Warm-Up If the average annual rate of inflation is 3% over 6 years, then the cost, c, of a house in any year of that period is c(t) = p (1.03) t where p is the present cost. a.If the cost is $100,000 now ( t =0), what is the cost in 2 years? b.How long will it be before the house will cost $110,000? c.Find the rate at which the cost is changing when t =5.

Review

Integrals of Inverse Trig Functions: Important Idea

Example Evaluate: Are these integrals the same?

6-1a: Slope Fields Objectives: Graphically solve first order differential equations © 2004 Roy L. Gover ( Solve initial value problems

Definition A differential equation solved for a particular value is called an initial value problem. The value of f for a particular value of x is the initial condition.

Important Idea If you can solve a differential equation that defines a rate of change and if you know an initial condition, you can find the amount present at any time t.

Example A company noticed that its sales had dropped from 100,000 units per month to 80,000 units per month over a 4 month period. If the sales follow this same pattern, what will be the sales in another 2 months?

Try This Between 1990 and 2000, the population of deer in a nature preserve increased at a average rate of 12.4% each year. If there were 6730 deer in 1995, what is the expected deer population in 2000? 12073

Example Suppose we have the initial value problem: and the initial condition: Find a formula for y at any time t. Problem: we can’t solve the differential equation!!!

Important Idea The slope (rate of change) at any point ( x, y ) on the solution curve is the x coordinate of the point minus the y coordinate.

(0,1) Solution Curve Rate of change at (0,1)= x - y =-1 Example

(2,1) Solution Curve Rate of change at (2,1)= x - y =1 Example

Warm-Up What are all values of x for which the function f defined by f(x) = (x 2 - 3)e -x is increasing? A) There are no such values of x B) x 3 C) -3 < x < 1 D) -1 < x < 3 E) All values of x

Problem of the Day What are all values of x for which the function f defined by f(x) = (x 2 - 3)e -x is increasing? A) There are no such values of x B) x 3 C) -3 < x < 1 D) -1 < x < 3 E) All values of x

Remember this? Between 1990 and 2000, the population of deer in a nature preserve increased at a average rate of 12.4% each year. If there were 6730 deer in 1995, what is the expected deer population in 2000?

Example can be represented by tangent line segments

Definition All such segments represent the slope field or direction field for

Example Using the slope field, sketch the solution curve through (0,1) Hint: start at (0,1). Sketch right then left,

Try This Using the slope field, sketch the solution curve through (1,0) (1,0) is the initial condition. Estimate the solution to the initial value problem at x =3.

Example For Sketch the tangent line segments (slope field) at each integer coordinate

Important Idea Sketching slope fields can be tedious. It is best done with a graphing program.

Important Idea This is the slope field for The TI-89 can do it, but these are usually on the no calculator part of the exam.

This is the slope field for 1)Press “Mode” Change the graph from “function” to “Diff Equations” 2)Go to “ y=“. 3)Plug in “-t/y1” 4)Press “graph” 5)Play with your “window”

Try This This is the slope field for Confirm that the solution curve is Hint: Solve the D.E.

Try This Which choice represents the slope field for A B

Try This This slope field is for which differential equation? A B C

Lesson Close What is the practical value of slope fields?

Assignment Slope Field Worksheet