Chapter 12: Q. 15. (page 964)

Sketch level curves $z = 19, 16, 21$ and $24$ for the function
$z = 25 - x^{2} - y^{2}$ . Include the graphs of three gradient
vectors on each level curve. What do you observe?
16. Sketch level curves $z = 9, 16, 21$ , and $16$ for the function.

Short Answer

Expert verified

The graph of level curves $z = 25 - x^{2} - y^{2}$ for $z = 9, 16, 21 a n d 24$ is as shown below.

The gradient and tangent vectors are orthogonal.

Step by step solution

The objective is to sketch the level curves z=9,16,21 and 24 for the function

The function is $z = 25 - x^{2} - y^{2}$

The graph of the function $z = 25 - x^{2} - y^{2}$ concentric circle, each of whose level curves is a circle centered at the origin.

So the gradient is

$z = 25 - x^{2} - y^{2} \nabla z = - 2 x i - 2 y j$

Hence, the magnitude of the gradient vectors grows.

Draw a graph of level curves z=25-x2-y2 for z=9,16,21 and 24and observe

The level of curves $z = 25 - x^{2} - y^{2}$ for $z = 9, 16, 21 a n d 24$ is

$x^{2} + y^{2} = 16 (\sin c e 9 = 25 - x^{2} - y^{2}) x^{2} + y^{2} = 9 (\sin c e 16 = 25 - x^{2} - y^{2}) x^{2} + y^{2} = 4 (\sin c e 21 = 25 - x^{2} - y^{2}) x^{2} + y^{2} = 1 (\sin c e 24 = 25 - x^{2} - y^{2})$

The graph of level curves $z = 25 - x^{2} - y^{2}$ for $z = 9, 16, 21 a n d 24$ is shown below:

Hence, the gradient and tangent vectors are orthogonal.

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Sketch level curves $z = 19, 16, 21$ and $24$ for the function
$z = 25 - x^{2} - y^{2}$ . Include the graphs of three gradient
vectors on each level curve. What do you observe?
16. Sketch level curves $z = 9, 16, 21$ , and $16$ for the function.

Short Answer

Step by step solution

The objective is to sketch the level curves z=9,16,21 and 24 for the function

Draw a graph of level curves z=25-x2-y2 for z=9,16,21 and 24and observe

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Sketch level curves z=19,16,21and 24for the functionz=25-x2-y2. Include the graphs of three gradientvectors on each level curve. What do you observe?16. Sketch level curvesz=9,16,21, and 16for the function.

Short Answer

Step by step solution

The objective is to sketch the level curves z=9,16,21 and 24 for the function

Draw a graph of level curves z=25-x2-y2 for z=9,16,21 and 24and observe

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Pure Maths

Geometry

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

Sketch level curves $z = 19, 16, 21$ and $24$ for the function
$z = 25 - x^{2} - y^{2}$ . Include the graphs of three gradient
vectors on each level curve. What do you observe?
16. Sketch level curves $z = 9, 16, 21$ , and $16$ for the function.