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 Chain Rule 1 Differentiate a 4x x2 100 e a3 cos3 x i tan2 3 b 1 x4 3 2 c 3 1 tan t f t4 1 3 t3 1 4 j ek tan d y 1 4 y 2 2y 5 g 101 x h 2 eu e u eu e u x k sin sin sin x 2 2 y l y 1 2 Suppose that f is differentiable on R Let F x f ex and G x ef x Find expressions for F 0 x and G0 x 3 For what values of r does the function y erx satisfy the differential equation y 00 5y 0 6y 0 4 Find the 50th derivative of y cos 2x Find the 1000th derivative of f x xe x 5 Air is being pumped into a spherical balloon At any time t the volume of the balloon is V t and the radius is r t What do the derivatives dV dr and dV dt represent What is the relationship between dV dt r and dr dt 6 Use the chain rule to prove that the derivative of an even function is an odd function and that the derivative of an odd function is an even function 7 If n is a positive integer prove that d sinn x cos nx n sinn 1 x cos n 1 x dx d Find a similar formula for dx cosn x cos nx 8 The Leibniz notation makes the chain rule very natural looking dy dy du dx du dx 1 d2 y d2 y du2 d2 y du 2 However the corresponding formula for the second derivative dx 2 du2 dx2 du2 dx is false Instead prove the following chain rule for second derivatives by using the chain and product rules d2 y d2 y du 2 dy d2 u dx2 du2 dx du dx2 9 a Let f be a differentiable function such that f 0 x 0 for all x By drawing a graph show that f is an increasing and hence one to one b Let g be the inverse of f so that f g x g f x x Use the chain rule to find the derivative of g in terms of the derivative of f c Show that ln0 x the derivative of ln x is 1 x Harder questions on earlier material 10 Write out the first few derivatives f f 0 f 00 of f x xex Do you notice a pattern 11 a Prove that if p is a polynomial of degree n then the derivative of p x ex is q x ex where q is also a polynomial of degree n b Let f x p x ex where p is a polynomial What is lim f n x x 12 Let r be a rational function so that r x p x q x for some polynomials p and q Define the degree of r to be deg p deg q where deg p is the degree of p i e the highest power of a non zero term in p Prove that r0 the derivative of r is a rational function and prove that deg r0 deg r 1 13 a Use the definition of derivative to prove the product rule b Use the product rule to prove the quotient rule c Let p be a polynomial Use the product rule but not the chain rule to prove that d 0 0 dx p q x p q x q x In fact from just the product rule you can prove the chain rule provided that the outer function is a rational function ratio of two polynomials 14 What s the derivative of sin2 x What s the derivative of cos2 x What happens when you add them together and why 15 Find numbers A and B so that d Aex cos x Bex sin x ex cos x dx 16 Let f x ax Then f 0 0 ln a Why Let s say we didn t know that Define the d x function a for a 0 by a dx a x 0 Use the product rule to show directly that ab a b Use the quotient rule to show directly that a b a b Show directly that 1 0 2
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