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**Unformatted text preview: **Calculus Integration page 1 Odds and Ends Slope Fields Calculating Integrals from areas Average value of a function 2 nd Fundamental Theorem More on the Integral as an Accumulator Differential equations revisited More Differential Equations and Exponential Growth and Decay More Techniques of Integration - Changing the limits of Integration Odds and Ends Integration Booklet 2009 Page 2 1. Slope Fields Sliding Down the Slippery Slope of Slope fields A differential equation is an equation containing a derivative. We will only concern ourselves with first-order differential equations in the form dy dx = f x , y ( ) . A slope field provides a graphical way to view a differential equation. The concept of a slope field uses the idea of local linearity; that is, if a function is differentiable at a point, then the tangent line approximates the function close to that point. By drawing a slope field, we are able to graphically see the family of functions or the general solution to the given differential equation. Example 1. Lets draw our first slope field. Consider the differential equation: 1- = x dx dy Determine the slope at the point you have been given then draw a small line segment with this slope passing through your point on the grid superimposed on the board. Trace a curve that passes through the point ( ) 2 , 1- to illustrate one solution to the differential equation. Now, solve the differential equation with this initial condition to obtain an algebraic representation of the solution. Calculus Integration page 3 Example 2. Draw the slope field for the differential equation: dy dx = 1 y What do you notice about this slope field compared to the one in Example 2? Solve the differential equation given the initial condition that the solution passes through the point ( ) 1 , . Draw the graph of your solution through the given point on the grid above. What observations can you make? Odds and Ends Integration Booklet 2009 Page 4 Example 3. Draw the slope field for the differential equation: dy dx = - x y What do you notice about this slope field compared to the ones in Example 1 and 2? Draw the solution to this differential equation that passes through the point ( ) 2 ,- , then solve the differential equation algebraically given this initial condition. Calculus Integration page 5 Example 4. Draw the slope field for the differential equation dy dx = x- y Where is the slope zero? Why? How is this slope field similar to the one in Example 3 and how is it different? Draw the solution that passes through the point (0, 1). Can you find an algebraic solution to this differential equation? Example 5. A calculator program has been used to create this slope field....

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