Let f(x) be differentiable at xo. Using the lim h→0 - ♪ language to prove: f(xo + h) − f(xo – h) - 2h - = = f'(xo).
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- 3. Let f (x) = (3x2 + 1)?. Find f'(x)in 3 different ways by following the instructions below in parts a, b and c: a) Algebraically multiply out the expression for f (x) and expand, then take the derivative. b) View f (x) as (3x? +1)(3x2 + 1) and use the product rule to find f' (x). C) Apply the chain rule directly to the expression f (x) = (3x² + 1)?. d) Are your answers in parts a, b, c the same? Why or why not?Estimate of f′(2):Assume that f'' exists and f'' (x) = 0 for all x. Prove that f (x) = mx + b, where m = f'(0) and b = f (0).
- a) Let n be a postive integer. In this part a) you will prove the Power Rule for the root functions y 1 Vx = x* on (0, ). Use inverse functions to show that if f(x) x" then the inverse dy function y = f-1(x) = x* has the derivative dx 1r-1. n b) Let n be a postive integer and m be an integer. In this part b) m т you will prove the Power Rule for rational exponents y = xn = on the interval (0, ∞). d Assume the result dx --1 of part a) and use the Chain Rule to show that d m m -1 dx n11. Let f(z)= 2x³ +1. a) Find f(x+ h): b) Find f(x + h) – f(x): c) Find +h)-f(x). d) Find f'(x): 12/15) search IA 近For the function f(x) = In (x + 3), find f"(x), f'"(0), f'(2), and f'"(- 4). 1 f"(x) = (x + 3)2 (Use integers or fractions for any numbers in the expression.)
- d) Consider the fourth-order truncation error (0(Ax )4) with the step size Ax and also (-Ax ) then, add them and finally obtain a finite difference approximation for the second-order derivative of U respect x (Uxx (x, t)).Use the product rule to differentiate the function h, where a and b are real numbers. h(x) = e¯ax cos (bx) W' (x) =Let f(x) be a function that is differentiable for all x. Let g(x) be defined by g(x) = f(x) + f(3 – x). Show that g'(x) has a root in the interval (0,3). (a) (b) Let f(x) be continuous and differentiable on the interval [-8,0]. Suppose f(-8) = -2 and f'(x) < 3 for all x. What is the largest possible value for f (0)? Justify your answer.
