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Polynomial Approximations. BC Calculus. Intro:. REM: Logarithms were useful because highly involved problems like Could be worked using only add, subtract, multiply, and divide.
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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: Requires a Polynomial with: a) the same y – intercept: b) the same slope: c) the same concavity: the same rate of change of concavity: 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: • 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 fat 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: the Polynomial (Series) representation of a function is unique. Theorem: If a function has a polynomial (Series) representation that representation will be the TAYLOR POLYNOMIAL (Series)
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 F-3 Calc #9 taylor ( taylor ( f (x) , x , order , point) taylor ( sin (x) , x , 3 , )
