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Lecture on Antiderivatives, Differential Equations and Slope Fields

Explore the concepts of antiderivatives, differential equations, and slope fields, including examples and explanations. Learn about inverse operations, solving equations, and constructing slope fields. Find detailed solutions at www.assignmentpoint.com.

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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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