1
Section 2.1 How Do We Measure Speed?
Consider an object that moves along a line. Let s(t) denote the position of the object at time
t. The average velocity of the object over the interval [a, b] is given by
Average velocity =
Change in position
s(b) − s(a)
=
.
Change in time
b−a
Example 1 In a time of t seconds, a particle moves a distance of s(t) meters from its starting point
along a line, where s(t) = 4t2 . Find the average velocity between
a. t = 1 and t = 3
b. t = 2 and t = 5
c. What is the geometric meaning of the average velocity you obtain in part a and part b above?
Now we want to define the instantaneous velocity. Again consider an object that moves along a line.
Let s(t) denote the position of the object at time t, then the average velocity between t = 2 and
t = 2.1 is given by
s(2.1) − s(2)
s(2.1) − s(2)
=
.
2.1 − 2
0.1
Similarly, the average velocity between t = 2 and t = 2.01 is given by
s(2.01) − s(2)
s(2.01) − s(2)
=
.
2.01 − 2
0.01
In general, the average velocity between t = 2 and t = 2 + h is given by
s(2 + h) − s(2)
s(2 + h) − s(2)
=
.
2+h−2
h
As h gets smaller and smaller, the average velocity between t = 2 and t = 2+h represents the velocity
of the object near time t = 2 better and better. This leads to the following general definition.
2
Definition Let s(t) be the position of an object at time t. Then the instantaneous velocity of the
object at time t = a is defined as
s(a + h) − s(a)
lim
.
h→0
h
Example 2 Suppose s(t) denotes the position of an object at time t, where s(t) = t2 . Find the
instantaneous velocity of the object
a. at time t = 2
b. at time t = 3
c. at time t = a
3
Section 2.2 The Derivative at a Point
Definition Let f be a function. The difference quotient between x = a and x = b is defined
by
f (b) − f (a)
.
difference quotient =
b−a
Remark The difference quotient of f between x = a and x = b is the slope of the line that connects
(a, f (a)) and (b, f (b)).
Example 1 Let f (x) = x2 − 2x, then the difference quotient of f between x = 1 and x = 3 is
f (3) − f (1)
=
3−1
=
=
Definition Let f be a function. Consider the limit
f (a + h) − f (a)
.
h→0
h
lim
If the limit exists, then we say that the function is differentiable at a and write
f 0 (a) = lim
h→0
f (a + h) − f (a)
.
h
If the limit does not exist, then we say that the function is not differentiable at a.
Remark
a. f 0 (a) is called the derivative of f at a.
b. We use
df
(a) and f 0 (a) interchangeably.
dx
c. If f is differentiable at a, then f is continuous at a. The converse is not true.
d. Geometric meaning of the derivative at a point:
4
Example 2 Calculate f 0 (5), if f is differentiable at 5, where f (x) = x2 − 2x + 3. Also calculate
f 0 (−1), f 0 (0), and f 0 (1).
Example 3 Let g(x) = −2x + 4. Calculate g 0 (0), g 0 (1), and g 0 (2).
Example 4 Let h(x) = 3 be a constant function. Calculate h0 (0), h0 (1), and h0 (2).
Example 5 Let f (t) =
t−2
.
t+1
Calculate f 0 (0) and f 0 (2).
5
Example 6 Expand (a + b)2 , (a + b)3 , and (a + b)4 . Using this, compute g 0 (3), where g(x) = x4 .
Example 7 Is the function f (x) = |x| differentiable at 0?
Example 8 Let g(x) = sin x. Calculate g 0 (0). What can you say about g 0 ( π2 ) and g 0 (π)?
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Section 2.3 The Derivative Function
In previous section, we computed the derivative of f (x) = x2 − 2x + 3 at various points, say f 0 (5) = 8,
f 0 (−1) = −4, f 0 (0) = −2, and f 0 (1) = 0. Suppose we also need to find f 0 (6), f 0 (7), f 0 (8), · · · , f 0 (20).
Instead of finding individual derivative, let us compute the derivative of the function at general point
x. According to the definition,
f (x + h) − f (x)
=
h→0
h
f 0 (x) = lim
In general, we have the following definition.
Definition Let f be a function. The derivative of f , denoted by f 0 or
f (x + h) − f (x)
.
h→0
h
f 0 (x) = lim
Example 1 Find the derivative of f (x) = 41 x2 − 4.
Example 2 Compute g 0 (x), where g(x) = x1 .
df
, is a function defined by
dx
7
We now compare the graphs of f and f 0 using f (x) = 14 x2 − 4 and its derivative f 0 (x) = 21 x.
y = f (x)
y
4
3
2
1
x
-7
-6
-5
-4
-3
-2
-1
1
O
2
3
4
5
6
7
-1
-2
-3
-4
y
4
y = f 0 (x)
3
2
1
x
O
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
-1
-2
-3
-4
In the graph of y = f (x), the tangent line to the graph at the point (2, −3) is given. It turns out
that the slope of the tangent line is 1. In other words, it means f 0 (2) = 1.
Example 3 Without looking at the graph of y = f 0 (x) above, but using the graph of y = f (x),
estimate f 0 (0) and f 0 (−4). Compare your answer with the true value of f 0 (0) and f 0 (−4).
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Example 4 Using the graph of y = f (x),
y
4
3
2
1
x
O
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-1
6
7
y = f (x)
-2
-3
-4
sketch the graph of y = f 0 (x) below.
y
4
3
2
1
x
-7
-6
-5
-4
-3
-2
-1
O
1
2
3
4
-1
-2
-3
-4
Remark
a. If f 0 > 0 on an interval, then f is increasing over that interval.
b. If f 0 < 0 on an interval, then f is decreasing over that interval.
5
6
7
9
Now we study the derivative of some elementary functions.
Example 5 Let f be a constant function. To be more precise, let’s assume that f (x) = k for
some constant k. Find f 0 (x).
Example 6 Let f be a linear function defined by f (x) = mx + b, where m and b are constants. Find
f 0 (x).
Example 7 Find the derivative of each of the following functions:
a. f (x) = x2
b. g(x) = x3
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Section 2.4 Interpretation of the Derivative
In previous section, we have seen that f 0 (a) denotes the slope of the tangent to the curve y = f (x)
at the point (a, f (a)). f 0 (a) gives some information about the curve near the point (a, f (a)). For
example, if f 0 (a) > 0, then it means the function is increasing near the point. If f 0 (a) < 0, then the
function is decreasing near the point. As another example, suppose f 0 (a) = 6 and f 0 (b) = 3, then
it means the function is increasing faster at (a, f (a)) then at (b, f (b)). In general, f 0 (a) can be also
interpreted as the rate of change of the function f at x = a. That is,
f (a + h) − f (a)
h→0
h
= the slope of the tangent line to the curve y = f (x) at (a, f (a))
= the rate of change of the function f at x = a.
f 0 (a) = lim
f (a + h) − f (a)
f (a + h) − f (a)
, if h is small, then f 0 (a) will be close to
.
h→0
h
h
In particular, if h = 1, then f 0 (a) is an approximation of f (a + 1) − f (a).
Example 1 Since f 0 (a) = lim
Example 2 The cost of extracting T tons of ore from a copper mine is C = f (T ) dollars.
a. Which one would be bigger, f (100) or f (101)?
b. Suppose that f (150) = 1200. What does this mean?
c. Suppose that f 0 (200) = 10. What does this mean?
Remark In practical applications, if y is a function of x, say y = f (x), then
unit of f 0 =
unit of y
.
unit of x
Example 3 What is the unit of f 0 (200) in Example 2 above?
Example 4 Let f (x) be the elevation in feet of the Connecticut river x miles from its source.
a. What are the units of f 0 (x)?
b. What can you say about the sign of f 0 (x)?
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Section 2.5 The Second Derivative
Since the derivative is itself a function, we can consider its derivative. For a function f , the derivativeof its
is called the second derivative, and written f 00 . We also use notations such as
derivative
d2 f
d df
or
.
dx dx
dx2
Example 1 Let f (x) = x3 . Find the second derivative f 00 of f .
Recall that
a. If f 0 > 0 on an interval, then f is increasing over that interval.
b. If f 0 < 0 on an interval, then f is decreasing over that interval.
Applying these to f 0 and f 00 , we get the following:
a. If f 00 > 0 on an interval, then f 0 is increasing over that interval.
b. If f 00 < 0 on an interval, then f 0 is decreasing over that interval.
What does it mean for f 0 to be increasing? We will use the function below as an example.
y
4
3
2
1
x
O
-7
-6
-5
-4
-3
-2
-1
1
-1
2
3
4
5
6
7
y = f (x)
-2
-3
-4
Observe that f 0 > 0 on intervals (−7, −3.5) and (3, 7). We can also see that f 0 < 0 on the interval (−3.5, 3). Now, on which interval(s) is the function f 0 increasing? On which interval(s) is f 0
decreasing? What can you say about the graph of y = f (x) on those intervals?
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Remark
a. If f 00 > 0 on an interval, then the graph of f is concave up (f bends upward) there.
b. If f 00 < 0 on an interval, then the graph of f is concave down (f bends downward) there.
Example 2 Consider a parabola y = ax2 . Show that the parabola is concave up if a > 0 and concave
down if a < 0.
Definition Let s(t) denote the position of an object at time t. We defined its instantaneous velocity
v(t) at time t to be v(t) = s0 (t). We also define instantaneous acceleration a(t) at time t to be
a(t) = v 0 (t) = s00 (t).
Example 3 The position of a particle moving along the x-axis is given by s(t) = 2t2 + 3.
a. Find the velocity v(t).
b. Find the acceleration a(t).