Chain Rule, Motion along a line, and R.O.C. of the Area of a circle with respect to the Radius.

R.O.C. of the area of a circle with respect to the radius:
-The formula for the are of a circle is A= πr^2
-Using the power rule, take the derivative of A= πr^2. to get dA/dr= 2πr.

Example:
When R=5
dA/dr=2π5= 10π

Motion Along a Line:
-This is the relationship between position, velocity, and acceleration.
-Position: Where an object is with respect to time. S(t)
-Velocity: Is measured by both the magnitude and direction. V(t)=S'(t)
-Acceleration: Change in velocity with respect to time. A(t)=V'(t)=S"(t)

Example:
A rock (named ethan) is shot straight up into the air with a velocity of 160ft/sec. Using the equation S=160t-16t^2. A)Find how high ethan goes. B)How long it takes ethan to reach the vertex. C) Find the time when ethan is at 256 feet above the ground. D) Find the speed of ethan when it is 256 feet above the ground. E) Find the acceleration of ethan.

A) S=160t-16t^2
S=160• 5-16•5^2
S=400 feet.

B) First take the derivative of original equation which= ds/dt=160-32t. Then simply set the equation equal to zero.
160-32t=0
t=5 seconds

C)256=160t-16t^2
16t^2-160t+256=0
t^2-10t+16=0
(t-8)(t-2)=0
At time=8 seconds and time=2 seconds ethan will reach 256 feet.

D) To find the speed at those previous points, use the velocity equation.
V(8)= 160-32•8
V(8)=-96 ft/s
V(2)=160-32•2
V(2)=96ft/s

E) To find the acceleration take the derivative of the velocity equation.
a(t)=V'(t)= -32 ft/sec

Chain Rule
This rule is used to take the derivative of a function when that function, according to Mr. Marchetti, "is very ugly."

Chain Rule: dy/du=(dy/du)•(du/dx)

Example:
f(x)=(3x+5)^10

-First take the derivative of the "inside equation" which is u=3x+5, the derivative of that then equals du/dx=3
-Then take the derivative of the "outside equation" which is y=u^10 the derivative of that then equals dy/du=10u^9
-Then simply place those derivatives into the chain rule equation to get 10(3x+5)•3

Homework Due Tuesday September 18: Pg 129, #2,4,9,12,14,16,25 and pg 146 #1-19 odd