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6.1 day 1: Antiderivatives and Slope Fields Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2009 Kitt Peak National Observatory, Arizona

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First, a little review: Consider: then: or It doesnt matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given:findWe dont know what the constant is, so we put C in the answer to remind us that there might have been a constant.

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If we have some more information we can find C. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and are asked to find the original equation.

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Initial value problems and differential equations can be illustrated with a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand.

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Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 000 010 00 00 2 3 10 2 112 204 0 -2 0-4

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If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

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Go to: and enter the equation as:Y= For more challenging differential equations, we will use the calculator to draw the slope field. (Notice that we have to replace x with t, and y with y1.) (Leave yi1 blank.) On the TI-89: Push MODE and change the Graph type to DIFF EQUATIONS. MODE Go to:Y= Press and make sure FIELDS is set to SLPFLD. I

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Set the viewing window: Then draw the graph: WINDOW GRAPH

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Be sure to change the Graph type back to FUNCTION when you are done graphing slope fields.

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Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!

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Integrals such as are called indefinite integrals because we can not find a definite value for the answer. When finding indefinite integrals, we always include the plus C.

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Many of the integral formulas are listed in your book. The first ones that we will be using are just the derivative formulas in reverse. Our book shows a technique to graph the integral of a function using the numerical integration function of the calculator (NINT). or This is extremely slow and usually not worth the trouble. A better way is to use the calculator to find the indefinite integral and plot the resulting expression.

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To find the indefinite integral on the TI-89, use: The calculator will return: Notice that it leaves out the +C. Use and to put this expression in the screen, and then plot the graph. COPYPASTE Y=

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[-10,10] by [-10,10]

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Differential equations and Slope Fields Greg Kelly, Hanford High School, Richland, Washington.

Differential equations and Slope Fields Greg Kelly, Hanford High School, Richland, Washington.

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