1 / 15

AP Calculus AB

AP Calculus AB. Antiderivatives, Differential Equations, and Slope Fields. Find. Review. Consider the equation. Solution. Antiderivatives. What is an inverse operation?. Examples include:. Addition and subtraction. Multiplication and division. Exponents and logarithms.

gundersonj
Download Presentation

AP Calculus AB

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. AP Calculus AB Antiderivatives, Differential Equations, and Slope Fields

  2. Find Review • Consider the equation Solution

  3. Antiderivatives • What is an inverse operation? • Examples include: • Addition and subtraction • Multiplication and division • Exponents and logarithms

  4. Antiderivatives • Differentiation also has an inverse… antidefferentiation

  5. Antiderivatives • Consider the function whose derivative is given by . • What is ? Solution • We say that is an antiderivative of .

  6. Antiderivatives • Notice that we say is an antiderivative and not the antiderivative. Why? • Since is an antiderivative of , we can say that . • If and , find and .

  7. Differential Equations • Recall the earlier equation . • This is called a differential equation and could also be written as . • We can think of solving a differential equation as being similar to solving any other equation.

  8. Differential Equations • Trying to find y as a function of x • Can only find indefinite solutions

  9. Differential Equations • There are two basic steps to follow: 1. Isolate the differential • Invert both sides…in other words, find the antiderivative

  10. Differential Equations • Since we are only finding indefinite solutions, we must indicate the ambiguity of the constant. • Normally, this is done through using a letter to represent any constant. Generally, we use C.

  11. Differential Equations • Solve Solution

  12. Slope Fields • Consider the following: HippoCampus

  13. Slope Fields • A slope field shows the general “flow” of a differential equation’s solution. • Often, slope fields are used in lieu of actually solving differential equations.

  14. Slope Fields • To construct a slope field, start with a differential equation. For simplicity’s sake we’ll use Slope Fields • Rather than solving the differential equation, we’ll construct a slope field • Pick points in the coordinate plane • Plug in the x and y values • The result is the slope of the tangent line at that point

  15. Slope Fields • Notice that since there is no y in our equation, horizontal rows all contain parallel segments. The same would be true for vertical columns if there were no x. • Construct a slope field for .

More Related