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 The Fundamental Theorem of Calculus part 2 1 Use the Fundamental Theorem of Calculus to evaluate the following integrals Z 5 a Z 6 dx 1 2 Z x4 5 dx e 0 d Z Z 4 5 2t 3t2 dt h 1 1 9 4 sec2 t dt j 0 Z 1 e2u du k 0 1 y 1 2y 1 dy 0 Z 2 Z 1 3 1 4 2 9 dt b 1 u u du f 4 2 5 1 t 0 Z 2 Z 2 3 x 2 x5 dx g c x 2x dx Z 2 i 1 x 1 dx x Z 3 2 l 1 2 2 What s wrong with the following equations 1 Z x 4 dx 2 Z 3 x 3 3 1 2 3 8 sec tan d sec 3 3 3 What s wrong with the following equations 4 Z 6 Z 1 1 dt cos t sin t 2 3 x dx 0 r 4 2 2 6 x2 3x 2 2 2 0 4 a What is the derivative of g x x ln x b Use your answer to part a to find an antiderivative for f x ln x R2 c Use your answer to part b to evaluate 1 ln x dx 6 dt 1 t2 n X i3 by first recognizing the sum as a Rieman sum for a function defined n n4 i 1 on 0 1 and then evaluating the corresponding integral using the Fundamental Theorem of Calculus 5 Evaluate lim 6 a Prove that cos x2 cos x for 0 x 1 Hint What s the relationship between x and x2 R 6 b Deduce that 0 cos x2 dx 12 Z 10 x2 7 Prove that 0 dx 0 1 by comparing the integrand to a simpler function x4 x2 1 5 8 If f is continuous on a b prove that Z 2 b f x f 0 x dx f b 2 f a 2 a 9 A manufacturing company owns a major piece of equipment that depreciates at the continuous rate f f t where t is the time measured in months since its last overhaul Because a fixed cost A is incurred each time the machine is overhauled the company wants to determine the optimal time T in months between overhauls Rt a Explain why 0 f s ds represents the loss in value of the machine over a period of time t since the last overhaul b Let C C t be given by 1 C t t Z A t f s ds 0 What does C represent and why would the company want to minimize C c Show that C has a minimum value at the numbers t T where C T f T Rx 10 If f is a differentiable function on 0 such that f x is never 0 and 0 f t dt f x 2 for all x 0 find f Z 1 x 11 Evaluate lim 1 tan 2t 1 t dt x 0 x 0 12 Evaluate the following limit by interpreting it as a Riemann sum for a continuous function on the interval 1 2 and evaluating the integral using by the Fundamental Theorem of Calculus 1 1 1 1 1 lim p n n n n n n 1 n n 2 n n 3 n n n 1 2
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