Mathematics 31B Practice Final Questions Exam Date December 8 2012 Instructor D E Weisbart NAME please print legibly Your University ID Number Your Discussion Section and TA Signature There are FOURTEEN questions on this examination QUESTION VALUE SCORE 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 TOTAL 70 1 1 10 points Inverse Function Theorem Problems a Use the inverse function theorem to compute f 0 x if f x tan 1 x Note that the domain of tan x is given to be 2 2 What is the domain of tan 1 x Answer b Use the inverse function theorem to compute f 0 x if f x sec 1 x Note that the domain of sec x is given to be 0 2 2 What is the domain of sec 1 x Answer c Suppose that f is an invertible function on an open interval containing 5 Use the inverse function theorem to compute the equation of the line tangent to the graph of the function f 1 x at the point 3 5 if f 5 3 and if f 0 5 4 Answer 2 2 10 points Differentiating logs and exponential functions a Compute the derivative of the function f x log5 cos2 x x2 1 Answer b Compute the derivative of the function f x xx 2 sin x Answer c Compute the derivative of the function x f x xx Answer d Suppose that x 1 Compute the derivative of the function f x logx x2 Answer e Suppose that x 1 Compute the derivative of the function f x logx sin x 2 Answer 3 3 10 points Differential Equations a A radioactive isotope has half life 200 years If you initially have 1000 grams of the material how much will you have after 900 years Answer b Your friend gives you an unknown amount of a radioactive isotope It is noon At 3pm you have 200 grams and at 5pm you have 170 grams remaining What is the half life of the substance How much will you have when you wake up at 6 00am the next morning Why are you accepting gifts of radioactive substances from your friends Answer c Your better friend gives you an unknown amount of money in a bank account It is January 1st 2012 Your interest is continuously compounding at an unknown rate On January 1st 2014 you have 10 000 in the account and on January 1st 2018 you have 12 000 in the account What is the interest rate How much will you have in your account on January 1st 2036 The year 2036 is an important one because on April 13 2036 there is a small chance that asteroid Apophis will collide with the earth If it does the impact will release more than two and a half times the energy released during 1883 Krakatoa eruption and more than 10 times the energy released by the Tsar Bomba perhaps some of you should start thinking about ways to deflect it Answer 4 4 10 points Limits and L Hopital s rule a lim x 2 x 3 x 2 Answer b ex 1 x 0 x lim Answer c lim x 0 2 1 sin x x Answer d lim xx x 0 Answer 5 5 10 points Integrating Exponentials and Logarithms a Z xe2x dx Answer b Z sin 2x e3x dx Answer c Z ln x dx Answer d Z x2 ln x dx Answer e Z 1 dx ln x Answer f Z 1 dx x ln x Answer g Z 1 dx x ln x 2 Answer 6 6 10 points Integration a 1 Z 0 1 dx 16 4x2 Answer b Z 1 x 9 x2 dx 0 Answer c Z 1 dx 4 9x2 Answer d Z 4 9x2 dx Answer e Z x2 4 9x2 dx Answer f Z x3 1 4x2 dx Answer 7 7 10 points Harder Trigonometric Integrals a Z 4 sec2 x dx 0 Answer b Z 4 sec5 x dx 0 Answer c Z csc4 x dx Answer d Z sin2 2x cos x dx Answer e Z 4 sin2 x cos2 x dx 0 Answer f Z sin 1 x dx Answer 8 8 10 points Improper Integrals a Is the following integral convergent or divergent Z x2 dx 1 x4 0 Answer b Compute Z 2 1 dx x ln x 2 Answer c Is the following integral convergent or divergent Z 1 sin x x2 1 dx x x 0 Answer d Is the following integral convergent or divergent Z 1 dx x ln x 2 Answer 9 9 10 points Numerical Methods for Integration a Use the midpoint rule to estimate the value of equal to 5 Do the same using the trapezoid rule R1 0 2 ex dx with the number of intervals Answer b Estimate the error in both of the approximations above How many intervals do you need to ensure the accuracy is better than 001 Answer 10 10 10 points Partial Fractions a Calculate the following antiderivative Z x 1 dx x 1 2x 1 Answer b Calculate the following antiderivative Z dx x 1 x2 x 2 Answer c Calculate the following antiderivative Z x3 1 dx x 1 x 2 Answer 11 11 10 points Taylor s Theorem 1 a Use Taylor s Theorem to estimate the value of e 2 by approximating ex by a degree 5 polynomial Answer b Find an upper bound for the error in the above approximation Answer 12 12 10 points Sequences a Calculate the limit of the sequence an where an 2n2 5 3n2 n 8 Answer b Calculate the limit of the sequence an where q 4 n1 2 an 1 n Answer c Calculate the limit of the sequence an where an 1 n sin n Answer 1 d Calculate the limit of the sequence an where an 4n 7n n Answer 13 13 10 points Series a Is the series P 1 n 1 n convergent or divergent hint integral test Answer b Is the series P n3 n 1 n convergent or divergent hint ratio test Answer c Compute the infinite sum given by P 1 n 1 2n hint your lecture notes Answer d Is the series P 1 n 2 n ln n convergent or divergent hint integral test Answer e Is the series P 1 n 2 n ln n 2 convergent or divergent hint integral test Answer f Is the series P 1 n 2 n sin n convergent or divergent hint comparison test Answer g Is the series P 1 n 2 n3 1 convergent or divergent hint comparison test Answer h Is the series P 5n n 1 n convergent or divergent hint root test Answer 14 14 10 points Taylor s Series a Calculate the Taylor Series for the function f x ln 1 x with a 0 What is the radius of convergence of this series Answer b Calculate the Taylor Series …