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Lecture on Antiderivatives, Differential Equations and Slope Fields

Lecture on 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. Antiderivatives.

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Lecture on Antiderivatives, Differential Equations and Slope Fields

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  1. Lecture on Antiderivatives, Differential Equations and Slope Fields www.assignmentpoint.com

  2. Find Review • Consider the equation Solution www.assignmentpoint.com

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

  4. Antiderivatives • Differentiation also has an inverse… antidefferentiation www.assignmentpoint.com

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

  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 . www.assignmentpoint.com

  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. www.assignmentpoint.com

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

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

  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. www.assignmentpoint.com

  11. Differential Equations • Solve Solution www.assignmentpoint.com

  12. Slope Fields • Consider the following: HippoCampus www.assignmentpoint.com

  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. www.assignmentpoint.com

  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 www.assignmentpoint.com

  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 . www.assignmentpoint.com

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