Limits, continuity, and derivatives
- 5. Graph
- 7. Graph
- 9. Graph
- 10. Solving Limits
- 11. graph
- 12. Solving Limits
- 13. Graph
- 14. Solving Limits
- 15. Graph
- 17. Limit of a Constant The limit of a constant is itself. 𝒍𝒊𝒎 𝒙→𝒂 𝒃 = 𝒃
- 18. Substitution Law X is substituted by the value X approaches. 𝒍𝒊𝒎 𝒙→𝒎 𝒙 = 𝒎 In this case, the limit of x as x approaches m is m.
- 19. Constant Multiple Theorem The limit of a multiple of a function is simply that multiple of the limit of the function.
- 20. Addition Theorem The limit of a sum of functions is the sum of the limits of the individual functions and the limit of a difference of functions is the difference of their limits.
- 21. Multiplication Theorem • The limit of a product of functions is equal to the product of their limits
- 22. Division Theorem The limit of a quotient of functions is equal to the quotient of the limits of the individual functions, provided the denominator limit is not equal to 0.
- 23. Power Theorem The limit of an integer power p of a function is just that power of the limit of the function.
- 24. Radical Theorem If n is a positive integer, the limit of the nth root of a function is just the nth root of the limit of the function, provided the nth root of the limit is a real number. Thus, it is important to keep in mind that if n is even, the limit of the function must be positive.
- 32. Continuity in Real Life
- 34. Tangent Lines
- 36. Secant Lines
- 37. Solving for the equation of Tangent Lines
- 38. Derivatives
- 40. The Constant Rule The derivative of f(x) = c where c is a constant is given by f '(x) = 0
- 41. The Power Rule The derivative of f(x) = 𝑥 𝑟 where r is a constant real number is given by f '(x) = r∗ 𝒙 𝒓−𝟏
- 42. The Constant Multiple Rule The derivative of f(x) = c g(x) is given by f '(x) = c∗g'(x)
- 43. The Sum and Difference Rule The derivative of f(x) = g(x) + h(x) is given by f '(x) = g '(x) + h '(x) The derivative of f(x) = g(x) - h(x) is given by f '(x) = g '(x) - h '(x)
- 44. The Product Rule The derivative of f(x) = g(x) h(x) is given by f '(x) = g(x) h '(x) + h(x) g '(x)
- 45. The Quotient Rule The derivative of f(x) = g(x) / h(x) is given by f '(x) = 𝐡 𝐱 𝐠′(𝐱) − 𝐠 𝐱 𝐡′(𝐱) 𝒉(𝒙) 𝟐
- 47. 1.AUTOMOBILES In a automobile there is always an odometer and a speedometer. These 2 gauges work in tandem and allow to drive to determine his speed and his distance that he has traveled. Electronic versionsof there gauges simply use derivatives to transform the data sent to the tiresto miles pen Hour (MPH) and distance (KM).
- 48. 2. BUSINESS In the business world application is when the data has been charted on graph or data table such as excel. Once it has been input,the data can be graphed and with the applications of derivatives you can estimate the profit and loss point for certain ventures.
- 49. 3.CLIMATE CHANGE In measuring how quickly the temperature is changing or recaps are melting, scientist can determine the severity of various situations.
- 50. 4.BIOLOGY