Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Spring1B Find two or three classmates and a few feet of chalkboard Introduce yourself to your new friends and write all of your names at the top of the 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 some are 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 Review and Preview 1 a What is the definition of a definite integral b What is the definition of an indefinite integral c What is the statement of the Fundamental Theorem of Calculus FTC R2 2 a Use the definition of a Riemann sum to approximate 0 x2 x dx with four subintervals taking the sample points to be right endpoints R2 b Use FTC to calculate 0 x2 x dx exactly What is the error in your estimation from part a c Repeat steps a and b with n subintervals where n is an arbitrary number You may want to use the following facts n X i 1 i n X n n 1 2 i2 i 1 n n 12 n 1 3 Remark We will develop approximation methods better than the Right or Left Endpoint method later in this course You ve met one already in Math 1A the Midpoint approximation which takes the sample points for calculating a Riemann sum to be the midpoints of each subinterval Particularly ambitious students are invited to repeat steps a and b with the Midpoint approximation Rx 3 a Use geometry to evaluate f x 0 1 t2 dt b Check the Fundamental Theorem of Calculus by showing directly that f 0 x 1 x2 c Use the substitution t sin to evaluate f x You will need the Pythagorean Theorem and the Double Angle Formula 1 cos2 cos 2 1 2 1 sin2 cos2 Check that you get the same answer as in part a Remark We will develop methods similar to the one in part c with which to evaluate many integrals that involve terms like 1 t2 4 a State the Chain Rule for differentiation 1 b Explain how the Chain Rule leads directly via FTC to the Substitution Formula for integration 5 a What s wrong with the following calculations Z Z Z x x 1 e e dx x u u 1 du u3 2 u1 2 du u 1 e 2 2 2 2 u5 2 u3 2 C 1 ex 5 2 1 ex 3 2 C 5 3 5 3 Z Z u3 3 03 3 2 2 cos x sin x dx u du u cos x 0 3 0 3 0 3 0 b Evaluate the two integrals above correctly 6 Use a substitution to evaluate the following integrals Z 2 p Z 5 dt 2 3 a y y 1 dy b 2 0 1 t 4 Z Z x e dx d sin t cos t dt e x 7 a Use an integral to estimate the sum 1000 X Z c 1 sin 3 t dt 0 Z f cos ln x dx x i i 1 b Is your estimate an overestimate or an underestimate c Estimate the error in your answer to part a When estimating errors you should always provide an overestimate rather than an underestimate d Without a calculator how could you improve the estimate of the value of the sum Remark In addition to developing techniques through which we can estimate the value of an integral using a sum as in question 2 we will develop techniques to estimate the values of sums including infinite sums using integrals 8 Sketch the curves x y 0 and x y 2 3y and find the enclosed area 9 Sketch the curves y x2 1 and y 9 x2 and find the volume obtained by rotating the enclosed region around the line y 1 10 Let p x be a cubic function such that the curves y p x and y x2 intersect when x 0 x a and x b a Is the function p x uniquely determined by the above condition b Express the above condition algebraically in terms of the function p x x2 c The two curves inclose two regions If these two regions have the same area how is b related to a Remark there are various cases here depending on the order of 0 a b Up to switching the names of a and b and switching x 7 x there are two possibilities either 0 a b or a 0 b Consider those two cases 2
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