Unit 5 Applications of Antidifferentiation

Transcription

1 Warmup 1) If f ( ) cos(ln ) for > 0, then f () (a) sin(ln ) (b) sin(ln ) (c) sin(ln ) (d) sin(ln ) (e) ln sin 2) If f ( ) 2, then f () (a) 2 ( ln 2) (b) 2 (1 ln 2) (c) 2 ln 2 (d) 2 (1 ln 2) (e) 2 (1 ln 2) 3) Consider the function f ( ) (ln ). (a) Find the instantaneous rate of change of f at = e. (b) Find the average rate of change of f over the interval [2.5, 3].

2 Recall: Solving Differential Equations E 1: Find the general solution to the differential equation: dy 2 2 e d Steps for Solving D.Eq s 1. SEPARATE the variables 2. INTEGRATE both sides 3. + C only on the right side 4. SOLVE for y in terms of (1) What is a differential equation telling me? The at any point (2) Then what is the solution of a differential equation? A with the D.E. as its derivative! Finding Particular Solutions from Eample 1 Given that y(0) 1, find the particular solution to this differential equation. Some Other Eamples Solve the differential equations: 2 dy dy E 2: 3y 0 E 3: (sin ) cos d d

3 Your Turn E 4: Find the particular solution for 2 2 ( 1) 0 given that (1) 0 yd e y dy y. dy E 5: Find the particular solution for 2y d given that 2 y(1) e. Slope Fields What are they? A slope field is the graphical representation of a differential equation; a graph of short line segments whose slope is determined by evaluating the derivative at the midpoint of a segment Why are they important? Writing an equation is one thing, but SEEING the solutions by plotting slope fields removes the abstractions from the symbolic representation. Some DE s can be solved algebraically, so slope fields serve to verify Some DE s CANNOT, so we can solve them graphically with slope fields.

4 A Graphical Perspective? A Numerical Perspective? E 1: Draw the slope field for the differential equation dy d y

5 E 2: Draw the slope field for the differential equations shown. Then answer the questions that follow! dy 1 d dy 2 y d What differences do you notice in these equations? What differences do you notice in their slope fields? Can you reach a general conclusion or establish a rule based on these observations? Solution Curves Given dy d (a) Draw the slope field. (b) Draw a possible solution curve through the point (0, 1).

6 Euler s Method What is Euler s Method? A clever way of approimating function values at a point ( y, ) by constructing a local approimation of the curve around this point. Euler s Method: The Process 1 Begin at the point (,y) specified by the initial condition, as required. 2 Use the differential equation to find the slope dy at the point. d 3 Increase by a small amount,. Increase y by a small amount, y, where dy y. This defines a new point, (, y y) that lies along the linearization. d 4 Using this new point, return to step 2. Repeating the process will bring you closer to your desired point. Eamples (1) Given the differential equation dy y and f (2) 0, use Euler s method and increments of d 0.2 to approimate f (3). Table??? Linearization???

7 dy (2) Given the differential equation 2 y and f (2) 3, use Euler s method with five equal d steps to approimate f (1.5). Table??? Linearization??? And Now..IT S AP QUESTION TIME!!! 2000 AB6 No Calculator

8 2002 BC5 No Calculator

9 2005 BC4 No Calculator

10 1989 AB6

11 The Logistic Differential Equation We ve seen an eponential model for growth before (Remember?) But in times that we want the growth rate to approach 0 as the population approaches a maimal carrying capacity (M), we introduce a limiting factor, and arrive at the Logistic Differential Equation shown here. dp kp ( M P ) dt A Basic Eample: Find the particular solution to this differential equation: dp P(200 P), P(0) 8 dt

12 Who Doesn t Like Moose? In 1985 and 1987, the Michigan Department of Natural Resources airlifted 61 moose from Algonquin Park, Ontario to Marquette County in the Upper Peninsula. It was originally hoped that the population (P) dp would reach carrying capacity in about 25 years with a growth rate of P(1000 P) dt (a) According to the model, what is the carrying capacity? (b) Let s generate a slope field and then solve the differential equation with the initial condition presented and show that it conforms to the slope field.

13 Scoring Guide to 2000 AB6 Scoring Guide to 2002 BC5

14 Scoring Guide to 2005 BC4