Math 4 Review 3(aswers) kt A t A e. If the half-life of radium is 690 years, ad you have 0 grams ow, how much will be preset i 50 years (rouded to three decimal places)?. The decay of radium is modeled by 0 A t 5 0 0e kt k690 e k l 50 690 l 690 A 50 0 9.797053... 9.797 grams e. The logistic growth model Pt.439 bacteria after t hours. 000 represets the populatio (i grams) of t 3.33e a) What is the carryig capacity of the eviromet?,000 grams b) What is the iitial populatio size (rouded to two decimal places)? 30 grams c) Whe will the populatio be 700 grams (rouded to two decimal places)? 000.439 t.439t 300 700 300 700 3.33e e.439t 3.33e 700 3.33 300 l 300 700 3.33.439t l t 9.848048866... 9.85 hours 700 3.33.439
x 3. The equatio x 3 e has two solutios. I approximatig them usig the Method of Successive Approximatios(see the graph), which solutio is a repellig solutio: solutio#, solutio#, or both? x y e 3 y x solutio# solutio# Solutio # is the repellig solutio. 4. Use the method of fiite differeces to fid a formula for geeratig the terms of the sequece,3,7,3,,3,43,57,. 3 7 3 3 43 57 st differece 4 6 8 0 4 d differece a A B C A A 3A B B A B C C a
5. Write out the first five terms of the followig sequeces: a a) a b) a, a ; 3 5,,,,, 3 5 3 7,,,,, 6 4 6. Use the followig formulas: k 3 k k k 4 9 6 3 3 k 8 7 64 k to fid the exact values of the followig series: 6 a) 00 k 00 0 k 0,00 40 40 40 k k k k b) k 4 40 48 4 4 40,300 6 c) 34 34 3 3 3 3 34 35 34 k k k k4 k k 353,989 7. Cosider the arithmetic sequece 5,,, 4, 7, 0,. a) Write a formula that will geerate the terms of the sequece. a 5 3 8 3 b) Fid the sum of the first 00 terms of this sequece. 5 9 S00 00 4,350
8. Fid the first term ad the commo differece for each of the followig arithmetic sequeces: a) 5 th term is 0 ad the 40 th term is -50 a 4d 0 a 39d 50 b) th term is 4 ad the 8 th term is 8 a d 4 5d 50 d, a 8 a 7d 8 6d 4 d 4, a 40 9. Fid x so that x,3x,5x 3 are cosecutive terms of a arithmetic sequece. 3x x 5x 3 3x x x x 0. How may terms must be added i a arithmetic sequece whose first term is 78 ad whose commo differece is -4 to get a sum of 70? 78 78 4 70 404 60 4 4 60 404 0. Express the sum of the series 000 40 35 0 3 or 7 000 as a sigle fractio.,000 3 3 4,000,00,00,00. Cosider the geometric sequece, 4,8, 6,3, 64,. a) Write a formula that will geerate the terms of the sequece. a b) Fid the sum of the first terms of this sequece. S 730 3 3. If x, x, x 4 are the first three terms of a geometric sequece, the what is the value of x? x x4 x x 4x x 4x x x x x
4. Determie if the followig geometric series coverge or diverge. If a series coverges, write what it coverges to. a) Diverges b) 4 8 Coverges to. 3 5. Use Mathematical Iductio to prove that 3 5 7 umbers,. For, the left-side is 3 ad the right-side is 3. Assume that the equatio is true for k: 3 5 7 k k k for all atural Ad we ll add k k 3 3 5 7 k k 3 k k k 3 to both sides to get k k 4 3 k k 3 k k So it s true for k. Therefore, 3 5 7 for all atural umbers,, by Mathematical Iductio. 6. Use Mathematical Iductio to prove that for all 3 atural umbers,. For, the left-side is ad the right-side is. Assume that the equatio is true for k: k 3 k Ad we ll multiply by o both sides to get k k 3 k k k k So it s true for k. Therefore, for all atural umbers,., by Mathematical 3 Iductio.
7. Use Mathematical Iductio to prove that For,, which is divisible by. Assume it s true for Therefore, k: k k k k divisible by k is divisible by. divisible by divisible by is divisible by for all atural umbers. k k k k k k k k So it s true for k. is divisible by for all atural umbers,, by Mathematical Iductio.