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Kenneth ArlandoFollow

May. 30, 2023β’0 likesβ’10 views

May. 30, 2023β’0 likesβ’10 views

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Science

Calculus

Kenneth ArlandoFollow

- 1. Review of the Previous Lesson
- 2. ACTIVITY
- 4. JEOPARDY GAME: THE TEACHER WILL GIVE YOU THE ANSWER, THE STUDENT WILL IDENTIFY THE QUESTION.
- 5. +1 POINT -1 POINT Its derivative is fβ² x = 2π₯.
- 6. +1 POINT -1 POINT Its derivative is fβ² x = 2π₯. What is the derivative of π π₯ = π₯2 ?
- 7. +1 POINT -1 POINT Its derivative is fβ² x = 3π₯2 .
- 8. +1 POINT -1 POINT Its derivative is fβ² x = 3π₯2 . What is the derivative of π π₯ = π₯3 ?
- 9. +1 POINT -1 POINT Its derivative is fβ² x = 4π₯.
- 10. +1 POINT -1 POINT Its derivative is fβ² x = 4π₯. What is the derivative of π π₯ = 2π₯2 ?
- 11. Its derivative is fβ² x = 1 π₯2. +2 POINTS -2 POINTS
- 12. Its derivative is fβ² x = 1 π₯2. +2 POINTS -2 POINTS What is the derivative of π π₯ = β 1 π₯ ?
- 13. Its derivative is fβ² x = π₯. +2 POINTS -2 POINTS
- 14. Its derivative is fβ² x = π₯. +2 POINTS -2 POINTS What is the derivative of π π₯ = 2 3 π₯ 3 2?
- 16. Learning Objectives: 1.Illustrate an antiderivative of a function. 2.Compute the general antiderivatives (indefinite integrals) of polynomial, radical, rational, exponential, logarithmic, and trigonometric functions.
- 17. Integral Calculus is the branch of calculus where we study integrals and their properties. Integration is an essential concept which is the inverse process of differentiation.
- 18. Let us have an intuitive approach in finding the antiderivative of a function. Let us consider this example. πβ² π± = π π π ππ + π = π π π ππ + π π π π πβ² π± = π π π ππ + π = π β ππβπ + π πβ² π± = π π π ππ + π = ππ
- 19. First, we will add one to the exponent of x since we subtract one from x during the process of differentiation. π π± = πππ+π = πππ
- 20. Second, we will divide π π± = πππ by its exponent 2 since we multiply the exponent during the process of differentiation. π π± = πππ π π π± = ππ
- 21. Did we already recover the original function?
- 22. The Derivative of a Constant Let π(π₯) be a constant function defined by π¦ = π(π₯) = π, where c is a constant, then π π π π = π π π π = π
- 23. We will add C (a constant arbitrary constant) to π π± = ππ . (Note that the derivative of a constant is zero.) π π± = ππ + πͺ If πͺ = π, then π π± = ππ + π
- 24. Hence, the antiderivative of πβ² π± = ππ is π π± = ππ + πͺ.
- 25. ANTIDIFFERENTIATION β’This operation of determining the original function from its derivative is the inverse operation of differentiation and is called antidifferentiation. β’Antidifferentiation is a process or operation that reverses differentiation.
- 26. ANTIDIFFERENTIATION β’Up to this point in Calculus, you have been concerned primarily with this problem: given a function, find its derivative β’Many important applications of calculus involve the inverse problem: given the derivative of a function, find the function
- 27. The relationship between derivatives and antiderivatives can be represented schematically:
- 28. Definition of the Antiderivative A function π π is called an antiderivative of a function π on an interval π° if πβ² π = π π for every value of π in π°.
- 29. Another term for antidifferentiation is integration. Another term for antiderivative is integral.
- 30. Integration can be classified into two different categories, namely, β’Definite Integral β’Indefinite Integral
- 31. The general solution is denoted by The expression β«f(x)dx is read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative. NOTATION FOR ANTIDERIVATIVES
- 32. Indefinite Integral Indefinite integrals are not defined using the upper and lower limits. The indefinite integrals represent the family of the given function whose derivatives are f, and it returns a function of the independent variable.
- 33. From our previous example, the antiderivative of πβ² π = ππ is π π± = ππ + πͺ, where C is a constant. The derivative of a constant is zero, so C can be any constant, positive or negative. The graph of π π± = ππ + πͺ is the graph of π π± = ππ shifted vertically by C units as shown in Figure 1.
- 34. Definition of the Indefinite Integral The family of antiderivatives of the function f is called the indefinite integral of f with respect to x. In symbols, this is written as π π π π Thus, if πΉ π₯ is the simplest antiderivative of f and C is any arbitrary constant, then π π π π = π π + πͺ
- 35. The symbol β« is just an elongated S meaning sum. This integral symbol was devised by Gottfried Wilhelm Leibniz. The dx refers to the fact that the function π π₯ is to be antidifferentiated or integrated with respect to the variable x. Note: β« π π π π is read as βthe indefinite integral of π π₯ with respect to xβ.
- 36. Definite Integral An integral that contains the upper and lower limits (i.e.) start and end value is known as a definite integral. The value of x is restricted to lie on a real line, and a definite Integral is also called a Riemann Integral when it is bound to lie on the real line. π π π π₯ ππ₯
- 37. Indefinite Integration Rules of Algebraic Function
- 38. The Power Rule If n is any number other than β1, then πππ π = ππ+π π + π + πͺ In words, when π₯π is integrated, the exponent n of x is increased by 1 and then π₯π+1 is divided by the new exponent n+1. Notice that the above formula cannot be used for π = β1.
- 39. Example 1. Evaluate β« ππ π. ππ π π ππ π π = ππ+π π + π + πͺ ππ π π = ππ+π π + π + πͺ ππ π π = π π + πͺ ππ π π = π + πͺ
- 40. Example 2. Evaluate β« π±π ππ±.
- 42. Example 3. Evaluate β« πππ π.
- 43. Example 4. Evaluate β« πππ π π.
- 45. Example 5. Evaluate β« πππ + ππ β π π π.
- 46. Example 6. Evaluate β« π π π β πβπ + ππ π π.
- 47. Example 7. Evaluate β« π + π π π π.
- 48. Indefinite Integration Rules of Exponential and Logarithmic Functions
- 50. Example 8. Evaluate β« ππ π π.
- 51. Example 9. Evaluate β« ππ+π π π.
- 52. Example 10. Evaluate β« π π π π.
- 53. Example 11. Evaluate β« ππ β π ππ π π.
- 54. Indefinite Integration Rules of Trigonometric Functions
- 56. Example 12. Evaluateβ« πππ π + πππ π π π.
- 57. Example 13. Evaluateβ« π ππ¬ππ π β π π¬πππ π π π.
- 58. Example 14. Evaluate β« πππ§π π± ππ±.
- 59. Example 15. Evaluate β« π+ππ¨π¬π π ππ¨π¬ π± π π
- 60. 1.β« ππ π π 2.β« ππ+π π π 3.β« βπ ππ¨π¬ π π π 4.β« ππ π π π 5.β« πβπ π π 6.β« ππ + π π π 7.β« ππ β π ππ π π 8.β« πππ+πππ+π ππ π π 9.β« π πππ§ π β π ππ¬ππ π π π 10.β« π π ππ¬ππ π π π Evaluate the following integrals below. Show your complete solution.