xy))
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
64/130
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
65/130
(x0, y0)
u
a
b
θ
If the unit vector u = a, b, makes an angle θ with the x-axis,then
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
66/130
(x0, y0)
u
a
b
θ
If the unit vector u = a, b, makes an angle θ with the x-axis,then
a = u cos θ
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
67/130
(x0, y0)
u
a
b
θ
If the unit vector u = a, b, makes an angle θ with the x-axis,then
a = u cos θ = cos θ
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
68/130
(x0, y0)
u
a
b
θ
If the unit vector u = a, b, makes an angle θ with the x-axis,then
a = u cos θ = cos θb = u sin θ
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
69/130
(x0, y0)
u
a
b
θ
If the unit vector u = a, b, makes an angle θ with the x-axis,then
a = u cos θ = cos θb = u sin θ = sin θ
Directional Derivatives & Gradient
Directional Derivative
8/18/2019 01 Directional Derivatives and Gradient
70/130
(x0, y0)
u
a
b
θ
If the unit vector u = a, b, makes an angle θ with the x-axis,then
a = u cos θ = cos θb = u sin θ = sin θ
and the formula in the previous theorem becomes
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ.
Directional Derivatives & Gradient
Example 4
8/18/2019 01 Directional Derivatives and Gradient
71/130
Example
Find the directional derivative of f (x, y) = ex2
y − 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
Directional Derivatives & Gradient
Example 4
8/18/2019 01 Directional Derivatives and Gradient
72/130
Example
Find the directional derivative of f (x, y) = ex2
y − 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
Directional Derivatives & Gradient
Example 4
8/18/2019 01 Directional Derivatives and Gradient
73/130
Example
Find the directional derivative of f (x, y) = ex2
y − 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
Directional Derivatives & Gradient
Example 4
8/18/2019 01 Directional Derivatives and Gradient
74/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3
Directional Derivatives & Gradient
Example 4
8/18/2019 01 Directional Derivatives and Gradient
75/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
Directional Derivatives & Gradient
Example 4
E l
8/18/2019 01 Directional Derivatives and Gradient
76/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
sin π
3
Directional Derivatives & Gradient
Example 4
E l
8/18/2019 01 Directional Derivatives and Gradient
77/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
sin π
3
Hence,
Duf (−1, 0)
Directional Derivatives & Gradient
Example 4
E l
8/18/2019 01 Directional Derivatives and Gradient
78/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
sin π
3
Hence,
Duf (−1, 0) = −2
Directional Derivatives & Gradient
Example 4
E l
8/18/2019 01 Directional Derivatives and Gradient
79/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
sin π
3
Hence,
Duf (−1, 0) = −2
1
2
Directional Derivatives & Gradient
Example 4
Example
8/18/2019 01 Directional Derivatives and Gradient
80/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
sin π
3
Hence,
Duf (−1, 0) = −2
1
2
+ 2
Directional Derivatives & Gradient
Example 4
Example
8/18/2019 01 Directional Derivatives and Gradient
81/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
sin π
3
Hence,
Duf (−1, 0) = −2
1
2
+ 2
√ 3
2
Directional Derivatives & Gradient
Example 4
Example
8/18/2019 01 Directional Derivatives and Gradient
82/130
Example
Find the directional derivative of f (x, y) = ex2y
− 2x + y at thepoint (−1, 0) in the direction of the unit vector given by θ = π
3.
From the previous formula,
Duf (x, y) = f x(x, y)cos θ + f y(x, y)sin θ
=
2xyex2y − 2
cos
π
3 +x2ex
2y + 1
sin π
3
Hence,
Duf (−1, 0) = −2
1
2
+ 2
√ 3
2
= −1 +
√ 3.
Directional Derivatives & Gradient
The Gradient Vector
8/18/2019 01 Directional Derivatives and Gradient
83/130
From the previous theorem,
Duf (x, y) = f x(x, y)a + f y(x, y)b
Directional Derivatives & Gradient
The Gradient Vector
8/18/2019 01 Directional Derivatives and Gradient
84/130
From the previous theorem,
Duf (x, y) = f x(x, y)a + f y(x, y)b
= f x(x, y), f y(x, y) · a, b
Directional Derivatives & Gradient
The Gradient Vector
8/18/2019 01 Directional Derivatives and Gradient
85/130
From the previous theorem,
Duf (x, y) = f x(x, y)a + f y(x, y)b
= f x(x, y), f y(x, y) · a, b= f x(x, y), f y(x, y) · u
Directional Derivatives & Gradient
The Gradient Vector
8/18/2019 01 Directional Derivatives and Gradient
86/130
From the previous theorem,
Duf (x, y) = f x(x, y)a + f y(x, y)b
= f x(x, y), f y(x, y) · a, b= f x(x, y), f y(x, y) · u
Directional Derivatives & Gradient
The Gradient Vector
8/18/2019 01 Directional Derivatives and Gradient
87/130
From the previous theorem,
Duf (x, y) = f x(x, y)a + f y(x, y)b
= f x(x, y), f y(x, y) · a, b= f x(x, y), f y(x, y) · u
The vector f x(x, y), f y(x, y) appears not only in the formulafor directional derivative but in many applications as well.
Directional Derivatives & Gradient
The Gradient Vector
8/18/2019 01 Directional Derivatives and Gradient
88/130
From the previous theorem,
Duf (x, y) = f x(x, y)a + f y(x, y)b
= f x(x, y), f y(x, y) · a, b= f x(x, y), f y(x, y) · u
The vector f x(x, y), f y(x, y) appears not only in the formulafor directional derivative but in many applications as well.
This vector is called the gradient of f , denoted grad f or ∇
f .
Directional Derivatives & Gradient
The Gradient Vector
Definition
8/18/2019 01 Directional Derivatives and Gradient
89/130
Definition
If f is a function of x and y, then the gradient of f is thevector function ∇f defined as
∇f (x, y) = f x(x, y), f y(x, y)
Directional Derivatives & Gradient
The Gradient Vector
Definition
8/18/2019 01 Directional Derivatives and Gradient
90/130
Definition
If f is a function of x and y, then the gradient of f is thevector function ∇f defined as
∇f (x, y) = f x(x, y), f y(x, y)
Thus, the directional derivative of f in the direction of a unitvector u = a, b can be written as
Duf (x, y) = ∇f (x, y) · u.
Example 5
Use gradient to find the directional derivative in Example 1.
Directional Derivatives & Gradient
Functions of Three Variables
Definition
Th di ti l d i ti f f( ) t ( ) i th
8/18/2019 01 Directional Derivatives and Gradient
91/130
The directional derivative of f (x,y,z) at (x0, y0, z0) in the
direction of the unit vector u = a,b,c islimh→0
f (x0 + ha, y0 + hb, z0 + hc) − f (x0, y0, z0)h
provided this limit exists.
Directional Derivatives & Gradient
Functions of Three Variables
Definition
Th di ti n l d i ti f f( ) t ( ) i th
8/18/2019 01 Directional Derivatives and Gradient
92/130
The directional derivative of f (x,y,z) at (x0, y0, z0) in the
direction of the unit vector u = a,b,c islimh→0
f (x0 + ha, y0 + hb, z0 + hc) − f (x0, y0, z0)h
provided this limit exists.
Definition
The gradient vector of f (x,y,z) is
∇f (x,y,z) =
f x(x,y,z), f y(x,y,z), f z(x,y,z)
Directional Derivatives & Gradient
Functions of Three Variables
Definition
The directional derivative of f(x y z) at (x y z ) in the
8/18/2019 01 Directional Derivatives and Gradient
93/130
The directional derivative of f (x,y,z) at (x0, y0, z0) in the
direction of the unit vector u = a,b,c islimh→0
f (x0 + ha, y0 + hb, z0 + hc) − f (x0, y0, z0)h
provided this limit exists.
Definition
The gradient vector of f (x,y,z) is
∇f (x,y,z) =
f x(x,y,z), f y(x,y,z), f z(x,y,z)
Duf (x,y,z) = f x(x,y,z)a + f y(x,y,z)b + f z(x,y,z)c
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
94/130
The Gradient Vector
Theorem
8/18/2019 01 Directional Derivatives and Gradient
95/130
Suppose f is a differentiable function of x and y. The
maximum value of Duf is ∇f and it occurs when u is in the same direction as ∇f . The minimum value of Duf is −∇f and it occurs when u is in the opposite direction as ∇f
Directional Derivatives & Gradient
The Gradient Vector
Theorem
S f i diff i bl f i f d Th
8/18/2019 01 Directional Derivatives and Gradient
96/130
Suppose f is a differentiable function of x and y. The
maximum value of Duf is ∇f and it occurs when u is in the same direction as ∇f . The minimum value of Duf is −∇f and it occurs when u is in the opposite direction as ∇f
Proof. Let θ be the angle between ∇f and u.
Directional Derivatives & Gradient
The Gradient Vector
Theorem
S f i diff ti bl f ti f d Th
8/18/2019 01 Directional Derivatives and Gradient
97/130
Suppose f is a differentiable function of x and y. The
maximum value of Duf is ∇f and it occurs when u is in the same direction as ∇f . The minimum value of Duf is −∇f and it occurs when u is in the opposite direction as ∇f
Proof. Let θ be the angle between ∇f and u.
Duf (x, y) = ∇f · u
Di ti l D i ti s & G di t
The Gradient Vector
Theorem
S f i diff ti bl f ti f d Th
8/18/2019 01 Directional Derivatives and Gradient
98/130
Suppose f is a differentiable function of x and y. The
maximum value of Duf is ∇f and it occurs when u is in the same direction as ∇f . The minimum value of Duf is −∇f and it occurs when u is in the opposite direction as ∇f
Proof. Let θ be the angle between ∇f and u.
Duf (x, y) = ∇f · u= ∇f u cos θ
Di ti l D i ti & G di t
The Gradient Vector
Theorem
S f i diff ti bl f ti f d Th
8/18/2019 01 Directional Derivatives and Gradient
99/130
Suppose f is a differentiable function of x and y. The
maximum value of Duf is ∇f and it occurs when u is in the same direction as ∇f . The minimum value of Duf is −∇f and it occurs when u is in the opposite direction as ∇f
Proof.
Let θ be the angle between ∇f and u.Duf (x, y) = ∇f · u
= ∇f u cos θ=
∇f
cos θ
Di ti l D i ti & G di t
The Gradient Vector
Theorem
Suppose f is a differentiable function of x and y The
8/18/2019 01 Directional Derivatives and Gradient
100/130
Suppose f is a differentiable function of x and y. The
maximum value of Duf is ∇f and it occurs when u is in the same direction as ∇f . The minimum value of Duf is −∇f and it occurs when u is in the opposite direction as ∇f
Proof.
Let θ be the angle between ∇f and u.Duf (x, y) = ∇f · u
= ∇f u cos θ=
∇f
cos θ
Result follows since the maximum value of cos θ is 1 and isattained when θ = 0, i.e., ∇f and u are in the samedirection
Di ti l D i ti & G di t
The Gradient Vector
Theorem
Suppose f is a differentiable function of x and y The
8/18/2019 01 Directional Derivatives and Gradient
101/130
Suppose f is a differentiable function of x and y. The
maximum value of Duf is ∇f and it occurs when u is in the same direction as ∇f . The minimum value of Duf is −∇f and it occurs when u is in the opposite direction as ∇f
Proof.
Let θ be the angle between ∇f and u.Duf (x, y) = ∇f · u
= ∇f u cos θ=
∇f
cos θ
Result follows since the maximum value of cos θ is 1 and isattained when θ = 0, i.e., ∇f and u are in the samedirection(minimum value occurs at cos θ = −1, i.e., θ = π).
Directional Derivatives & Gradient
Example 6
Example
8/18/2019 01 Directional Derivatives and Gradient
102/130
Find the minimum directional derivative of f (x,y,z) = xy2 − x2z + y3 − 1 at the point (3, 0, 2).
Directional Derivatives & Gradient
Example 6
Example
8/18/2019 01 Directional Derivatives and Gradient
103/130
Find the minimum directional derivative of f (x,y,z) = xy2 − x2z + y3 − 1 at the point (3, 0, 2).
∇f (x,y,z)
Directional Derivatives & Gradient
Example 6
Example
8/18/2019 01 Directional Derivatives and Gradient
104/130
Find the minimum directional derivative of f (x,y,z) = xy2 − x2z + y3 − 1 at the point (3, 0, 2).
∇f (x,y,z) = f x(x,y,z), f y(x,y,z), f z(x,y,z)=
y2 − 2xz,
Directional Derivatives & Gradient
Example 6
Example
8/18/2019 01 Directional Derivatives and Gradient
105/130
Find the minimum directional derivative of f (x,y,z) = xy2 − x2z + y3 − 1 at the point (3, 0, 2).
∇f (x,y,z) = f x(x,y,z), f y(x,y,z), f z(x,y,z)=
y2 − 2xz, 2xy + 3y2,
Directional Derivatives & Gradient
Example 6
Example
8/18/2019 01 Directional Derivatives and Gradient
106/130
Find the minimum directional derivative of f (x,y,z) = xy2 − x2z + y3 − 1 at the point (3, 0, 2).
∇f (x,y,z) = f x(x,y,z), f y(x,y,z), f z(x,y,z)=
y2 − 2xz, 2xy + 3y2, − x2
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
107/130
8/18/2019 01 Directional Derivatives and Gradient
108/130
Example 6
Example
8/18/2019 01 Directional Derivatives and Gradient
109/130
Find the minimum directional derivative of f (x,y,z) = xy2 − x2z + y3 − 1 at the point (3, 0, 2).
∇f (x,y,z) = f x(x,y,z), f y(x,y,z), f z(x,y,z)=
y2 − 2xz, 2xy + 3y2, − x2
⇒ ∇f (3, 0,−2) = 12, 0,−9
Hence, the minimum directional derivative of f at (3, 0, 2) is
Directional Derivatives & Gradient
Example 6
Example
8/18/2019 01 Directional Derivatives and Gradient
110/130
Find the minimum directional derivative of f (x,y,z) = xy2 − x2z + y3 − 1 at the point (3, 0, 2).
∇f (x,y,z) = f x(x,y,z), f y(x,y,z), f z(x,y,z)=
y2 − 2xz, 2xy + 3y2, − x2
⇒ ∇f (3, 0,−2) = 12, 0,−9
Hence, the minimum directional derivative of f at (3, 0, 2) is
−∇f (3, 0,−2)
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
111/130
Example 7
Example
Suppose that the shape of a hill above sea level is given by the
8/18/2019 01 Directional Derivatives and Gradient
112/130
equation h(x, y) = 1000 − 0.02x2 − 0.01y2, where x, y, andh(x, y) are measured in meters. The positive x−axis points eastand the positive y−axis points north. Suppose you are standingat a point with coordinates (50, 80, 886).
1
If you walk due west, will you start to ascend or descend?At what rate?
2 If you walk due southeast, will you start to ascend ordescend? At what rate?
3 In which direction is the elevation changing fastest (inwhat direction should you proceed initially in order toreach the top of the hill fastest)? What is the rate of ascentin that direction?
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
113/130
Example 7
∇h(x, y)
8/18/2019 01 Directional Derivatives and Gradient
114/130
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x,
8/18/2019 01 Directional Derivatives and Gradient
115/130
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y
8/18/2019 01 Directional Derivatives and Gradient
116/130
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y
8/18/2019 01 Directional Derivatives and Gradient
117/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80)
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y
8/18/2019 01 Directional Derivatives and Gradient
118/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.6
Directional Derivatives & Gradient
8/18/2019 01 Directional Derivatives and Gradient
119/130
Example 7
∇h(x, y) = −0.04x, − 0.02y
8/18/2019 01 Directional Derivatives and Gradient
120/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y
8/18/2019 01 Directional Derivatives and Gradient
121/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80)
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y
8/18/2019 01 Directional Derivatives and Gradient
122/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y( ) ( ) ( )
8/18/2019 01 Directional Derivatives and Gradient
123/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02yh( ) ( ) ( )
8/18/2019 01 Directional Derivatives and Gradient
124/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)2 Due SE:
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y∇h(50 80) 0 04(50) 0 02(80) 2 1 6
8/18/2019 01 Directional Derivatives and Gradient
125/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)2 Due SE: u = 1√
21,−1
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y∇h(50 80) 0 04(50) 0 02(80) 2 1 6
8/18/2019 01 Directional Derivatives and Gradient
126/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)2 Due SE: u = 1√
21,−1 ⇒ Duh(50, 80)
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y∇h(50 80) 0 04(50) 0 02(80) 2 1 6
8/18/2019 01 Directional Derivatives and Gradient
127/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)2 Due SE: u = 1√
21,−1 ⇒ Duh(50, 80) = −
√ 2
5
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y⇒ ∇h(50 80) 0 04(50) 0 02(80) 2 1 6
8/18/2019 01 Directional Derivatives and Gradient
128/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)2 Due SE: u = 1√
21,−1 ⇒ Duh(50, 80) = −
√ 2
5 (descend)
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y⇒ ∇h(50 80) 0 04(50) 0 02(80) 2 1 6
8/18/2019 01 Directional Derivatives and Gradient
129/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)2 Due SE: u = 1√
21,−1 ⇒ Duh(50, 80) = −
√ 2
5 (descend)
3 The maximum rate of change occurs in the direction of ∇h(50, 80) and the rate of ascent is given by∇h(50, 80) =
(−2)2 + (−1.6)2 = 2.5612.
Directional Derivatives & Gradient
Example 7
∇h(x, y) = −0.04x, − 0.02y⇒ ∇h(50 80) 0 04(50) 0 02(80) 2 1 6
8/18/2019 01 Directional Derivatives and Gradient
130/130
⇒ ∇h(50, 80) = −0.04(50),−0.02(80) = −2,−1.61 Due west: u = −1, 0 ⇒ Duh(50, 80) = 2(ascend)2 Due SE: u = 1√
21,−1 ⇒ Duh(50, 80) = −
√ 2
5 (descend)
3 The maximum rate of change occurs in the direction of ∇h(50, 80) and the rate of ascent is given by∇h(50, 80) =
(−2)2 + (−1.6)2 = 2.5612.
Remark
We say that at (x0, y0), f will increase most rapidly in thedirection of ∇f (x0, y0), or that ∇f (x0, y0) points in thedirection of the steepest ascent.
Directional Derivatives & Gradient
Top Related