Chapter 3 Section 4

Free fall of an object:  s = (1/2)gt²
s is distance, t is time, g is constant acceleration given to an object by the force of gravity.  

g depends on the units to measure t and s:
g = 32 ft/sec²  s =(1/2)(32)t² = 16t²  (s in feet)

g = 9.8 m/sec²  s =(1/2)(9.80)t² = 4.9t²  (s in meters)

g = 980 cm/sec²  s =(1/2)(980)t² = 490t²  (s in cm)

Example:
Ball falls 144 ft during 1st 3 seconds. How long will it take to fall 180 ft using g = 32 for feet.
s(3) = 16(3)²  = 144 ft

s = 16t² 
180 = 16t² 
180/16 = t²
45/4 = t²
t = 3.35

Average velocity of a body moving along a line from position s=f(t) to position s +  Äs = f(t+Ät) is

Instantaneous velocity is the derivative of position, or distance.

v = gt         V = s'(t) derivative of distance

Absolute value of velocity.
Acceleration is the derivative of velocity.

The acceleration of a ball with velocity 80 - 32 ft/sec
is -32 ft/sec²
* the negative means downward acceleration.

a(t) = v'(t) = s''(t)

Average rate of change of f(x) over the interval x to x+h is:

Instantaneous rate of change is the derivative.

Economics: marginal value is the derivative of the function.

Example:

Suppose that the dollar cost of producing x computers is c(x) = 3000 + 100x - 0.1x²

a.) Find the average cost of producing 50 computers.
        Average cost is c(50)/50 which is $155.

b.) Find the marginal cost when 50 computers are
     produced.
        c'(x) = 100 - 0.2x then c'(50) = $98

Example:
The number of ounces of liquid in a dropper t milliseconds after the dropper has started to drip d(t) = 200(16-t)² . How fast is the liquid dropping at the end of 12 milliseconds?

d(t) = 200(16-t)²
                       200(256 - 32t + t²)
d'(t) = 200(-32+2t)
      400(t-16)
d'(12) = 400(12-16) = -1600 ounces/millisecond