Lesson Summary 1.We found general solutions of differential equations. 2.We used initial conditions to find particular solutions of differential equations. Multimedia Link The following applet allows you to set the initial equation for and then the slope field for that equation is displayed. In magenta you'll see one possible solution for . If you move the magenta point to the initial value, then you will see the graph of the solution to the initial value problem. Follow the directions on the page with the applet to explore this idea, and then try redoing the examples from this section on the applet. Slope Fields Applet. Review Questions In problems #1–3, solve the differential equation for 1.2.3.In problems #4–7, solve the differential equation for given the initial condition. 4.and . 5.and 6.and 7.and 132( )3f8.Suppose the graph of f includes the point (-2, 4) and that the slope of the tangent line to f at x is -2x+4. Find f(5).
10In problems #9–10, find the function that satisfies the given conditions. 9.with and 10.with (4)7fand (4)25fReview Answers 126.96.36.199.188.8.131.52.; 9.10.
11Initial Condition & Integration of Trig Functions Practice 1.Find the particular solution ( )yf xthat satisfies the differential equation and initial condition. 2.Find the equation of the function fwhose graph passes through the point. '( )610,4,2fxx3.Find the function fthat satisfies the given conditions. a. "( )2,'(2)5,(2)10fxffb. 2 3"( ),'(8)6,(0)0fxxff4.Integrate. a. (2sin3cos )xx dxb. 1csc cottt dtc. 2csccosdd. 2sintt dtAnswers: x
127.3 The Area Problem Learning Objectives Use sigma notation to evaluate sums of rectangular areas Find limits of upper and lower sums Use the limit definition of area to solve problems Introduction In The Lesson The Calculus we introduced the area problem that we consider in integral calculus. The basic problem was this: Suppose we are interested in finding the area between the axis and the curve of from to We approximated the area by constructing four rectangles, with the height of each rectangle equal to the maximum value of the function in the sub-interval. We then summed the areas of the rectangles as follows: and We call this the upper sumsince it is based on taking the maximum value of the function within each sub-interval. We noted that as we used more rectangles, our area approximation became more accurate. We would like to formalize this approach for both upper and lower sums. First we note that the lower sumsof the area of the rectangles results in Our intuition tells us that the true area lies