1 / 11

Chapter 6: Calculus~ Hughes-Hallett

Chapter 6: Calculus~ Hughes-Hallett. Constructing the Antiderivative Solving (Simple) Differential Equations The Fundamental Theorem of Calculus (Part 2). Review: The Definite Integral. Physically - is a summing up Geometrically - is an area under a curve

aman
Télécharger la présentation

Chapter 6: Calculus~ Hughes-Hallett

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6: Calculus~Hughes-Hallett • Constructing the Antiderivative • Solving (Simple) Differential Equations • The Fundamental Theorem of Calculus (Part 2)

  2. Review: The Definite Integral • Physically - is a summing up • Geometrically - is an area under a curve • Algebraically - is the limit of the sum of the rectangles as the number increases to infinity and the widths decrease to zero:

  3. Review of The Fundamental Theorem of Calculus (Part 1) If f is continuous on the interval [a,b] and f(t) = F’(t), then: • In words: the definite integral of a rate of change gives the total change.

  4. Properties of Antiderivative: 1. [f(x)  g(x)]dx = f(x)dx  g(x)dx (The antiderivative of a sum is the sum of the antiderivatives.) 2. cf(x)dx = cf(x)dx (The antiderivative of a constant times a function is the constant times the anti- derivative of the function.)

  5. The Definition of Differentials (given y = f(x)) 1. The Independent Differential dx: If x is the independent variable, then the change in x, x is dx; i.e. x = dx. 2. The Dependent Differential dy: If y is the dependent variable then: i.) dy = f ‘(x) dx, if dx  0 (dy is the derivative of the function times dx.) ii.) dy = 0, if dx = 0.

  6. Using the differential with the antiderivative.

  7. Solving First Order Ordinary Linear Differential Equations • To solve a differential equation of the form dy/dx = f(x) write the equation in differential form: dy = f(x) dx and integrate: dy = f(x)dx y = F(x) + C, given F’(x) = f(x) • If initial conditions are given y(x1) = y1 substi-tute the values into the function and solve for c: y = F(x) + C  y1 = F(x1) + C C = y1 - F(x1)

  8. Example: Solve, dr/dp = 3 sin pwith r(0)= 6, i.e. r= 6 when p = 0 • Solution:

  9. The Fundamental Theorem of Calculus (Part 2) If f is a continuous function on an interval, & if a is any number in that interval, then the function F, defined by F(x) =  ax f(t)dt is an antiderivative of f, and equivalently:

  10. Example:

  11. That’s all Folks! Have a good Christmas! God Bless

More Related