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Individual task in calculus 4 October 2018.

Edited at 4pm 4 October 2018.

s is your student number.

k = s mod 10000. T = s mod 100. m10= s mod 10.

T = 1. When does limit exist?





T = 2. List indeterminate forms.



T = 3. What the properties of the limit? (times constant, sum, product, quotient)

T = 4. Give the main methods for calculating limits.

T = 5. What are the great limits of calculus? 1. 2.

T = 6. [pic]=

T = 7. [pic]

T = 8. [pic] =

T = 9. [pic]

T = 10. [pic]

T = 11. [pic]

T = 12. [pic]

T = 13. [pic]

Use





Continuity:

Investigate continuity of the function:

T = 14. x

T = 15. [pic]

T = 16. [pic]

T = 17. [pic]

T = 18. [pic]

T = 19. [pic]

T = 20. [pic]

Derivative:

T = 21. Give the properties of derivative: times constant, sum, product, quotient.

T = 22. Prove expression for derivative of x2 using limit.

Find derivatives of these functions:

T = 23. ex

T = 24. xp

T = 25. cos(x)

T = 26. [pic]



Applications of derivatives:

Increasing or decreasing:

T = 27. -6x

T = 28. 9x

T = 29. sin(x)

T = 30. cos(x)

T = 31. tan(x)

Theorems:

T = 32. What is Mean Value Theorem?



T = 33. Explain Rolle theorem.



T = 34. Give Fermat theorem.

(stationary_points)

Concave or convex:

T = 35. x3

T = 36.-x3

T = 37. cos(x)

T = 38. sin(x)

Find inflection point:

T = 39. x3

T = 40.-x3

m10 = 1: 41. cos(x)

m10 = 2: 42. sin(x)

m10 = 3: 43. What is curvature?



m10 = 4: 44. Explain linear approximation.



m10 = 5: 45. What is chain rule?



Multivariate derivatives:

Calculate partial derivatives of the function:

m10 = 6: 46. xy

m10 = 7: 47. yz

m10 = 8: 48. xz

m10 = 9: 49. xyz

m10 = 0: 50. Find linear least-square approximation for your dataset.

51. Calculate (1 + 1/T)T.

52. [pic]. f(x) = Tx + k. For any ε find δ, using ε – δ definition of the limit.

Find derivatives of these functions:

53. xT.

54. sin Tx

55. Find the largest area rectangle with perimeter of T meters.

56. Calculate the largest area right-angled triangle with perimeter of T meters.

57. Find maximum volume cylinder for surface area of T meters square.

58. Calculate maximum volume cone for surface area of T meters square.

59. Calculate maximum area scalene triangle with perimeter of T meters.

60. Find the best song using calculus.

Other October tasks:





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