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This pdf includes the following topics:-

Applications of the Gradient

Evaluating the Gradient

Practice Problems 1 and 2

Meaning of the Gradient

Examples

Applications of the Gradient

Evaluating the Gradient

Practice Problems 1 and 2

Meaning of the Gradient

Examples

1.
Applications of the Gradient

• Evaluating the Gradient at a Point

• Meaning of the Gradient

• Evaluating the Gradient at a Point

• Meaning of the Gradient

2.
Evaluating the Gradient

In 1-variable calculus, the derivative gives you an

equation for the slope at any x-value along f(x).

You can then plug in an x-value to find the actual

slope at that point. 2 f(x) = x

f’(x) = 2x

Actual tangent line slope is… 10 when x = 5

-4 when x = -2 5 when x = 2.5

0 when x = 0

In 1-variable calculus, the derivative gives you an

equation for the slope at any x-value along f(x).

You can then plug in an x-value to find the actual

slope at that point. 2 f(x) = x

f’(x) = 2x

Actual tangent line slope is… 10 when x = 5

-4 when x = -2 5 when x = 2.5

0 when x = 0

3.
Evaluating the Gradient

Similarly, the gradient gives you an equation for

the slope of the tangent plane at any point (x, y)

or (x, y, z) or whatever. You can then plug in the

actual values at any point to find the slope of

the tangent plane at that point.

The slope of the tangent plane will be written as

a vector, composed of the slopes along each

Similarly, the gradient gives you an equation for

the slope of the tangent plane at any point (x, y)

or (x, y, z) or whatever. You can then plug in the

actual values at any point to find the slope of

the tangent plane at that point.

The slope of the tangent plane will be written as

a vector, composed of the slopes along each

4.
Evaluating the Gradient

As an example, given the function f(x, y) = 3x2y –

2x and the point (4, -3), the gradient can be

calculated as:

[6xy – 2 3x2]

Plugging in the values of x and y at (4, -3) gives

[-74 48]

which is the value of the gradient at that point.

As an example, given the function f(x, y) = 3x2y –

2x and the point (4, -3), the gradient can be

calculated as:

[6xy – 2 3x2]

Plugging in the values of x and y at (4, -3) gives

[-74 48]

which is the value of the gradient at that point.

5.
Practice Problems 1 and 2

Evaluate the gradient of…

1. f(x, y) = x2 + y2 at

a) (0, 0) b) (1, 3) c) (-1, -5)

2. f(x, y, z) = x3z – 2y2x + 5z at

a) (1, 1, -4) b) (0, 1, 0) c) (-3, -2, 1)

Evaluate the gradient of…

1. f(x, y) = x2 + y2 at

a) (0, 0) b) (1, 3) c) (-1, -5)

2. f(x, y, z) = x3z – 2y2x + 5z at

a) (1, 1, -4) b) (0, 1, 0) c) (-3, -2, 1)

6.
Meaning of the Gradient

In the previous example, the function f(x, y) =

3x2y – 2x had a gradient of [6xy – 2 3x2], which

at the point (4, -3) came out to [-74 48].

500

The tangent plane at that

400

300

200

100 (4, -3) point will have a slope of

0

-100

-200

-300

-400

-74 in the x direction and

-500

-600

-700

-800

+48 in the y direction.

3 -1

y axis x axis

-5 7

Even more important is the vector itself, [-74 48].

In the previous example, the function f(x, y) =

3x2y – 2x had a gradient of [6xy – 2 3x2], which

at the point (4, -3) came out to [-74 48].

500

The tangent plane at that

400

300

200

100 (4, -3) point will have a slope of

0

-100

-200

-300

-400

-74 in the x direction and

-500

-600

-700

-800

+48 in the y direction.

3 -1

y axis x axis

-5 7

Even more important is the vector itself, [-74 48].

7.
Meaning of the Gradient

Here is the graph again, with the vector drawn in

as a vector rather than two sloped lines:

Recall that vectors give us

500

400

300

direction as well as magnitude.

200

100

0

-100

-200

The direction of the gradient

-300

-400

-500

-600

-700

vector will always point in the

3

-800

-1 direction of steepest increase for

y axis x axis

-5 7 the function.

And, its magnitude will give us the slope of the plane in

that direction.

Here is the graph again, with the vector drawn in

as a vector rather than two sloped lines:

Recall that vectors give us

500

400

300

direction as well as magnitude.

200

100

0

-100

-200

The direction of the gradient

-300

-400

-500

-600

-700

vector will always point in the

3

-800

-1 direction of steepest increase for

y axis x axis

-5 7 the function.

And, its magnitude will give us the slope of the plane in

that direction.

8.
Meaning of the Gradient

That’s a lot of different slopes!

16 Each component of the gradient

14

vector gives the slope in one

12

10

dimension only.

8

The magnitude of the gradient

6

4

vector gives the steepest possible

2 5

4

slope of the plane.

3

0 2

0 1

1 2 0

3 4 5 6

Recall that the magnitude can be found using the Pythagorean

Theorem, c2 = a2 + b2, where c is the magnitude and a and b are

the components of the vector.

That’s a lot of different slopes!

16 Each component of the gradient

14

vector gives the slope in one

12

10

dimension only.

8

The magnitude of the gradient

6

4

vector gives the steepest possible

2 5

4

slope of the plane.

3

0 2

0 1

1 2 0

3 4 5 6

Recall that the magnitude can be found using the Pythagorean

Theorem, c2 = a2 + b2, where c is the magnitude and a and b are

the components of the vector.

9.
Practice Problem 3

Find the gradient of f(x, y) = 2xy – 2y, and the

magnitude of the gradient, at…

a) (0, 0) 40

30

b) (5, -3) 20

10

0

-10

-20

c) (20, 10) -30

-40

d) (-5, 4)

e) Find where the gradient = 0.

Find the gradient of f(x, y) = 2xy – 2y, and the

magnitude of the gradient, at…

a) (0, 0) 40

30

b) (5, -3) 20

10

0

-10

-20

c) (20, 10) -30

-40

d) (-5, 4)

e) Find where the gradient = 0.

10.
Practice Problem 4

Find the gradient of

f(x, y, z, w) = 3xy – 2xw + 5xz – 2yw

and the magnitude of the gradient at (0, 1, -1, 2).

Find the gradient of

f(x, y, z, w) = 3xy – 2xw + 5xz – 2yw

and the magnitude of the gradient at (0, 1, -1, 2).

11.
Practice Problem 5

Suppose we are maximizing the function

f(x, y) = 4x + 2y – x2 – 3y2

Find the gradient and its magnitude from

a) (1, 5)

b) (3, -2)

c) (2, 0)

d) (-4, -6)

e) Find where the gradient is 0.

Suppose we are maximizing the function

f(x, y) = 4x + 2y – x2 – 3y2

Find the gradient and its magnitude from

a) (1, 5)

b) (3, -2)

c) (2, 0)

d) (-4, -6)

e) Find where the gradient is 0.

12.
Practice Problem 6

Suppose you were trying to minimize f(x, y) = x2

+ 2y + 2y2. Along what vector should you travel

from (5, 12)?

Suppose you were trying to minimize f(x, y) = x2

+ 2y + 2y2. Along what vector should you travel

from (5, 12)?