AP Calculus Unit 7 Differential Equations | Whitman College

University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2022
Views:

291

Pages:

2
Author:

Payton Perry

Unit 7: Differential Equations ● Sketching Slope Fields and solution curves ○ Slope fields: a way to visualize the graph of a differential equation ■ You will be given a function of dx/dy ■ 1) Make a chart with columns x, y, and dy/dx ■ 2) Plug in all x and y coordinates into the dy/dx function ● The resulting number will give you the dy/dx value or slope ■ 3) At each point (x,y), make a tangent line that has the slope that you found before ■ 4) Repeat for all points ● It will look something like this: o Solution curve: the actual set of points that the function goes through ● Exponential Growth and Decay ○ dp/dt= KP ■ The population grows at the rate that is proportional to the population ■ Dp/dt is the population growth rate ■ The function for exponential growth and decay is derived from this function. ■ P is size of the population o The function we use for exponential growth and decay is P(t)= P0 eKt ■ P0 is the initial population ■ K is relative growth rate ■ T is time o Remember in order to bring an exponent down, take the natural log ● Separable Differential Equations: finding the equation that goes through a specific point ○ 1) Separate the equation so that all of y and dy are together and all x and dx are together. ○ 2) Integrate both sides ■ On the Y side, integrate with respect to y ■ On the X side, integrate with respect to x o 3) Plug in the point that was initially given to us to solve for the C value ○ 4) If your result is a ± square root: ■ Look at the initial point they gave to us ● If the Y value is positive, we will use the positive square root ● If the Y value is negative, we will use the negative square root