Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
EXERCISES 6.2 In Exercises 1-14, evaluate the given integral. 1. ' (2x - 3) dx 3. [*(3√₁ +41) di 5. 1. ²³5-2 +²³ d 7. dx •· [(36²³t- 9. 3e³t+1)dt 11. 1. [₁²/3/₁ dx x²-√x 8. . 12 d 12 [1+%dx 10. dx 1. f' 13. et + e0.5x 15. Given I'r [₁ f(x)dx = 0 and id 16. Given 17. Given find [²(2563)-3 18. Given dx (2f(x) - 3g(x))dx. f(x)dx= 3.5 and f(x)dx= 3 and Los 14. nd frox -10 g(x)dx. [₁f(x)dx = 0 and -0.5 2 عام 1 [²8(x) f(x)dx = 5, find f(x)dx = 4, find g(x)dx= -1, find L₂ -0.5 dx [Fox f(x)dx. 10 [ Foxydx. (2g(x) + f(x))dx= -4,

EXERCISES 6.2 In Exercises 1-14, evaluate the given integral. 1. ' (2x - 3) dx 3. [*(3√₁ +41) di 5. 1. ²³5-2 +²³ d 7. dx •· [(36²³t- 9. 3e³t+1)dt 11. 1. [₁²/3/₁ dx x²-√x 8. . 12 d 12 [1+%dx 10. dx 1. f' 13. et + e0.5x 15. Given I'r [₁ f(x)dx = 0 and id 16. Given 17. Given find [²(2563)-3 18. Given dx (2f(x) - 3g(x))dx. f(x)dx= 3.5 and f(x)dx= 3 and Los 14. nd frox -10 g(x)dx. [₁f(x)dx = 0 and -0.5 2 عام 1 [²8(x) f(x)dx = 5, find f(x)dx = 4, find g(x)dx= -1, find L₂ -0.5 dx [Fox f(x)dx. 10 [ Foxydx. (2g(x) + f(x))dx= -4,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Q11, 13,15 and Q17 Needed to be solved correctly in the order to get positive feedback These are easy questios please solve all.
308 CHAPTER 6 The Definite Integral In Exercises 19-22, combine the integrals into one integral, then evaluate the integral. 9. ²2 / ² ( 3x + = x ² − x ³) dx + 3 / ₁²(x²2 (x² - 2x + 7)dx (x - 1)dx +3 / 0x x² + x²de 20. (4x-2 22. :-2)dx +3 21. Lx² + ²x + [²√x +4 :+4)dx + •L(7x + In Exercises 23-26, use formula (8) to help you answer the question. 23. Given f'(x) = -2x + 3, compute f(3)-f(1). 24. Given f'(x) = 73, compute f(4) -ƒ(2). 25. Given f'(t)=-.5t + e-2, compute f(1)-f(-1). 1 26. Given f'(t)=-12- compute f(3)-f(0). froxydx. 27. Refer to Fig. 4 and evaluate 31 y² y 2 0 Figure 5 1 0 Figure 4 28. Refer to Fig. 5 and evaluate y1 y=1-2²2² 1 -1 +5)dx f(x) y=1+t 1 Figure 6 frosdx. f(x) 10 11 29. Refer to Fig. 6 and evaluate f(t)dt. 0 30. Refer to Fig. 7 and evaluate y=(1-x)(x-3) 3 x y=1-1 1 f(t)dt. 1 0 SO y=12²-1 y=1-1 2 Figure 7 31. Net Change in Position A rock is dropped from the top of a 400-foot cliff. Its velocity at time 1 seconds is v(t) = -321 feet per second. Find the displacement of the rock during the time interval 2 SS4. 32. Net Change in Position The velocity at timer seconds of a ball thrown up into the air is v(t) = -321 + 75 feet per second. (a) Find the displacement of the ball during the time interval 0≤1≤ 3. (b) Given that the initial position of the ball is s(0) = 6 feet, use (a) to determine its position at time t = 3. 33. Net Change in Position The velocity at time t seconds of a ball thrown up into the air is v(t) = -321 + 75 feet per second. (a) Compute the displacement of the ball during the time in- terval 1 ≤ ≤ 3. (b) Is the position of the ball at time t = 3 higher than its position at time = 1? Justify your answer. (c) Repeat part (a) using the time interval 1 st≤ 5. 34. Velocity of a Skydiver The velocity of a skydiver at time / sec- onds is v(1) = 45-45e-0.21 meters per second. Find the dis- tance traveled by the skydiver the first 9 seconds. 35. Net Change in Cost A company's marginal cost function is .1x²-x + 12 dollars, where x denotes the number of units produced in 1 day. (a) Determine the increase in cost if the production level is raised from x = 1 to x = 3 units. (b) If C(1) 15, determine C(3) using your answer in (a). 36. Cost Increase A company's marginal cost function is given by C'(x) = 32+, where x denotes the number of items pro- duced in 1 day and C(x) is in thousands of dollars. Determine the increase in cost if the company goes from a production level of 15 to 20 items per day. 37. Net Increase of an Investment An investment grew at an expo- nential rate R(1) 700e0.07 + 1000, where t is in years and R(1) is in dollars per year. Approximate the net increase in value of the investment after the first 10 years (as / varies from 0 to 10). 38. Depreciation of Real Estate A property with appraised value of $200,000 in 2015 is depreciating at the rate R(t) = -8e-0.04t, where t is in years since 2015 and R(t) is in thousands of dollars per year. Estimate the loss in value of the property between 2015 and 2021 (ast varies from 0 to 6). 39. Population Model with Emigration The rate of change of a pop- ulation with emigration is given by P'(t)=3/25 - ¹/16 where P(t) is the population in millions, 7 years after the year 2000. (a) Estimate the change in population as t varies from 2000 to 2010. (b) Estimate the change in population as t varies from 2010 to 2040. Compare and explain your answers in (a) and (b).
Transcribed Image Text:308 CHAPTER 6 The Definite Integral In Exercises 19-22, combine the integrals into one integral, then evaluate the integral. 9. ²2 / ² ( 3x + = x ² − x ³) dx + 3 / ₁²(x²2 (x² - 2x + 7)dx (x - 1)dx +3 / 0x x² + x²de 20. (4x-2 22. :-2)dx +3 21. Lx² + ²x + [²√x +4 :+4)dx + •L(7x + In Exercises 23-26, use formula (8) to help you answer the question. 23. Given f'(x) = -2x + 3, compute f(3)-f(1). 24. Given f'(x) = 73, compute f(4) -ƒ(2). 25. Given f'(t)=-.5t + e-2, compute f(1)-f(-1). 1 26. Given f'(t)=-12- compute f(3)-f(0). froxydx. 27. Refer to Fig. 4 and evaluate 31 y² y 2 0 Figure 5 1 0 Figure 4 28. Refer to Fig. 5 and evaluate y1 y=1-2²2² 1 -1 +5)dx f(x) y=1+t 1 Figure 6 frosdx. f(x) 10 11 29. Refer to Fig. 6 and evaluate f(t)dt. 0 30. Refer to Fig. 7 and evaluate y=(1-x)(x-3) 3 x y=1-1 1 f(t)dt. 1 0 SO y=12²-1 y=1-1 2 Figure 7 31. Net Change in Position A rock is dropped from the top of a 400-foot cliff. Its velocity at time 1 seconds is v(t) = -321 feet per second. Find the displacement of the rock during the time interval 2 SS4. 32. Net Change in Position The velocity at timer seconds of a ball thrown up into the air is v(t) = -321 + 75 feet per second. (a) Find the displacement of the ball during the time interval 0≤1≤ 3. (b) Given that the initial position of the ball is s(0) = 6 feet, use (a) to determine its position at time t = 3. 33. Net Change in Position The velocity at time t seconds of a ball thrown up into the air is v(t) = -321 + 75 feet per second. (a) Compute the displacement of the ball during the time in- terval 1 ≤ ≤ 3. (b) Is the position of the ball at time t = 3 higher than its position at time = 1? Justify your answer. (c) Repeat part (a) using the time interval 1 st≤ 5. 34. Velocity of a Skydiver The velocity of a skydiver at time / sec- onds is v(1) = 45-45e-0.21 meters per second. Find the dis- tance traveled by the skydiver the first 9 seconds. 35. Net Change in Cost A company's marginal cost function is .1x²-x + 12 dollars, where x denotes the number of units produced in 1 day. (a) Determine the increase in cost if the production level is raised from x = 1 to x = 3 units. (b) If C(1) 15, determine C(3) using your answer in (a). 36. Cost Increase A company's marginal cost function is given by C'(x) = 32+, where x denotes the number of items pro- duced in 1 day and C(x) is in thousands of dollars. Determine the increase in cost if the company goes from a production level of 15 to 20 items per day. 37. Net Increase of an Investment An investment grew at an expo- nential rate R(1) 700e0.07 + 1000, where t is in years and R(1) is in dollars per year. Approximate the net increase in value of the investment after the first 10 years (as / varies from 0 to 10). 38. Depreciation of Real Estate A property with appraised value of $200,000 in 2015 is depreciating at the rate R(t) = -8e-0.04t, where t is in years since 2015 and R(t) is in thousands of dollars per year. Estimate the loss in value of the property between 2015 and 2021 (ast varies from 0 to 6). 39. Population Model with Emigration The rate of change of a pop- ulation with emigration is given by P'(t)=3/25 - ¹/16 where P(t) is the population in millions, 7 years after the year 2000. (a) Estimate the change in population as t varies from 2000 to 2010. (b) Estimate the change in population as t varies from 2010 to 2040. Compare and explain your answers in (a) and (b).
INCORPORATING 1. Evaluate SOLUTION TECHNOLOGY 3. Lovi+ 400 5. 1²-23/3dx 7. [²5-2³ dx 9. [02²+14 + 1)dt · [²/3/1dx Check Your Understanding 6.2 e²x - 1 11. EXERCISES 6.2 In Exercises 1-14, evaluate the given integral. 1. ((2x-1) dr dx 6. Find the total amount of federal health expenditures from the year 2000 (t = 0) to the year 2010 (t = 10). 8. Let T(t) denote the federal health expenditures from time 0 (2000) until time t. So, T'(t) = R(t). We wish to calculate 7(10), the health expenditures from 2000 to 2010. Because T(0) represents the expenditures from time 0 to time 0, we have 7(0) = 0, and so T(10) = T(10) T(0). This shows that 7(10) is the net change in T as t varies from 0 to 10. This net change is given by 12. 2. ^ ( 3² - ²1 x) dx 4. 1. 11/1dx Vx [^(-x + √x)dx 4x²-√x X 10. // * dt T(10) T(0) = -2 6.2 The Definite Integral and Net Change of a Function 307 1+x (e0.12(10) - 1). Thus, 7(10) = 3166.67(e0.12(10)-1)= 7347 billion dollars. Hence, the federal health expenditures from 2000 to 2010 exceeded 7 trillion dollars. 10 - L." dx dx 10 = [" T' (t)dt = 380 .12 Computing a Definite Integral The definite integral in Example 7 is evaluated in Fig. 3. To do this evaluation, select MATH 9. Complete the integral, as shown in Fig. 3. Then press ENTER. 380 380e0.121 dt = 0.12 .12 -0.12(10) Figure 3 13. NORMAL FLOAT FRAC REAL RADIAN MP ₂. (.03x²-2x+25) dx Jo ex+0.5x e²r S 1 L'exse for 15. Given L". 16. Given 17. Given 380 .12 18. Given 2. If f'() 1-1, find f(2)-f(0). find R(1) dt -0.12(0)= -0.5 -0.5 dx (2f(x) - 3g(x))dx. lo 380 .12 f(x)dx = 0 and -95 Solutions can be found following the section exercises. f(x)dx = 3 and 10 f(x)dx = 3.5 and 0 g(x)dx. In 2 14. 6th S -1 ¹1₁'s f(x)dx = 0 and ex + ex 2 f(x)dx = 4, find dx f(x)dx = 5, find g(x)dx= -1, find [₁ 10 f(x)dx. f(x)dx. (2g(x) + f(x))dx= -4,
Transcribed Image Text:INCORPORATING 1. Evaluate SOLUTION TECHNOLOGY 3. Lovi+ 400 5. 1²-23/3dx 7. [²5-2³ dx 9. [02²+14 + 1)dt · [²/3/1dx Check Your Understanding 6.2 e²x - 1 11. EXERCISES 6.2 In Exercises 1-14, evaluate the given integral. 1. ((2x-1) dr dx 6. Find the total amount of federal health expenditures from the year 2000 (t = 0) to the year 2010 (t = 10). 8. Let T(t) denote the federal health expenditures from time 0 (2000) until time t. So, T'(t) = R(t). We wish to calculate 7(10), the health expenditures from 2000 to 2010. Because T(0) represents the expenditures from time 0 to time 0, we have 7(0) = 0, and so T(10) = T(10) T(0). This shows that 7(10) is the net change in T as t varies from 0 to 10. This net change is given by 12. 2. ^ ( 3² - ²1 x) dx 4. 1. 11/1dx Vx [^(-x + √x)dx 4x²-√x X 10. // * dt T(10) T(0) = -2 6.2 The Definite Integral and Net Change of a Function 307 1+x (e0.12(10) - 1). Thus, 7(10) = 3166.67(e0.12(10)-1)= 7347 billion dollars. Hence, the federal health expenditures from 2000 to 2010 exceeded 7 trillion dollars. 10 - L." dx dx 10 = [" T' (t)dt = 380 .12 Computing a Definite Integral The definite integral in Example 7 is evaluated in Fig. 3. To do this evaluation, select MATH 9. Complete the integral, as shown in Fig. 3. Then press ENTER. 380 380e0.121 dt = 0.12 .12 -0.12(10) Figure 3 13. NORMAL FLOAT FRAC REAL RADIAN MP ₂. (.03x²-2x+25) dx Jo ex+0.5x e²r S 1 L'exse for 15. Given L". 16. Given 17. Given 380 .12 18. Given 2. If f'() 1-1, find f(2)-f(0). find R(1) dt -0.12(0)= -0.5 -0.5 dx (2f(x) - 3g(x))dx. lo 380 .12 f(x)dx = 0 and -95 Solutions can be found following the section exercises. f(x)dx = 3 and 10 f(x)dx = 3.5 and 0 g(x)dx. In 2 14. 6th S -1 ¹1₁'s f(x)dx = 0 and ex + ex 2 f(x)dx = 4, find dx f(x)dx = 5, find g(x)dx= -1, find [₁ 10 f(x)dx. f(x)dx. (2g(x) + f(x))dx= -4,
Expert Solution
steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS