Math 141 Quiz 1 Sept 4 1 True or False The function f x ln x 2 is one to one on 0 A True B False Since for example f 1 2 f 2 2 2 Circle the correct answer If y A xx sin x e x 1 then dy dx x2x cos x e x 1 2 e x e x 1 e x C cot x x 2x log x x e 1 2 xx sin x e x D x cot x x 2x log x x e 1 e 1 2 xx sin x e x E x cot x x 2x log x x e 1 e 1 Use logarithmic differentiation B cot x x 2x log x Re 3 Evaluate the definite integral 1 x xdxln x Factor and use the substitution u 1 ln x ln 2 4 Evaluate the indefinite integral Use the substitution u 2x 1 R x 1 2x 1 dx 1 x 1 ln 2x 1 4 2 4 Math 141 Quiz 2 Sept 11 2 Circle the correct answer Simplify sin tan 1 x 1 x2 A x 1 x2 B C 1 x x D 1 x2 E 1 x 1 2 Circle the correct answer Evaluate limx tan 1 ex A B C D E 4 4 2 3 Evaluate the indefinite integral Use the substitution u 1 ex R x e x dx 1 e 2 1 ex C 4 Evaluate the indefinite integral R cos x dx 1 sin x 2 Use u sin x tan 1 sin x C Math 141 Quiz 3 2 Circle the correct answer Evaluate A B C D E 2 15 8 15 2 3 4 5 Use u sin x 4 5 Sept 18 R 2 0 cos x 5 dx Also written R 2 0 cos5 x dx 2 Circle the correct answer Evaluate A B C D E 1 6 x ln x 6 1 6 x ln x 6 1 6 x ln x 6 1 6 x ln x 6 1 6 x ln x 6 R x5 ln x dx x6 C 36 x6 C 36 x6 C 6 x6 C 6 Use u ln x dv x5 dx in integration by parts 3 Evaluate the indefinite integral 2x e x 1 2 4 xe2x dx R 2 sin 2t 2 dt Also written 2 sin2 2t dt 4 Evaluate the indefinite integral t R 1 sin 4t 4 Math 141 Quiz 4 1 Circle ALL of the correct answers 5 points L Hospital s rule applies directly to which of the following limits A B C sin x x 0 x sin x lim x x x e lim x x lim 2 Evaluate limx ln ex 1 x 1 cos x 1 3 Evaluate limx 0 ex 1 x 1 R Sept 25 5 4 Evaluate limx 0 e2xx 1 0 Math 141 Quiz 5 Sept 25 x3 1 A B C 1 There are constants A B C such that x x 1 x 2 x x 1 x 2 A True B False Everything is fine except that the rational function on l h s is not proper 2 Circle the correct answer With the substitution x becomes Z A 2 tan d Z B 2 tan 2 d Z C 2 sec 2 d Z D 2 sec 3 d Z E 2 sin 2 d 3 Evaluate the indefinite integral R R 2 tan the integral x2 2 dx tan t 3 sec t 6 dt sec8 t sec6 t C 8 6 R x 3 4 Evaluate the definite integral x 0 9 x2 dx 9 4 Math 141 Quiz 6 Oct 9 1 Circle the correct answer 5 points n 1 The sequence defined by an 1 n for n 1 matches which of the following 1 1 1 1 1 2 3 4 5 1 1 1 1 B 1 2 4 8 32 1 1 1 1 C 1 2 3 4 5 1 1 1 1 D 1 2 4 8 32 A 2 Evaluate A R1 dx 1 x2 1 B 0 C 1 D 2 E does not exist integral diverges R 4 Evaluate the improper integral 2 e1x dx If it does not converge write does not exist 1 e2 R 1 1 4 Evaluate the improper integral 1 1 x 5 x dx If it does not converge write does not exist ln 3 Math 141 Quiz 7 sin n n n 1 Since lim Divergent Series Test A True B False 0 the convergence of Oct 9 P n 1 sin n n may be deduced directly from the 2 Suppose that it is known that m P n 1 3n2 n 1 n2 n 1 2 2m2 m m 1 2 Evaluate P n 1 3n2 n 1 n2 n 1 2 A 0 B 1 C 2 D 3 E the series diverges 18 18 3 Evaluate the infinite geometric series 18 10 100 1000 A 0 B 1 C 2 D 3 E the series diverges 4 Let 0 p 1 and let A be the series P n 1 A 1 np and let B be the series P n 1 1 n 1 np A diverges and B converges B A converges and B converges C A diverges and B diverges D A converges and B diverges P Bonus Evaluate 1 1 2 n 2 n 1 n 1 A 0 B C 1 2 D 2 E the series diverges F I don t know Math 141 Quiz 9 P Oct 29 n n 1 en2 converges Say which test you use Integral Test Root Test 1 n 1 n 1 Since limn enn2 limn nen 0 1 the Root Test says it is convergent 1 Show that P n n n 1 1 en2 A does not converge and does not converge absolutely 2 Cicle the correct answer The series B converges but only converges conditionally C converges and converges absolutely D does not converge but does converge absolutely P n n 1 2n converges Ratio test n Since limn 2n 2 limn 2n 2 2n 1 0 1 the Ratio Test says it is convergent n 1 2n n 1 P n 4 Does n 1 n5 n converge Limit Comparison Test P 1 P n Yes We may do a limit comparison with n 1 n3 2 which is a convergent n 1 n5 p series 3 Show that Math 141 Quiz 10 Oct 30 P 1 n 2n 1 n 1 3 n Diverges by Ratio Test P n ln n 2 n 2 1 n Converges by AST but only conditionally by Integral Test or DCT 3 P tan 1 n n n 1 2 Converges absolutely by Root Test 4 Cicle the correct 5answer 5 It is known that 1536 1 315 515 715 915 1115 Which of the following is a reasonable estimation of the number of decimal digits correct in the approximation 5 5 1 1 1 1 1 5 …