MAT 1236 Calculus III Section 14.5 The Chain Rule

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Presentation transcript:

MAT 1236 Calculus III Section 14.5 The Chain Rule

HW.. WebAssign 14.5 Quiz: 14.5

Preview The Chain Rule for multivariable functions Look only at how to memorize the formulas Why the formulas actually work depends on 14.4 which we do not cover.

The Doppler Effect

1 variable Vs 2 variables

The Chain Rule

The Chain Rule: Case 1

Example 1 Find

The Chain Rule: Case 2

Example 2 Find

Other Cases Similar

Example 3 Find in terms of partial derivatives in x and y (i.e. ) Note that the function f is not given explicitly.

Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation What is the relationship between

Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation What is the relationship between

Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation

Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation Show that

Where to use? (If time permitted) Taken from MAT 3724 Applied Analysis (Mathematical Physics)

Wave Equation: Set Up Assumptions:

Thought Experiment

What would you expect to happen?

Thought Experiment

Modeling Wave Propagation in a String

Solution: d’Alembert’s Formula