Polynomial Approximations BC Calculus. Intro: REM: Logarithms were useful because highly involved problems like Could be worked using only add, subtract,

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Presentation transcript:

Polynomial Approximations BC Calculus

Intro: REM: Logarithms were useful because highly involved problems like Could be worked using only add, subtract, multiply, and divide THE SAME APPLIES TO FUNCTIONS - The easiest to evaluate are polynomials since they only involve add, subtract, multiply and divide.

Polynomial Approximations To approximate near x = 0: a) the same y – intercept: b) the same slope: c) the same concavity: d)the same rate of change of concavity: Requires a Polynomial with: e) the same.....

Polynomial Approximations To approximate near x = 0: same y – intercept:

Polynomial Approximations To approximate near x = 0: same y – intercept: the same slope: We want the First Derivative of the Polynomial to be equal to the derivative of the function at x = a

Polynomial Approximations To approximate near x = 0: same y – intercept: the same slope: the same concavity:

Polynomial Approximations To approximate near x = 0: same y – intercept: the same slope: the same concavity: the same rate of change of concavity.

Called a Taylor Polynomial (or a Maclaurin Polynomial if centered at 0)

Method: (A)Find the indicated number of derivatives ( for n = ). Beginning point Slope: Concavity: etc…….. (B) Evaluate the derivatives at the indicated center. ( x = a ) (C) Fill in the polynomial using the Taylor Formula

Example:: Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = 0.

Example:: Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = 0.

Example: Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = a.

Example: Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = a.

Taylor and Maclaurin Polynomials In General (for any a ) Taylor Polynomial Maclaurin if a = 0 Theorem: If a function has a polynomial (Series) representation that representation will be the TAYLOR POLYNOMIAL (Series) Theorem: the Polynomial (Series) representation of a function is unique.

Example:: Find the Taylor (Maclaurin) Polynomial of degree 3 approximation of f at x = 0. Use it to approximate f (.2)

Example:: Find the Taylor (Maclaurin) Polynomial of degree 3 approximation of f at x = 0. Use it to approximate f (.2)

Taylor’s on TI - 89 taylor ( f (x), x, order, point) F-3 Calc #9 taylor ( taylor ( sin (x), x, 3, )

Last update: 4/10/2012 Assignment: Wksht:DW 6053