MATH 131-505 Spring 2015
c
Wen
Liu
2.6
2.6 Derivatives and Rates of Change
The tangent line to the curve y = f (x) at the point
P (a, f (a)) is the line through P with slope
m = lim
x→a
f (x) − f (a)
x−a
provided that this limit exists.
Notice that as x approaches a, h approaches 0 (because
h = x − a) and so the expression for the slope of the
tangent line becomes
f (a + h) − f (a)
h→0
h
m = lim
Example 1: (p. 136)
(a) Find the slope of the tangent line to the hyperbola f (x) =
3
at the point where x = a > 0.
x
(b) Find the equation of the tangent line at the point (3, 1).
Page 1 of 5
MATH 131-505 Spring 2015
2.6
c
Wen
Liu
Suppose an object moves along a straight line according to an equation of motion s = f (t), where
s is the displacement (direct distance) of the object from the origin at time t. The function f that
describe the motion is called the position function of the object. In the interval from t = a to
t = a + h the change in position is f (a + h) − f (a). The average velocity over this time interval is
average velocity =
displacement
f (a + h) − f (a)
=
time
h
We define the velocity (or instantaneous velocity) v(a) at time t = a to be
f (a + h) − f (a)
h→0
h
v(a) = lim
Example 2: (p. 137) Suppose that a ball is dropped from the upper observation deck of the CN
Tower, 450 m above the ground. Given s = f (t) = 4.9t2 .
(a) What is the velocity of the ball after t = a seconds?
(b) What is the velocity of the ball after 5 seconds?
(c) When will the ball hit the surface?
(d) How fast is the ball traveling when it hits the ground?
Page 2 of 5
MATH 131-505 Spring 2015
2.6
c
Wen
Liu
The derivative of a function f at a number a, denoted by f 0 (a), is
f (a + h) − f (a)
h→0
h
f 0 (a) = lim
if this limit exists.
Similarly, we can use
f 0 (a) = lim
x→a
f (x) − f (a)
x−a
Therefore, the tangent line to y = f (x) at (a, f (a)) is the line through (a, f (a)) whose slope is equal
to f 0 (a), the derivative of f at a.
Examples:
cos(x + h) + 1
represents the derivative of some function f (x) at some
h→0
h
number a. Find the appropriate f (x) and a.
3. (p. 143) The limit lim
4. (p. 143) Find f 0 (a) if f (x) =
√
1 − 2x
Suppose y = f (x). If x changes from x1 to x2 , then the change in x (also called the increment of
Page 3 of 5
MATH 131-505 Spring 2015
2.6
c
Wen
Liu
x) is
M x = x 2 − x1
and the corresponding change in y is
M y = f (x2 ) − f (x1 )
The difference quotient
My
f (x2 ) − f (x1 )
=
Mx
x2 − x1
is called the average rate of change of y with respect to x over the interval [x1 , x2 ] and can be
interpreted as the slope of the secant line between points (x1 , f (x1 )) and (x2 , f (x2 )).
By analogy with velocity, we consider the average rate of change over smaller and smaller intervals
by letting x2 approach x1 and therefore letting M x approach 0. The limit of these average rates of
change is called the (instantaneous) rate of change of y with respect to x at x = x1 , which is
interpreted as the slope of the tangent to the curve y = f (x) at P (x1 , f (x1 )):
My
f (x2 ) − f (x1 )
= lim
= f 0 (x1 )
x2 →x1
Mx→0 M x
x2 − x1
instantaneous rate of change = lim
From the last equality, we can say that the derivative f 0 (a) is the instantaneous rate of change of
y = f (x) with respect to x when x = a.
If s = f (t) is the position function of a particle that moves along a straight line, then f 0 (a) is the rate
of change of the displacement s with respect to the time t. In other words, f 0 (a) is the velocity of the
particle at time t = a. The speed of the particle is the absolute value of the velocity, that is, | f 0 (a) |.
Examples:
5. (p. 140) A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x
yards of this fabric is C = f (x) dollars.
(a) What is the meaning of the derivative f 0 (x)? What are its units?
Page 4 of 5
MATH 131-505 Spring 2015
2.6
c
Wen
Liu
(b) What does it mean to say that f 0 (1000) = 9?
(c) Which do you think is greater, f 0 (50) or f 0 (500)? What about f 0 (5000)?
6. (p. 142) A particle starts by moving to the right along a horizontal line; the graph of its position
function is shown. When is the particle moving to the right? Moving to the left? Standing still?
Draw a graph of the velocity function.
Page 5 of 5