Riva - AP Calc AB
A difference quotient for a function determines an
average rate of change for a function.
Note: Speed is the rate of change of distance traveled
(it's always positive!!!)
Velocity is the rate of change of position
(it can be positive or negative)
Example: If an arrow is shot upward on the moon with a velocity
of 58 m/sec, its height in meters after t seconds is given by
Find the average velocity over the given time intervals
(i) [1,2]
(ii) [1,1.5]
(iii) [1,1.1]
Riva - AP Calc AB
A function is said to be locally linear over an interval, if the
difference quotient is constant over that interval.
Most of the functions you will encounter are "nearly linear"
over very small intervals. That is, most functions are locally
linear. Thus, when we "zoom in" on a point on the graph of a
function, we are very likely to "see" what appears to be a
straight line.
Instantaneous Rate of Change is the measure of the limit of
average rates of change as the size of the domain interval
approaches zero
Example: If a ball is thrown into the air with a velocity of
40 ft/sec, its height after t seconds is given by
Find the instantaneous velocity when t=2.
Riva - AP Calc AB
Calculus is built on two ideas....
(1) The derivative
(2) The integral
The thickness of a calculus text is do to applying these two ideas in many
various ways, applications.
The "meta-idea" above the derivative and integral is this idea of a limit.
Riva - AP Calc AB
Use the properties of limits to evaluate the following four limits.
Riva - AP Calc AB
The Sandwich Theorem allows us to indirectly find a limit.
Use the Sandwich Theorem to determine the following limits.
Note: One limit that the textbook gives you is:
This limit is used to determine many other limits; so, it
is a limit that you should memorize.
Riva - AP Calc AB
Homework 9/21
p.66 #1-4, 30-34, 65, 67, 69, 70
p.92 #1-6, 23-26
Hint for #30: sin(2x)=2sin(x)cos(x)