Chapter 1. calc_tutorial_14_4_007

1.1 Problem Statement

{3,4,5,6,7,8,9}
pow($r,2)+1
2*$r
-((pow($r,2)+6)/2)
((pow($r,2)+6)/2)
((pow($r,2)-8)/2)

Find an equation of the tangent plane to

at the point .

1.2 Step 1

Question Sequence

Question 1.1

Recall how to find the equation of the tangent plane to a surface at a given point. Assume that f(x, y) is defined in a disk D containing (a, b) and that fx(a, b) and fy(a, b) exist. Then the tangent plane to the graph at (a, b, f(a, b)) is the plane defined by the following equation.

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

Note that it is necessary that fx(a, b) and fy(a, b) exist to use this equation. We can check that they exist by ensuring that f(x, y) is differentiable on an open disk D containing (a, b). To do so, we use the fact that f(x, y) is differentiable on an open disk D provided that fx(x, y) and fy(x, y) exist and are continuous on an open disk D.

Given , find and .

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

and are both defined for 9CjeBIv4OujlCR0oRgsjdw/QTzp8AgHOFxIrGg== and sXTdFZMJ6/SarqUax8ecUu3VmuqgxTyJvyv63Q==.

Therefore zVpudauK/UPx/KjQc3EOeIN8nEc= be differentiable on an open disk containing the point ($r, 1).

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Find F($r, 1), Fr($r, 1) and Fs($r, 1).

F($r, 1) = or6dmYWrEbA=

Fr($r, 1) = mpgA8Zi4UKU=

Fs($r, 1) = Hs0oCnDH0eo=

Correct.
Incorrect.

1.4 Step 2

Question Sequence

Question 1.3

Use information from the previous steps to write the equation of the tangent plane to

at the point .

= mpgA8Zi4UKU=r - zDsgB75cNfg=s - bEz/iYnXpnk=

Correct.
Incorrect.