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Chapter 12: Q. 15. (page 964)

Sketch level curves z=19,16,21and 24for the function

z=25-x2-y2. Include the graphs of three gradient

vectors on each level curve. What do you observe?

16. Sketch level curvesz=9,16,21, and 16for the function.

Short Answer

Expert verified

The graph of level curves z=25-x2-y2for z=9,16,21and24is as shown below.

The gradient and tangent vectors are orthogonal.

Step by step solution

01

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

The function isz=25-x2-y2

The graph of the function z=25-x2-y2concentric circle, each of whose level curves is a circle centered at the origin.

So the gradient is

z=25-x2-y2鈭�z=-2xi-2yj

Hence, the magnitude of the gradient vectors grows.

02

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

The level of curves z=25-x2-y2for z=9,16,21and24is

x2+y2=16since9=25-x2-y2x2+y2=9since16=25-x2-y2x2+y2=4since21=25-x2-y2x2+y2=1since24=25-x2-y2

The graph of level curves z=25-x2-y2for z=9,16,21and24is shown below:

Hence, the gradient and tangent vectors are orthogonal.

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Most popular questions from this chapter

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.

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