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Exercise 2
Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)
Chapter 3: Many Variables; Section 1: Differentiation; page 2
Partial Derivatives, Exercise 2
Question
The function u of three (3) variables is defined as:
- Calculate the partial derivatives
- What is the value of the function at the point
- Obtain the first approximation of the function, about this point!
Reminder
We define the partial derivative of a function
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(3.1.2.1)
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with respect to the variable xk, as the limit
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(3.1.2.2)
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The first approximation of a function of n variables
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(3.1.2.30)
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around the point
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(3.1.2.31)
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has the form of:
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(3.1.2.32)
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Parts 1-3
Solution of question 1
-
-
-
Part 4
Solution of question 2
- The substitution of
into
yields
Part 5-9
Solution of question 3
- According to (3.1.2.31) the first approximation of u about the point (x, y, z)0 is
- The substitutions of the point's coordinates in the results obtained in the parts 1, 2, 3 yield
-
-
- From parts 4-8 one obtains the final result
Score
By parts.
Parts 1, 2, 3, 5, 6, 7, 8, 9 are worth 1 point each.
Part 4 is worth 2 points.
By questions.
Question 1 is worth 3 points.
Question 2 is worth 2 points.
Question 3 is worth 5 points.