30
1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
estimate of P ′ (2), include units on your answer, and write one sentence (using
everyday language) to explain the meaning of the value you found.
(e) On your graph above, sketch two lines: one whose slope represents the average
rate of change of P on [2, 4], the other whose slope represents the instantaneous
rate of change of P at the instant a = 2.
(f) In a carefully-worded sentence, describe the behavior of P ′ (a) as a increases in
value. What does this reflect about the behavior of the given function P?
⊳
Summary
In this section, we encountered the following important ideas:
• The average rate of change of a function f on the interval [a, b] is
f (b) − f (a)
b − a
. The
units on the average rate of change are units of f per unit of x, and the numerical
value of the average rate of change represents the slope of the secant line between the
points (a, f (a)) and (b, f (b)) on the graph of y = f (x). If we view the interval as being
[a, a + h] instead of [a, b], the meaning is still the same, but the average rate of change
is now computed by
f (a + h) − f (a)
h
.
• The instantaneous rate of change with respect to x of a function f at a value x = a
is denoted f ′ (a) (read “the derivative of f evaluated at a” or “ f -prime at a”) and is
defined by the formula
f
′ (a) = lim
h→0
f (a + h) − f (a)
h
,
provided the limit exists. Note particularly that the instantaneous rate of change at
x = a is the limit of the average rate of change on [a, a + h] as h → 0.
• Provided the derivative f ′ (a) exists, its value tells us the instantaneous rate of change
of f with respect to x at x = a, which geometrically is the slope of the tangent line to
the curve y = f (x) at the point (a, f (a)). We even say that f ′ (a) is the slope of the curve
y = f (x) at the point (a, f (a)).
• Limits are the link between average rate of change and instantaneous rate of change:
they allow us to move from the rate of change over an interval to the rate of change at
a single point.
Exercises
1. Consider the graph of y = f (x) provided in Figure 1.16.
(a) On the graph of y = f (x), sketch and label the following quantities:
1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
estimate of P ′ (2), include units on your answer, and write one sentence (using
everyday language) to explain the meaning of the value you found.
(e) On your graph above, sketch two lines: one whose slope represents the average
rate of change of P on [2, 4], the other whose slope represents the instantaneous
rate of change of P at the instant a = 2.
(f) In a carefully-worded sentence, describe the behavior of P ′ (a) as a increases in
value. What does this reflect about the behavior of the given function P?
⊳
Summary
In this section, we encountered the following important ideas:
• The average rate of change of a function f on the interval [a, b] is
f (b) − f (a)
b − a
. The
units on the average rate of change are units of f per unit of x, and the numerical
value of the average rate of change represents the slope of the secant line between the
points (a, f (a)) and (b, f (b)) on the graph of y = f (x). If we view the interval as being
[a, a + h] instead of [a, b], the meaning is still the same, but the average rate of change
is now computed by
f (a + h) − f (a)
h
.
• The instantaneous rate of change with respect to x of a function f at a value x = a
is denoted f ′ (a) (read “the derivative of f evaluated at a” or “ f -prime at a”) and is
defined by the formula
f
′ (a) = lim
h→0
f (a + h) − f (a)
h
,
provided the limit exists. Note particularly that the instantaneous rate of change at
x = a is the limit of the average rate of change on [a, a + h] as h → 0.
• Provided the derivative f ′ (a) exists, its value tells us the instantaneous rate of change
of f with respect to x at x = a, which geometrically is the slope of the tangent line to
the curve y = f (x) at the point (a, f (a)). We even say that f ′ (a) is the slope of the curve
y = f (x) at the point (a, f (a)).
• Limits are the link between average rate of change and instantaneous rate of change:
they allow us to move from the rate of change over an interval to the rate of change at
a single point.
Exercises
1. Consider the graph of y = f (x) provided in Figure 1.16.
(a) On the graph of y = f (x), sketch and label the following quantities: