4.1-linearization-and-differentials.pdf - 4.1 Linearization...

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4.1 Linearization & Differentials Daniel James Linear Ap- proximation Example 1 Example 2 Linearization Example 3 Example 4 Example 5 Differentials Example 6 4.1 Linearization & Differentials Daniel James University of Kansas Department of Mathematics 1 / 19
4.1 Linearization & Differentials Daniel James Linear Ap- proximation Example 1 Example 2 Linearization Example 3 Example 4 Example 5 Differentials Example 6 Introduction Edward Lorenz - American mathematician and meteorologist who established the theoretical basis for weather prediction. In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the higher-precision 0.506127 value. The result was a completely different weather scenario. Butterfly Effect 2 / 19
4.1 Linearization & Differentials Daniel James Linear Ap- proximation Example 1 Example 2 Linearization Example 3 Example 4 Example 5 Differentials Example 6 Introduction Sometimes, we are interested in how a small change in the input value of a function can effect a change in the value of the function itself. Mathematically, we have a function f and we’re interested in the change Δ f = f ( a + Δ x ) - f ( a ) where Δ x is small. This value appears in the definition of one of the big tools we’ve developed so far: f 0 ( a ) = lim Δ x 0 f ( a + Δ x ) - f ( a ) Δ x = lim Δ x 0 Δ f Δ x When Δ x is small, we have Δ f/ Δ x f 0 ( a ) , and thus, Δ f f 0 ( a x 3 / 19
4.1 Linearization & Differentials Daniel James Linear Ap- proximation Example 1 Example 2 Linearization Example 3 Example 4 Example 5 Differentials Example 6 Introduction Sometimes, we are interested in how a small change in the input value of a function can effect a change in the value of the function itself. Mathematically, we have a function f and we’re interested in the change Δ f = f ( a + Δ x ) - f ( a ) where Δ x is small. This value appears in the definition of one of the big tools we’ve developed so far: f 0 ( a ) = lim Δ x 0 f ( a + Δ x ) - f ( a ) Δ x = lim Δ x 0 Δ f Δ x When Δ x is small, we have Δ f/ Δ x f 0 ( a ) , and thus, Δ f f 0 ( a x 3 / 19
4.1 Linearization & Differentials Daniel James Linear Ap- proximation Example 1 Example 2 Linearization Example 3 Example 4 Example 5 Differentials Example 6 Introduction Sometimes, we are interested in how a small change in the input value of a function can effect a change in the value of the function itself. Mathematically, we have a function f and we’re interested in the change Δ f = f ( a + Δ x ) - f ( a ) where Δ x is small. This value appears in the definition of one of the big tools we’ve developed so far: f 0 ( a ) = lim Δ x 0 f ( a + Δ x ) - f ( a ) Δ x = lim Δ x 0 Δ f Δ x When Δ x is small, we have Δ f/ Δ x f 0 ( a ) , and thus, Δ f f 0 ( a x 3 / 19
4.1 Linearization & Differentials Daniel James Linear Ap- proximation Example 1 Example 2 Linearization Example 3 Example 4 Example 5 Differentials Example 6 Introduction Sometimes, we are interested in how a small change in the input value of a function can effect a change in the value of the function itself.
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