c
Amy
Austin, September 22, 2015
1
Section 2.7: Tangents, Velocities, and Other Rates of Change
Tangents Definition: The slope of the tangent line to the graph of f (x) at x = x0 is
f (x0 + ∆x) − f (x0 )
m = lim
∆x→0
∆x
EXAMPLE 1: Find the slope of the tangent line to the graph of f (x) = x2 + 2x at
the point (1, 3).
2
c
Amy
Austin, September 22, 2015
EXAMPLE 2: Find the equation of the tangent line to the graph of f (x) =
at the point were x = 2.
√
2x + 5
Velocities If f (t) is the position of an object at time t, then
(a) The Average Velocity of the object from t = a to t = b is
f (b) − f (a)
b−a
(b) The Instantaneous Velocity of the object at time t = a is
f (a + h) − f (a)
h→0
h
v(a) = lim
EXAMPLE 3: The position (in meters) of an object moving in a straight path is
given by s(t) = t2 − 8t + 18, where t is measured in seconds.
(i) Find the average velocity over the time interval [3, 4].
(ii) Find the instantaneous velocity at time t = 3.
3
c
Amy
Austin, September 22, 2015
Other Rates of Change Let f (x) be a function.
(a) The Average Rate of change of f (x) from x = a to x = b is
f (b) − f (a)
b−a
(b) The Instantaneous Rate of Change of f (x) at x = a is
f (a + h) − f (a)
h→0
h
lim
1
, find
x+2
(i) the average rate of change of f (x) from x = 3 to x = 4.
EXAMPLE 4: f (x) =
(ii) The instantaneous rate of change of f (x) at x = 3.
c
Amy
Austin, September 22, 2015
4
EXAMPLE 5: The limit below represents the instantaneous rate of change of some
(2 + h)5 − 32
function f at some number a. State such an f and a: lim
h→0
h
EXAMPLE 6: The limit below represents the instantaneous rate of change of some
f (x) − 32
function f at some number a. State such an f and a: lim
x→2
x−2