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Unformatted text preview: e point (x0 , y0 , z0 ) =
Finding Partial Derivatives:
Example: Higher Order Partials Mixed Partials Read from Right to Left = Read from Left to Right In Class Example: Find
Theorem 3.1:
Mixed Partials are EQUAL IF they are both continuous. This holds for higher order partials also. Estimate Partials from a Table of Data and from Contour Plots
Interpret the meaning of a partial for a word problem
Mathematica Skills:
In one graph, show the function surface, the plane , the trace curve, and the tangent line to the trace curve 12.5: The Chain Rule
Derivatives of Composite Functions
101A Review: OR Example: where So 101C Version: Notice the similarity to the 101A version This is for the composite of a Multivariable function and a Vectorvalued function The Shorthand Notation: z Visual of the Chain Rule:
x y t Gradient of f ( is called the “del” operator ) These are Linear Operators (from Linear Algebra Class)
A linear operator satisfies: From 101A our operators were: Example:
So We could have got the same answer by substituting x and y in first. You would have a function of one variable and can
take the ordinary derivative of the function.
The derivative =
So what’s the point then? Sometimes we only know the rates of change of x and y at specific values and not the
functions themselves.
Example of This:
Solid Right Circular Cylinder
Its heated so r and h increase and thus so does
its Surface Area. We know AT r=10 and h=100 that r is increasing at a rate of 0.2 cm/hr and h is increasing at a rate of
0.5 cm/hr. How fast is the Surface Area S increasing then? Meaning what is So The Chain Rule can be extended to a multivariable function of 3 or more variables
where x , y , and z are all functions of t. w x y t z We can also have the functions x, y, and z be multivariable functions:
w
x s y z t
s t s t Word Problem Example:
The radius of the base of a cone is 6 cm and it is increasing at a rate of 2 cm/sec. The height of the cone is 10 cm and it is
increasing at a rate of 3 cm/sec. At what rate is the volume increasing? Finding Partials with Implicit Differentiation
Let z be a function of x and y...
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 Spring '08
 Loukianoff,V
 Calculus

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