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4.1 – Day 2 Notes Antiderivatives and Indefinite Integration Homework: Worksheet #2 Learning Target #2 I can solve a differential equation. I can write the general solution of a differential equation. I can find a particular solution of a differential equation. I can sketch approximate solution curves of differential equations on slope fields. Sketch the graphs of two functions that have the given derivative. Ex 1. Ex 2. A differential equation in x and y is an equation that involved x, y, and derivatives of y. Ex 3. Find an equation for y. dy 2 x dx Ex 5. Solve the differential equation: f ''( x) Ex 4. Find the particular solution to the differential equation given to the left given the initial condition that that graph passes through the point 1,1 . 1 given f '(1) 3 and f (0) 1 x Ex 6. Show that the height s above the ground of an object thrown upward from a point s0 feet above the ground with an initial velocity of v0 feet per second is given by the function s (t ) 16t 2 v0t s0 . (Recall, acceleration due to gravity is constant 32 ft/s 2 .) Slope Fields Are also called “directional fields” Are a collection of line segments with slopes given by the value of the differential equation at each indicated point Gives a visual perspective of the solutions of the differential equation using slope segments as linear approximations (i.e. the “flow” of the slope field maps out the solution curves) Ex 7. Generate a slope field for dy x , dx y Add tangent lines 3 2 1 0 -1 -2 -3 * -3 -2 -1 0 1 2 3 Ex 8. a. Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. b. Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the results with the sketches from part a. dy 1 x 1, 4, 2 dx 2