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 Definite Integral 1 Estimate the area under the graph f x cos x from x 0 to x 2 using four rectangles and either left or right endpoints Determine whether your estimate is an underestimate or an overestimate 2 Speedometer readings for a motorcycle at 12 second intervals are given in the table below Estimate the distance traveled by the motorcycle during the first minute of driving Can you determine whether the left or right estimates are over or underestimates t s 0 12 24 36 48 60 v ft s 30 28 25 22 24 27 3 Find an expression for the area under the graph of f as a limit Do not evaluate the limit a f x 4 x 1 x 16 b f x x cos x 0 x 2 4 Determine a region whose area is equal to the given limit Do not evaluate the limit n X i tan n 4n 4n a lim b lim n i 1 n 1 X i 0 ln 1 ni n i 5 a Find an expression for the area under the curve y x2 from 0 to 2 using right endpoints b Use the fact that 2 2 2 2 2 1 2 3 n 1 n n X i2 i 1 n n 1 2n 1 6 to evaluate your expression in part a c What happens if you use left endpoints 6 Evaluate the Riemann sum for f x 3 12 x 2 x 14 with six sub intervals taking the sample points to be the left endpoints What does the sum represent 7 Express the limit as a definite integral on the given interval a lim n n X i 1 xi ln 1 x2i x 2 6 b lim n n q X i 1 2x i x i 2 x 1 8 8 Using only P theRfollowing facts and the linearity Sum Difference and Constant Multiple rules of and evaluate the integrals below The facts n X 1 i 1 n n n n X n n 1 X 2 n n 12 n 1 X 3 n2 n 1 2 i i i 1 2 3 4 i 1 n 1 i 1 The integrals Z 5 Z 1 3x dx a 1 2 Z 2 2 x dx b 5 Z 2 1 2x dx c x3 dx 1 0 0 2 d 9 Using the facts in the previous problem prove the following formulas Z b a a b2 a2 x dx 2 Z b b3 a3 x dx 3 Z 2 b a b x3 dx c a b4 a4 4 Rb 10 Based on the formulas from the previous problem guess the formula for a xn dx Check that your formula works for n 0 What happens when n 1 For which n are you confident that your formula is correct 11 Evaluate each of the following integrals by interpreting them in terms of areas Z 3 a 0 Z 1 x 1 dx 2 Z c 3 2x dx 1 Z p 4 x2 dx Z 0 d 2 x dx e 2 3 b 2 1 1 p 9 x2 dx 3 Z 1 f x5 cos x dx 1 R 4 cos2 x dx 0 sin2 x dx R 4 b Use the previous problem and the Pythagorean theorem to find 0 sin2 x R4 R5 R5 13 If 1 f x dx 12 and 4 f x dx 3 6 find 1 f x dx R5 3 x 3 14 Find 0 f x dx if f x x x 3 12 a Use the fact that cos x sin 4 x to prove that 2 R 4 0
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