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 1 For each of the following functions find f 0 a You may use only the definition of the derivative and Derivative Laws allowed on next week s midterm sum difference constant multiple power exponential 2t 1 1 a f t b f x c 3x 1 d f x 3 2x 4x2 t 3 x 2 2 Make a careful sketch of the graph of y sin x and sketch the graph of the derivative sin0 x In particular what are the zeros of sin0 and where is it positive and negative Can you guess the formula for sin0 based on the graph 3 What is the domain of the function f x x What is the domain of its derivative f 0 x 4 Suppose that f is a function with the property that f x x2 for all x Show that f 0 0 Then show that f 0 0 0 5 Differentiate the following functions using only the Derivative Laws allowed on the midterm 5 a f x 30 g F x 12 x d f x 5ex 3 b f t 21 t6 3t4 t h y ax2 bx c e y 3 x 2 1 4 x2 2 x 1 3 c g t t 8 f h x i v x 4 x x 6 Find equations for the tangent line and the normal line to the curve y 1 2x 2 at the point 1 9 1 7 Graph the function f x x and also find and graph the derivative F 0 x Are your x graphs consistent 8 Find all derivatives of the function f x x4 3x3 16x I e find the first derivative the second derivative etc until you get some derivative that is identically 0 9 What happens if you take the function 1 x and start differentiating Does the sequence of functions ever stop in the sense of eventually becoming identically zero Justify your answer 10 For what values of x does the graph of y x3 3x2 x 3 have a horizontal tangent 1 11 Show that the curve y 6x3 5x 3 has no tangent line with slope 4 12 Find an equation of the tangent line to the curve y x x that is parallel to the line y 1 3x 13 Find equations of both lines that are tangent to the curve y 1 x3 and are perpendicular to the line x 12y 1 14 Draw a picture to show that there are two tangent lines to the parabola y x2 that pass through the point 0 4 Find the coordinates of the points where these tangent lines meet the parabola 15 a Find equations of both lines through the point 2 3 that are tangent to the parabola y x2 x b Provide two proofs one using algebra and the other using a graph to show that there is no line through the point 2 7 that is tangent to the parabola 16 Find a second degree i e quadratic polynomial P such that P 2 5 P 0 2 3 and P 00 2 2 17 Let f x x2 if x 2 mx b if x 2 For what values of m and b is f x differentiable everywhere 18 Find the derivative of the function f x x x Be sure to specify at what points f is differentiable Product and Quotient Rules 19 Suppose that f 5 1 f 0 5 6 g 5 3 and g 0 5 2 Find f g 0 5 f g 0 5 and g f 0 5 20 Differentiate You may use the product and quotient rules x 1 a x3 2x ex c 3 x x 2 2 3 5 2 b u u u 2u t d t 1 2 e 2t 4 t2 f ax b cx d 21 Find f 0 x and f 00 x a f x x4 ex b f x x5 2 ex c f x x2 1 2x 22 How many tangent lines to the curve y x x 1 pass through the point 1 2 At what points do these tangent lines touch the curve 23 Use the Product Rule twice to prove that if f g h are differentiable then f gh 0 f 0 gh 3 2 d f x 3 f x f 0 x and use this to f g 0 h f gh0 Then take f g h to show that dx differentiate y e3x 24 If f and g are differentiable show that f g 00 f 00 g 2f 0 g 0 f g 00 Find similar formulas for f g 000 and f g 4 Do you notice a pattern Guess a formula for f g n 2
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