Download 4.1 Day 2 Notes

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