super bowl sunday post

Substitution takes the position of derivative rules when integrating. The steps for substitution are:
1. Find a derivative of something else inside of the integral.
2. Set u equal to the non derivative found in the integral
3. Then take the derivative of u4. Substitute back in
5. Solve

I am going to talk about related rates. The steps for related rates are:
1. Identify all of the variables and equations
2. Identify the things that you are looking for
3. Sketch a graph and then label that graph
4. Create and write an equation using all of the variables
5. Take the derivative of this equation with respect to time
6. Substitute everything back in
7. Solve the equation
Example problem:
The variables x and y are differentiable functions of t and are related by the equation y= 2x^3-x+4when x=2 dx/dt=1. Find dy/dt when x=2.
dy/dt=?
dy/dt=6x^2 dx/dt-1 dx/dt
x=2 dx/dt=1 dy/dt=6(2)^2(-1)-(-1)
y=2x^3-x+4 dy/dt=-23

The angle of elevation is pretty much the same thing as rate of change except the difference is you look for an angle of a triange instead of a rate.
Example problem: A balloon rises at a rate of 5 meters/second from a point on the ground 50 meters from the observer. Find the rate of change of the angle of elevation of the balloon to the observer when the balloon is 50 meters from the ground.
tan(theta)=4/x
tan(theta)=5/50
sec^2(theta)=1/10 d(theta)/dt=1/10 dy/dt
d(theta)/dt=cos^2(theta)=1/10(4)
1/2(1/10)(4)= 1/5 radians/second

Small errors are my main problem on the AP tests.