Mathematics 31B Practice Midterm 1 Exam Date October 22 2012 Instructor D E Weisbart NAME please print legibly Your University ID Number Your Discussion Section and TA Signature There are SEVEN questions on this examination Calculators notes and books may not be used in this examination You may not receive full credit for a correct answer if insufficient work is shown QUESTION VALUE SCORE 1 12 2 12 3 12 4 12 5 12 6 12 7 12 TOTAL 84 1 1 12 points a Simplify the following expression in terms of ln 2 and ln 5 Compute ln 209 ln 209 9 ln 20 9 ln 4 5 9 2 ln 2 ln 5 Answer b Your answer to the following question should be expressed as a whole number Compute 3 e ln 10 ln 2 3 e ln 10 ln 2 3 eln 10 eln 2 10 8 80 Answer c Compute lim n 1 4 3n 2n lim n 1 4 3n 2n 1 4 since if n 4 3n 31 Note that this was not the intended question I had a typo and intended to write 2n 8 4 lim 1 e3 n 3n Answer 2 2 12 points a Suppose that f x is differentiable that f 3 8 and that f 0 3 4 Then 0 1 f 1 8 4 Answer b Suppose that f x is differentiable that f 2 4 and that f 0 2 21 Find the equation for the line tangent to the graph of f 1 x at the point 4 2 y 2x 6 Answer 3 3 12 points a Compute the derivative of the function f x ln x2 2 e2x 2 x 2x 2 4x 1 e2x x 2 x2 Answer b Compute the derivative of the function 2 f x log4 xe3x 3x2 ln xe 3x2 2 3x2 d d 1 e 6x e 1 6x2 3x2 log4 xe dx dx ln 4 ln 4 x ln 4 xe3x2 Answer c Compute the derivative of the function f x x2 5x 1 2x 1 d ln x2 5x 1 2x 1 d 2x 1 ln x2 5x 1 d 2 x 5x 1 2x 1 e e dx dx dx d 2 2x 1 ln x2 5x 1 e 2x 1 ln x 5x 1 dx 2x 1 2x 5 2 2x 1 2 x 5x 1 2 ln x 5x 1 x2 5x 1 Answer 4 4 12 points a Suppose that y 0 x 2y 1 and y 0 10 Compute y 4 We have that Z dy 2y 1 Z dt so 1 ln 2y 1 t c 2 Therefore y t 1 19e2t 1 2 so y 4 1 19e8 1 2 Answer b A radioactive isotope has half life 100 years If you initially have 100 grams of the material how much will you have after 750 years 750 100 1 100 2 Answer c Your friend gives you and unknown amount of a radioactive isotope After 2 years you have 100 grams and after 4 years you have 80 grams How much will you have after 10 years 1 100 y 10 100e10 2 ln 80 Answer 5 5 12 points a Can you use l Hopital s rule to calculate the following limit x2 1 x 1 x 1 lim No The denominator does not tend to 0 although the numerator does and so the algebraic expression is not an indeterminate form Answer b 2 ex e lim x 1 x 1 By l Hopital s rule we have 2 2 ex e 2x ex lim lim 2e x 1 x 1 x 1 1 Answer c ln x x x lim By l Hopital s rule we have 1 ln x 1 lim x lim 0 x x x 1 x x lim Answer 6 6 12 points a Z 2x2 e3x dx Do integration by parts two times letting x2 be the guy differentiated on the first round and x be the guy differentiated on the second Answer b Z ln 3x dx Do integration by parts with ln 3x as the term to be differentiated and integrate 1 dx x ln 3x x C Answer 7 7 12 points a Z e2x sin 3x dx Integrate twice by parts as done in class Answer b 1 Z 0 Set x q 2 9 1 dx 2 9x2 tan so that dx q 2 9 sec2 d Plug in to get 1 2 r 2 tan 1 9 r 2 9 Answer c Z 1 1 Set x 1 dx 2 x2 2 sin so that dx 2 cos d Plug in to get 2 Answer 8