6.1 antiderivatives and slope fields objectives swbat: 1)construct antiderivatives using the...
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6.1 Antiderivatives and Slope Fields
ObjectivesSWBAT:
1) construct antiderivatives using the fundamental theorem of calculus2) solve initial value problems
3) construct slope fields
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Indefinite Integrals• Last chapter we dealt with definite integrals
(we took an antiderivative and evaluated it between two points).
• Definite integrals have limits.• Indefinite integrals DO NOT have limits.• The set of all antiderivatives of a function f(x)
is the indefinite integral of f with respect to x and is denoted:
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Example 1: Evaluate each integral.
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Differential Equations• A differential equation is an equation containing a
derivative.• Just like in Algebra, when you want to solve an
equation, you use an inverse operation. To “undo” a derivative, we take an antiderivative (integral).
• Functions have many antiderivatives, all of which vary by a constant.
• Solving a differential equation involves finding a unique equation that satisfies some initial conditions or initial values (will help us solve for C).
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We don’t have to write the “C” on both sides. When we integrate, we would get a C on the left and a C on the right. We would then need to move the C from the left over to the right to combine our constants. We can think of the C currently on the right as that combination of constants already taking place.
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A Graphical Look at Differential Equations
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Example 7: Suppose that you know that the point (0,-1) is on a particular solution of the differential equation from the previous slide. By following the slopes, draw on the diagram what you think the particular solution looks like. (Note: The graph should follow the pattern of the slope field, but may go between the points rather than through them.
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