use Integration by Parts to derive 12.6.1 THEOREM 12.6.1 TAYLOR'S THEOREMIff has n+1 continuous derivatives on an open interval I that contains 0,then for each x Є If" (0)f(n) (0)f(x):=f(0) + f'(0) x +-x² +2!+-x" + Rn(x)n!Xwith Rn(x)=1 * (+1) (10)(x = 0)" dt. We call R,(x) the remainder.n!

Step-by-Step Derivation Step 1: Start with the Integral Expression Step 2: Apply Integration by…

Answered: THEOREM 12.6.1 TAYLOR'S THEOREM Iff has…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

See similar textbooks

Similar questions

y = f(x) ++ 3. Exer. 33-36: Use right-hand and left-hand derivatives to prove that f is not differentiable at a. 33 f(x) = |x - 5|, a = 5 34 f(x) = |x + 2|, a = -2 35 f(x) = [x - 2; a = 2 36 f(x) = [x]- 2; a 2 Exer. 37-40: Use the graph of f to find the domain of f'.
2. a. f(x) = (x+2)/(x +4). Use the Mean Value Theorem on the interval [0, 4] to find all values of x in the interval (0, 4) such that the value of the first derivative of f at those points is (f(b) – f(a))/(b – a). b. For the interval [-5, 0], fb) - fa))(b — а)-GG0) — f-5)(0 — (-5)) = (½ – 3)/(0 + 5) = -½ but, since the first derivative of f(x) is always positive, nowhere on that interval does the first derivative of f(x) = -½. Thus the Mean Value Theorem does not hold for f(x) on this interval. Why?
Let fand g be the functions whose graphs are shown below. -4-3-2 4- 3 2 1 -1- -2 -3- -4- N f(x) 2 3 4 X g(x) (a) Let u(x) = f(x)g(x). Find u'(4). (b) Let v(x) = g(g(x)) . Find v'(-3). 10 5
Prove that the function ax+b cx+d ƒ(x)=· is constant (takes the same value for every x such that cx+d #0) if and only if la bl | (Hint: Find the derivative of f(x). The quotient rule for derivatives is: U = 0. u'v-uv' 1² .)
ƒ" (x) = x(x − 1)²(x + 2)³ g" (x) = x(x − 1)²(x + 2)³ + 1 h" (x) = x(x − 1)²(x + 2)³ - 1 The twice-differentiable functions f, g, and h have second derivatives given above. Which of the functions f, g, and h have a graph with exactly two points of inflection? A B D g only honly fand g only f, g, and h 12
#10 ch2 pt1

Recommended textbooks for you

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

THEOREM 12.6.1 TAYLOR'S THEOREM Iff has n+1 continuous derivatives on an open interval I that contains 0, then for each x Є I f" (0) f(n) (0) f(x): = f(0) + f'(0) x + -x² + 2! + -x" + Rn(x) n! X with Rn(x) = 1 * (+1) (10)(x = 0)" dt. We call R,(x) the remainder. n!