1 7. Define Taylor's Theorem and use it to find a second order approxima- tion to (1.1)¹/4. Suppre f(x) has a +1 devivatues at a het x be some other point. Taylo's Theorem saya: 2 • point. x=a. f(x) = f(a)+ f'(a)(x-a) + f "l(a)(x-²) + - + f(a)(x-²) + R₂ 2! n! To approximate of justure first 3 termand igne the rest. ntl R₁ = f(c)(x-a)" for some (n+1)! for some cin [a, x] or [24, α] depending xsa пна. Students may not see this so igine.

4. Let f be the function given by f(x) = 4 - x. g is a function with derivative given by g'(x) = f(x)ƒ'(x)(x - 2). On what intervals is g decreasing? a. (-∞0, 2] b. [2,00) C. [2,4] [4,00)

Q1) Check the continuity of the given function at the given c (or c's). 2 if z+3 (a) f(=) = { -3 c= 3 5 if 1= 3 1+z if zS -2 (b) f(x) = { 2-z if -2 2 Q2) Find the derivative of each function. f). f(e} = In(sec8 + tan 8) 8) f(z) = In () „a) f(z) = %3D b) f(z) = sin %3D %3D 1+cosz c) f(z) = tsinz d) f(z) = (1+z)*(1 – 2)² h) y = (In(ax))² 1+conz g(z) = In(r³ – 3z + 2) i) ) iy = %3D e) f(z) = 19 – 16 +-4 Q3) 27-30 Differentiate f and find the domain of f. 7. f(x) = 28. F(x) = /2 + In x (I – x)u – 1 22. f(x) = In(x² – 2x) 30. (x) = In In In x

please assist...

find f (x² + 1) da 2 can not be done 1 ¹ + ² + c ²x³ + x + 1 none of these answers

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 n

5. Let the derivative of the function f(r) be i od va av dait 11g(x) = x+sin(r-2) and ƒ(2+x)-f(2) I which of the following is the value of b? (a) -4 1 (b) € (c) (d) (e) 4 8 lim x-0 b 2

1. Use the graph of the function f(x) below to answer the following questions: a) For what values of a is the function not differentiable? b) What is f'(-3)? 4- c) What is f'(2) ? 2- d) What is f'(4)? 5 -4 -3 -2 -1.0 -2 e) Make a rough sketch the derivative of f below. -4 -4 +t

Use the graph of the derivative f' of a continuous function f is shown. (Assume f' continues to ∞.) y -4 y = f'(x) -2 X 6 Ⓡ (a) On what interval(s) is f increasing? (Enter your answer using interval notation.) [1,6]U[8,00) X On what interval(s) is f decreasing? (Enter your answer using interval notation.) [0,1] U [6,8] (b) At what value(s) of x does f have a local maximum? (Enter your answers as a comma-separated list.) X = 6 At what value(s) of x does f have a local minimum? (Enter your answers as a comma-separated list.) X = 1,8 (c) On what interval(s) is f concave upward? (Enter your answer using interval notation.) (0,2) U (3,5) U(7,00) X On what interval(s) is f concave downward? (Enter your answer using interval notation.) (2,3) U (5,7)

4.1(3)   Determines the derivative of each function.