Math 1A Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Spring1A Find two or three classmates and a few feet of chalkboard As a group try your hand at the following exercises Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises in particular the last few exercises may be very hard Many of the exercises are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart these are marked with an Others are my own or are independently marked Derivatives and graphing 1 To the right the graph of the derivative f 0 of a function f is shown On what intervals is f increasing Decreasing Where is f concave up Concave down At what values does f have a local maximum or minimum Inflection points X y f 0 x 2 4 6 2 Sketch the graph of a function f such that f 0 1 f 0 1 0 f 0 x 0 if x 1 f 0 x 0 if 1 x 2 f 0 x 1 if x 2 f 00 x 0 if 2 x 0 and such that 0 1 is an inflection point of y f x 3 Suppose that f x 2 f 0 3 21 and f 0 x 0 and f 00 x 0 for all x a Sketch a possible graph of f b How many solutions does the equation f x 0 have c Is it possible that f 0 2 13 4 For each of the following functions find intervals when f is increasing intervals when f is decreasing intervals when f is concave up intervals when f is concave down local extreme of f inflection points Then sketch a graph of the function a f x 4x3 3x2 6x 1 c f x xe x c f x x1 3 x 4 b f x x2 ln x d f x 200 8x3 x4 d f x x cos x 5 Sketch a careful graph of y x2 x 1 ex Label any interesting features intercepts asymptotes extrema points of inflection 6 a Let f x x r ex Use calculus to sketch a graph of f x and label the zeros local extrema and inflection points Also label the y intercept and any horizontal asymptotes b Let f x x2 bx c ex What is the behavior of f x as x Use the Mean Value Theorem to show that if f x has one or two zeros then it must have two local extrema 1 c More generally let f x p x ex where p x is a polynomial of degree n Show that if f x has exactly n real distinct zeros then it also has exactly n local extrema and exactly n inflection points d Harder Let s return to the case when f x x2 bx c ex Prove that the zeros of f correspond to the zeros of q0 x x2 bx c f x ex and the number of these is classified by the determinant b2 4c For n a non negative integer define qn x to be f n x ex the polynomial part of the nth derivative of ex Prove that qn x is a quadratic for any n How if the determinant of q1 x related to the determinant of q0 x Is this the same as the relationship between the determinants of qn x and qn 1 x for arbitrary n Why or why not Prove that for n large enough qn x and hence f n x will have two roots 7 Suppose the derivative of a function f is f 0 x x 1 2 x 3 5 x 6 4 On what intervals is f increasing What are the local maxima of f 8 Use calculus to sketch the family of curves y x3 3a2 x 2a3 where a is a positive constant x 1 9 Find the value of x such that f x increases most rapidly x2 1 10 Find a cubic function f x ax3 bx2 cx d that has a local maximum value of 3 at x 2 and a local minimum value of 0 at x 1 2 11 For what values of the numbers a and b does the function f x axebx have the maximum value f 2 1 12 Show that the curve y straight line 1 x has three points of inflection and that they all lie on one 1 x2 13 Show that the curves y e x and y e x touch the curve y e x sin x at its inflection points 14 Show that tan x x for 0 x 2 Hint show that f x tan x x is increasing on 0 2 15 Show that a cubic function always has precisely one point of inflection Show that if the graph has three x intercepts x1 x2 and x3 then the x coordinate of the inflection point is x1 x2 x3 2 What is the similar statement about local extrema of a quadratic function 16 a Show that f x x4 is such that f 00 0 0 but 0 0 is not an inflection point of the graph of f b Show that g x x x has an inflection point at 0 0 but g 00 0 does not exist c Let f be any function Use the First Derivative Test and Fermat s Theorem on the function g f 0 to show that if c f c is an inflection point and f 00 exists in an open interval that contains c then f 00 c 0 2