NEWTONS METHOD TANGENT APPROXIMATIONS Section 3 8 and

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NEWTON’S METHOD &TANGENT APPROXIMATIONS Section 3. 8 and 3. 9 Calculus AP/Dual, Revised ©

NEWTON’S METHOD &TANGENT APPROXIMATIONS Section 3. 8 and 3. 9 Calculus AP/Dual, Revised © 2013 viet. dang@humble. k 12. tx. us 12/30/2021 11: 02 PM 3. 9 - Tangent Line and Line Approximations 1

NEWTON’S METHOD A. Consider a function that is continuous on the closed interval [a,

NEWTON’S METHOD A. Consider a function that is continuous on the closed interval [a, b] and is differentiable at (a, b). If f(a) and f(b) differs in signs, then by the Intermediate Value Theorem, there must exist one zero in the interval (a, b). B. Newton’s Method is used to approximate the real zeros of a function C. Tangent Lines are used to approximate the graph of a function near the x-intercepts D. Successive applications of this procedure is called iteration. E. Newton’s Method fails if the derivative is zero for any xn. If it happens, find another x 1 and start over F. The xn does not converge (approaches to a limit) 12/30/2021 11: 02 PM 3. 9 - Tangent Line and Line Approximations 2

STEPS 12/30/2021 11: 02 PM 3. 9 - Tangent Line and Line Approximations 3

STEPS 12/30/2021 11: 02 PM 3. 9 - Tangent Line and Line Approximations 3

EXAMPLE 1 Complete the two iterations of Newton’s Method for f(x) = x 2

EXAMPLE 1 Complete the two iterations of Newton’s Method for f(x) = x 2 – 3 given that x 1 = 1. 7 12/30/2021 11: 02 PM 3. 9 - Tangent Line and Line Approximations 4

EXAMPLE 1 Complete the two iterations of Newton’s Method for f(x) = x 2

EXAMPLE 1 Complete the two iterations of Newton’s Method for f(x) = x 2 – 3 given that x 1 = 1. 7 12/30/2021 11: 02 PM 3. 9 - Tangent Line and Line Approximations 5

EXAMPLE 1 Complete the two iterations of Newton’s Method for f(x) = x 2

EXAMPLE 1 Complete the two iterations of Newton’s Method for f(x) = x 2 – 3 given that x 1 = 1. 7 12/30/2021 11: 02 PM 3. 9 - Tangent Line and Line Approximations 6

YOUR TURN Complete the two iterations of Newton’s Method for f(x) = x 3

YOUR TURN Complete the two iterations of Newton’s Method for f(x) = x 3 + x – 1 given that x 1 = 0. 5 and continue the process until two successive approximations differ by less than 0. 001 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations 7

EXAMPLE 2 The function f(x) = x 1/3 is not differentiable at x =

EXAMPLE 2 The function f(x) = x 1/3 is not differentiable at x = 0. Show that Newton’s Method fails to converge using x 1 = 0. 1. 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations 8

TANGENT LINE APPROXIMATION 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line

TANGENT LINE APPROXIMATION 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations 9

EXAMPLE 3 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations

EXAMPLE 3 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations 10

EXAMPLE 3 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations

EXAMPLE 3 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations 11

YOUR TURN 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations

YOUR TURN 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations 12

ASSIGNMENT Page 233: 1 -9 odd Page 240: 1 -6 all (tangent line only)

ASSIGNMENT Page 233: 1 -9 odd Page 240: 1 -6 all (tangent line only) 12/30/2021 11: 03 PM 3. 9 - Tangent Line and Line Approximations 13