Math 151
Section 2.7
Tangents, Velocities, and Other Rates of Change
Instantaneous Rate of Change The instantaneous rate of change of a function f ( x ) at x = a is the
slope of the tangent line at x = a.
Example: Use the graph to answer the following questions.
A. What is the estimated
instantaneous rate of change
at x = 1?
B. What is the equation of
the tangent line at x = 1?
C. For what value of x does
f ( x) have an instantaneous
rate of change of 0?
Math 151
Limit Definition To calculate the instantaneous rate of change, i.e. slope of the tangent line, for a
function f ( x ) at x = a, find the value of the limit:
mtan = lim
x!a
f ( x ) " f ( a)
x"a
= lim
h!0
f ( a + h) " f ( a)
h
Example: Calculate the instantaneous rate of change of f ( x) = 2x 2 ! x at x = 1.
Example: Calculate the slope and write the equation of the tangent line for f ( x) =
1
x
at x = 9.
Math 151
Tangent Vector If r (t ) is a vector function, then the tangent vector at t = a is given by
1 #
1
r (t ) " r ( a)&( = lim #%r ( a + h) " r ( a)&(
%
' h!0 h $
'
t!a t " a $
v = lim
Example:
A. Find a vector tangent to the curve r (t ) = t 2 + t,5t at t = 2.
B. What are the parametric equations corresponding to the tangent line to r (t ) at t = 2?
Math 151
Velocity Let f (t ) represent the position of an object at time t.
1. The average velocity of the object from t = a to t = b is
f (b) ! f ( a)
b! a
2. The instantaneous velocity of the object at t = a is
v ( a) = lim
t!a
f (t ) " f ( a )
t"a
= lim
h!0
f ( a + h) " f ( a)
h
Note: These are the same formulas as the slope of the tangent lines! These formulas also work for
any rate of change, not just velocity.
Example: The position (in meters) of an object moving in a straight path is given by
s(t ) = t 2 !8t +18 where t is measured in seconds.
A. Find the average velocity over the time interval [3, 4].
B. Find the instantaneous velocity at time t = 3.
Math 151
Example: A ball is thrown into the air. The position of the ball in feet at time t, measured in
seconds, is given by s(t ) = 5t,100t !16t 2 .
A. Find the velocity of the ball at time t = 2.
B. Find the speed of the ball at time t = 2.