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Chapter 12: Q. 27 (page 976)

In Exercises 27鈥�30, use the result from Example 4 to find the distance from the point P to the given plane.

P=3,5,-2,x-3y+4z=8

Short Answer

Expert verified

The distance is142613.

Step by step solution

01

Step 1. Given Information.

The point is P=3,5,-2and the equation of the plane isx-3y+4z=8.

02

Step 2. Explanation.

The distance between point Px0,y0,z0to the plane with equation localid="1650277378244" ax+by+cz=dis D=ax0+by0+cz0-da2+b2+c2.

So, the distance between point P=3,5,-2and the line is calculated as,

localid="1650367282409" role="math" D=3脳1+5脳-3+-2脳4-812+-32+42=142613

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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.

鈭�z鈭�twhenz=x2y3,x=tsins,andy=scost

Consider the function f(x, y) = 2x + 3y.

(a) Why is the graph of f a plane?

(b) In what direction is f increasing most rapidly at the

point (鈭�1, 4)?

(c) In what direction is f increasing most rapidly at the

point (x 0, y 0)?

(d) Why are your answers to parts (b) and (c) the same?

In Exercises 37-42, use the partial derivatives of role="math" localid="1650186853142" gx,y=tan-1xy2and the point role="math" localid="1650186870407" 1,0specified to

afind the equation of the line tangent to the surface defined by the function in the xdirection,

bfind the equation of the line tangent to the surface defined by the function in the ydirection, and

cfind the equation of the plane containing the lines you found in parts aandb.

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

f(x,y)=xyx2+y2,P=(鈭�2,1),v=35,鈭�45

Evaluate the following limits, or explain why the limit does not exist.

lim(x,y,z)鈫�(0,0,0)x2+y2x2+y2+z2
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