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CALCULUS 2 Name: _____________________________ WORKSHEET 7.1-1 1. Graph y 5 x and y log 5 x . 2. Graph y e x and y ln x . Convert to log form. Convert to exponential form. 3. y 5 x 7. y log 9 x 4. 243 35 8. 3 log 1000 5. x e 7 9. x ln 10 1 2 1 2 10. log 9 1 3 27 2 11. log 4 64 12. log 7 1 7 6. 4 Evaluate. 13. log 1 100 14. log 5 3 5 2 1 16 15. log 9 27 16. log 64 17. 11log11 21 18. 2log2 2 log2 4 CALCULUS 2 Name: _____________________________ WORKSHEET 7.1-2 Expand each logarithm. 1. log6 3x 2. log2 3. log4 xy2 4. log3 5. log3 xy4z 6. log2 (xy) 4 7. log5 2 z 8. log8 x 7 x 3y 2 3 3x 2 y z 34 3 x 2 y3 z4 Write each logarithmic expression as a single logarithm. 9. log3 7 – log3 x 10. 2 log5 x + log5 3 11. 2 3 log2 x – 3 log2 y 12. 13. 1 2 log7 x + 13 log7 y – 2 log7 z 14. log5 x – 4(log5 y + 2 log5 z) 15. log2 (x – 4) + 5 log2 (x+1) - 3 4 log2 (x-1) 16. 1 2 1 2 (log3 4 + log3 y) – 3 log3 z [log6 (x-2) + 2 log6 (x+1) – log6 (x+2) – 5 log6 x] Evaluate. 17. log 2 16 18. log a a 2 19. log 3 7 20. 6 log6 7 21. log 3 14.62 22. log a 23. 2 x 3 24. 3x 1 25 25. 23 x 4 5 26. 2 x 2 16 27. 3x 1 4 x 28. 31 x 52 3 x 29. e6 x 314 30. 225 31e .07 x 31. ln 2.1 ln e3.2 x 1 a Solve. CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-1 Find the derivative of each of the following: 1. y x ln x 2. y t ln t t 4. y ln x 2 5. y ln x 3 3x 1 7. y ln sin3t 1 8. y x 2 ln x 10. y ln ln x 11. y ln ln x x 13 y 3ln x 1 14. y ln( x 4)3 16. y x 2 ln( x 2 ) (ln x )3 17. y 19. y log2 x 20. y log10 x 3 x 2 3 3. y ln x 6. y ln2x 9. y 2 ln x x 12. y ln( x 2 3x ) 15. y ln 3x 2 3 ln x 1 ln 2 2 x ln x x 18. y ln( x x 2 1) 21. y log3 sin t Find the equation of the tangent line to the graph of f at the given point. 22. f ( x ) ln t ; t 5 24. f ( x ) lnsin x ; Find 23. f ( x ) ln(8 4t ); t 1 x 2 25. f ( x ) 3x 2 ln x ; x 1 dy using implicit differentiation. dx 26. x 2 3ln y y2 10 27. ln xy 5x 30 CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-2 Find dy using logarithmic differentiation. dx 1. y ( x 2)( x 4) 3. x( x 1)3 y (3x 1)2 5. y 2x 1 4 x 2 7. 9. 2. y ( x 1)( x 2)( x 4) x( x 2 1) 4. y 6. y= y ( x 2 1)( x 2 2)( x 2 3)2 8. y y x 2x 10. y x cos x 11. y x x 2 x 9 x 1 x( x 2) (2x 1)(2x 2) x cos x ( x 1)sin x 12. y e x x CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-3 Find the derivative of each of the following: 1. y 5x 4. y e x 7. y x 2e x 2 10. y ln 1 e 2x 13. y e x (sin x cos x ) 2 x 2. y 3x 5. y e2x 8. y e t et 3 3. y 4cos x 6. y e 9. y e 3/x 1 ex 11. y ln x 1e 12. y 14. y ln e x 15. y e x 2 e x e x 2 ln x 2 Find the equation of the tangent line to the graph of the function at the given point. 16. f ( x ) e1 x ; x 1 2 17. f ( x ) e2x x ; x 2 18. f ( x ) ln e x ; x 2 e x e x 19. f ( x ) ln 2 20. f ( x ) x 2e x 2xe x 2e x ; x 1 21. f ( x ) e x ln x ; x 1 2 Find ; x 0 dy using implicit differentiation. dx 22. xe y 10x 3 y 0 23. e xy x 2 y2 10 Find the equation of the tangent line to the graph of the function at the given point. 24. xe y ye x 1; (0,1) 25. 1 ln xy e x y ; (1,1) CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-4 Expand the logarithmic expression. x4 y 1. ln 4 2. ln 3 x 2 1 z 3. ln z( z 1)2 Write each expression as a single logarithm. 1 5. 2ln3 ln x 2 1 2 6. 2ln x ln( x 1) ln( x 1) 8. ln x ln4x 0 9. 7e5t 100 11. ln( x 4 ) ln( x 2 ) 2 12. log3 y 3log3 y 2 14 13. y ln x 14. y x ln x 15. y x 2e x ex 16. y ln 1 ex 17. y e e x2 19. y x e 1 20. y e sin2 x 2 4. 3ln x 2ln y 4ln z Solve. 7. ln x 1 2 10. 2x Find Find 2 2 x 8 dy for each of the following: dx 2x 2 x 18. y e x2 2 21. y 63x dy for each of the following using logarithmic differentiation: dx x 3 22. y 2 x 1 2 23. y x x2 1 x 4 24. y sin x x Find the equation of the tangent line to the graph of the function at the given point. 2 1 25. f ( x ) 4 x 2 ln x 1 ; x 0 26. f ( x ) 2e1 x ; x 1 2 Find dy for each of the following using implicit differentiation: dx 27. cos x 2 xe y 28. ye x xe y xy Find the equation of the tangent line to the graph of the function at the given point. 29. y ln x y2 0; (e , 1) 30. ln( x y ) x ; (0,1) CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-5 Find the intervals where f(x) is increasing and decreasing and the x-coordinates of any extrema. 1. f x x ln x 2. f x x ln x 3. f x ln x x 4. f x ln x 2 2x 3 5. f x 1 x 1 x e e 2 2 6. f x xe x 7. f x x 2 e x 8. f x 1 2 x e x Find the intervals where f(x) is concave up and concave down and the x-coordinates of any points of inflection. 9. f x e x x 2 2 10. f x xe x 11. f x xe x 12. f x 2 ln x x CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-6 Integrate: 1 1 1 1. x 1 dx 2. 3x 2 dx 3. 3 - 2x dx 4. 3x 2 1 x 3 x dx 5. x x dx 2 1 6. x 2 2x 3 x 3 3x 2 9x dx 9. x 12. 3x - 1ln 3x - 1 csc 2 x dx 14. cot x 15. 1 sin x 17. cot x dx 18. tan 2x dx 20. cot x ln sinx dx 21. 7. 10. 1 x 1 dx 1 xlnx dx sec 2 x dx 13. tan x 16. sec x tan x dx sec x - 1 19. cot 3x - 1 dx 8. 11. lnx 4 x dx ln2x 2 2x dx dx lnx dx cos x dx tan lnx dx x 2 CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-7 Integrate: 1. 5x 5e dx 2. 3 -x - 4x e dx 3. 4 e x dx x 1 4. e x 3 dx 5. e x 1 e x dx 6. e 2x 1 e 2 x dx 7. x x e 1 - e dx 8. e x - e x e x e x dx 9. e x e x e x e x dx x2 10. 13. 2e x - 2e x e x e x 2 2x -1 ln e dx dx 5-ex 11. 2x dx e 14. e 2x 2e x 1 dx ex 12. e -x tan e -x dx 15. ex 1 e x 4 dx CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-8 Evaluate the definite integrals: 1. 1 1 x 2dx 1 e2 6. 2 8x dx 4 9. 4 12 dx 1 x 8. x 4 0 11. 7. e 1 x 5 tan xdx 1 13. xe 16. 0 0 2 x 2 dx 14. xe x 2dx 17. 2 e 4 0 1 0 e 2 x dx 0 2 2 x3 2 xe 2 0 2 12. dx cosx esinx dx 4 1 1 0 1 cos x dx x sin x x 4 e 3 3 x 1 dx x dx e3 x dx x2 15. 3 18. 2 1 5 3x 1dx 3. 1 dx x ln x 5. 4. 10. 1 ln x 2 dx 2 7 x 4dx 9 2. 3 sec 2x tan 2xe sec 2 x dx CALCULUS 2 Name: _____________________________ WORKSHEET 7.2-9 1. Find the area of the region between the curve y 2. Find the area between the curves y 2x and the x-axis over the interval 2,2. 1 x2 1 1 and y over the interval 1, e . x x 3. Find the area of the region above the curve y 2 and below the x-axis over the interval 4,1 . x2 4. Find the volume of the solid obtained by rotating the region under the curve y 1 x 1 , over the interval 0,1 , about the x-axis. 5. Find the area of the region enclosed by y e x , y 3 and x 0. 6. Find the volume of the region enclosed by y e x , y 0, x ln 3 and x 0 , rotated about the x-axis. CALCULUS 2 Name: _____________________________ WORKSHEET 7.7 Assume exponential growth or decay for each of the following: 1. A bacterial culture starts with 500 bacteria. After 3 hours, there are 8000 bacteria. a. Find the number of bacteria after 4 hours. b. When will the population reach 30,000 bacteria? 2. A cell of a particular bacterium divides into 2 cells every 1/3 of an hour. The initial population of bacteria is 100 cells. a. Find the number of cells after 10 hours. b. When will the population reach 10,000 cells? 3. The population of a particular city doubled from 1890 to 1950. The population was 60,000 in 1950. What was the population in 2000? 4. The half life of carbon-14 is 5730 years. How old is a specimen when it contains 40% of its original quantity of carbon-14? 5. 30% of a radioactive substance disappears in 15 years. Find the half-life of the substance. 6. A particular substance triples in size every hour. At the end of 4 hours, the substance has a size of 10 units. What was the substance’s initial size? CALCULUS 2 Name: _____________________________ CHAPTER 7 PRACTICE TEST Expand the logarithmic expression: x 4 y 2 2 2. ln z4 Solve: 3. ln 4 2 ln x 2 Find Write the expression as a single logarithm: 1 2. 4 ln 2 ln x 1 5 ln y 3 4. 5e 3 t 11 5. 2 3x 4 8 dy for each of the following: dx 6. y ln x 3 2x 7. y ln 2x 12 3 3 2x 3 8. y 2 ln 4 x e 2x 9. y ln 1 e x 10. y e x 11. y 2 1 12. y 3xe cos 4 x Find x5 ex 13. y e 2 x e x 3 dy for each of the following using logarithmic differentiation: dx 14. y 3 x5 1 2x 2 15. y x sin x Find the equation of the tangent line to the graph of the function at the given point: 16. y ln x ; x 4 Find 17. y xe 2 x ; x 0 dy using implicit differentiation: dx 18. x 2 y y 2 e x Find the x coordinates of any maxima, minima and points of inflection: 19. y xe x 20. y x 2 ln x Integrate: 4 dx 21. 2x 1 ln x 3 23. 25. e 5 x e 5 x e 5x e 5x dx x dx 3 sec 2 x 22. dx tan x 24. e 3x e x 3 dx e 2x 26. x 2 e 43x dx 3 Evaluate the definite integrals: 1 27. e4 e2 1 dx x ln x 2 e x 28. 1 dx 3 2 x 1 29. Find the area enclosed by y e x , x = 0, y = 0 and x = ln4. 30. Find the volume formed by rotating the area enclosed by y 1 x 1 , x = 2, x = 5 and y = 0 about the x-axis. 31. The half life of a particular radioactive substance is 876 years. If the initial size was 13 grams, what will be the size in 500 years? 32. The population of a certain organism tripled between 1920 and 1980. In what year will the population have quadrupled?