a) How is the number e defined (b) Express e as a limit. (c) Why is the natural

Write an expression for the slope of the tangent line to the curve y fsxd at the point sa, fsadd.

Define the derivative f9sad. Discuss two ways of interpreting this number

If y fsxd and x changes from x1 to x2, write expressions for the following. (a) The average rate of change of y with respect to x over the interval fx1, x2 g (b) The instantaneous rate of change of y with respect to x at x x1

Define the second derivative of f. If fstd is the position function of a particle, how can you interpret the second derivative

a) What does it mean for f to be differentiable at a? (b) What is the relation between the differentiability and continuity of a function? (c) Sketch the graph of a function that is continuous but not differentiable at a 2.

Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

State each differentiation rule both in symbols and in words. (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f) The Quotient Rule (g) The Chain Rul

State the derivative of each function. (a) y x n (b) y ex (c) y bx (d) y ln x (e) y logb x (f) y sin x (g) y cos x (h) y tan x (i) y csc x (j) y sec x (k) y cot x (l) y tan21 x

a) How is the number e defined? (b) Express e as a limit. (c) Why is the natural exponential function y ex used more often in calculus than the other exponential functions y bx ? (d) Why is the natural logarithmic function y ln x used more often in calculus than the other logarithmic functions y logb x?

(a) Explain how implicit differentiation works. When should you use it? (b) Explain how logarithmic differentiation works. When should you use it?

Write an expression for the linearization of f at a.

Write an expression for the nth-degree Taylor polynomial of f centered at a.

If tsxd x 5 , then limxl2 tsxd 2 ts2d

An equation of the tangent line to the parab

1550 Calculate y9. 15. y sx 4 2 3x 2 1 5d 3 16. y cosstan xd 17. y sx 1 1 s 3 x 4 18. y 3x 2 2 s2x 1 1

1550 Calculate y9. 15. y sx 4 2 3x 2 1 5d 3 16. y cosstan xd 17. y sx 1 1 s 3 x 4 18. y 3x 2 2 s2x 1 2

1550 Calculate y9. 15. y sx 4 2 3x 2 1 5d 3 16. y cosstan xd 17. y sx 1 1 s 3 x 4 18. y 3x 2 2 s2x 1 3

1550 Calculate y9. 15. y sx 4 2 3x 2 1 5d 3 16. y cosstan xd 17. y sx 1 1 s 3 x 4 18. y 3x 2 2 s2x 1 4

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

1550 Calculate y9. 19. y 2xsx 2 1 1 20. y ex 1 1 x 2 21. y esin 2 22. y e2t st 2 2 2t 1 2d 23. y t 1 2 t 2 24. y emx cos nx 25. y e1yx x 2 26. y S u 2 1 u2 1 u 1 1 D 4 27. xy 4 1 x 2 y x 1 3y 28. y lnscsc 5xd 29. y sec 2 1 1 tan 2 30. x 2 cos y 1 sin 2y xy 31. y ecxsc sin x 2 cos xd 32. y lnsx 2 ex d 33. y log5s1 1 2xd 34. y sln xd cos x 35. sinsxyd x2 2 y 36. y st lnst 4d 37. y 3x ln x 38. xey y 2 1 39. y ln sin x 2 1 2 sin2 x 40. y sx 2 1 1d 4 s2x 1 1d 3 s3x 2 1d 5 41. y x tan21 s4xd 42. y ecos x 1 cossex d 43. y ln | sec 5x 1 tan 5x | 44. y 10tan  45. y tan2 ssin d 46. y ln Z x 2 2 4 2x 1 5 Z 47. y sinstan s1 1 x 3 d 48. y arctansarcsin sx d 49. y cos(estan 3x ) 50. y sin2 scosssin x

If f std s4t 1 1, find f99s2d.

If tsd  sin , find t99sy6d.

If fsxd 2x , find f snd sxd

Find f snd sxd if fsxd 1ys2 2 xd

5556 Find an equation of the tangent to the curve at the given point. 55. y 4 sin2 x, sy6, 1d 56. y x2 2 1 x2 1 1 , s0, 21d

5556 Find an equation of the tangent to the curve at the given point. 55. y 4 sin2 x, sy6, 1d 56. y x2 2 1 x2 1 1 , s0, 21d

5758 Find equations of the tangent line and normal line to the curve at the given point. 57. y s2 1 xde2x , s0, 2d 58. x2 1 4xy 1 y2 13, s2, 1d

5758 Find equations of the tangent line and normal line to the curve at the given point. 57. y s2 1 xde2x , s0, 2d 58. x2 1 4xy 1 y2 13, s2, 1d

(a) If f sxd xs5 2 x , find f9sxd. (b) Find equations of the tangent lines to the curve y xs5 2 x at the points s1, 2d and s4, 4d.

(a) Graph the function fsxd x 2 2 sin x in the viewing rectangle f0, 8g by f22, 8g. (b) On which interval is the average rate of change larger: f1, 2g or f2, 3g? (c) At which value of x is the instantaneous rate of change larger: x 2 or x 5? (d) Check your visual estimates in part (c) by computing f9sxd and comparing the numerical values of f9s2d and f9s5

Suppose that hsxd fsxd tsxd and Fsxd fs tsxdd, where fs2d 3, ts2d 5, t9s2d 4, f9s2d 22, and f9s5d 11. Find (a) h9s2d and (b) F9s2d.

If f and t are the functions whose graphs are shown, let Psxd fsxd tsxd, Qsxd fsxdytsxd, and Csxd fs tsxdd. Find (a) P9s2d, (b) Q9s2d, and (c) C9s2d

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

6370 Find f9 in terms of t9. 63. fsxd x 2 tsxd 64. fsxd tsx 2 d 65. fsxd f tsxdg2 66. fsxd ts tsxdd 67. fsxd tsex d 68. fsxd etsxd 69. fsxd ln | tsxd | 70. fsxd tsln x

At what point on the curve y flnsx 1 4dg2 is the tangent horizontal?

(a) Find an equation of the tangent to the curve y ex that is parallel to the line x 2 4y 1. (b) Find an equation of the tangent to the curve y ex that passes through the origin

Find the points on the ellipse x 2 1 2y 2 1 where the tangent line has slope 1.

Find a parabola y ax 2 1 bx 1 c that passes through the point s1, 4d and whose tangent lines at x 21 and x 5 have slopes 6 and 22, respectively

An equation of motion of the form s Ae2ct cosst 1 d represents damped oscillation of an object. Find the velocity and acceleration of the object.

A particle moves along a horizontal line so that its coordinate at time t is x sb2 1 c 2t 2 , t > 0, where b and c are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction

The volume of a right circular cone is V 1 3r 2 h, where r is the radius of the base and h is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constan

The Michaelis-Menten equation for the enzyme pepsin is v 0.50fSg 3.0 3 1024 1 fSg where v is the rate of an enzymatic reaction and fSg is the concentration of a substrate S. Calculate dvydfSg and interpret it

Health care expenditures The US health care expenditures for 19702008 have been modeled by the function Estd 101.35e 0.088128t where t is the number of years elapsed since 1970 and E is measured in billions of dollars. According to this model, at what rate were health care expenditures increasing in 1980? In 2000?

Drug concentration The function Cstd Kse2at 2 e2btd, where a, b, and K are positive constants and b . a, is used to model the concentration at time t of a drug injected into the bloodstream. (a) Show that limtl` Cstd 0. (b) Find C9std, the rate of change of drug concentration in the blood. (c) When is this rate equal to 0?

Bacteria growth A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after t hours. (b) Find the number of bacteria after 4 hours. (c) Find the rate of growth after 4 hours. (d) When will the population reach 10,000?

Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg?

Drug elimination Let Cstd be the concentration of a drug in the bloodstream. As the body eliminates the drug, Cstd decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus C9std 2kCstd,

A cup of hot chocolate has temperature 80C in a room kept at 20C. After half an hour the hot chocolate cools to 60C. (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to 408 C?

The volume of a cube is increasing at a rate of 10 cm3 ymin. How fast is the surface area increasing when the length of an edge is 30 cm?

Yeast population The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function n fstd a 1 1 be20.7t where t is measured in hours. At time t 0 the population is 20 cells and is increasing at a rate of 12 cellsyhour. Find the values of a and b. According to this model, what happens to the yeast population in the long run?

Use Newtons method to find the root of the equation x5 2 x4 1 3x2 2 3x 2 2 0 in the interval f1, 2g correct to six decimal places.

Use Newtons method to find all roots of the equation sin x x 2 2 3x 1 1 correct to six decimal place

(a) Find the linearization of f sxd s 3 1 1 3x at a 0. State the corresponding linear approximation and use it to give an approximate value for s 3 1.03 . ; (b) Determine the values of x for which the linear approximation given in part (a) is accurate to within 0.

(a) Find the first three Taylor polynomials for fsxd 4sx 2 2d 22 centered at 0. ; (b) Graph f and the Taylor polynomials from part (a) on the interval f21, 1g and comment on how well the polynomials approximate f

Dialysis The project on page 458 models the removal of urea from the bloodstream via dialysis. In certain situations the duration of dialysis required, given that the initial urea concentration is c, where c . 1, is given by the equation t lnS 3c 1 s9c 2 2 8c 2 D (a) Use a linear approximation to estimate the time required if the initial concentration is near c 1. (b) Use a second-order Taylor polynomial to give a more accurate approximation.

Infectious disease outbreak size We have worked with the model e2qA 1 2 A where A is the fraction of the population infected, q is a measure of disease transmissibility, and  is the fraction of the population that is initially susceptible to infection. (a) Use implicit differentiation to find the linear approximation of A as a function of q at q 0. (b) Find the second-order Taylor polynomial approximation for Asqd at q 0.

Express the limit lim  ly3 cos  2 0.5  2 y3 as a derivative and thus evaluate it

Find points P and Q on the parabola y 1 2 x 2 so that the triangle ABC formed by the x-axis and the tangent lines at P and Q is an equilateral triangle