62
1.6. THE SECOND DERIVATIVE
and f ′′ is
on the interval as well
but of course with the blanks filled in. Throughout, view the scale of the grid for the
graph of f as being 1 × 1, and assume the horizontal scale of the grid for the graph of
f ′ is identical to that for f . If you need to adjust the vertical scale on the axes for the
graph of f ′ or f ′′ , you should label that accordingly.
⊳
Summary
In this section, we encountered the following important ideas:
• A differentiable function f is increasing at a point or on an interval whenever its first
derivative is positive, and decreasing whenever its first derivative is negative.
• By taking the derivative of the derivative of a function f , we arrive at the second
derivative, f ′′ . The second derivative measures the instantaneous rate of change of the
first derivative, and thus the sign of the second derivative tells us whether or not the
slope of the tangent line to f is increasing or decreasing.
• A differentiable function is concave up whenever its first derivative is increasing (or
equivalently whenever its second derivative is positive), and concave down whenever its
first derivative is decreasing (or equivalently whenever its second derivative is negative).
Examples of functions that are everywhere concave up are y = x 2 and y = e x ; examples
of functions that are everywhere concave down are y = −x 2 and y = −e x .
• The units on the second derivative are “units of output per unit of input per unit of
input.” They tell us how the value of the derivative function is changing in response to
changes in the input. In other words, the second derivative tells us the rate of change
of the rate of change of the original function.
Exercises
1. Suppose that y = f (x) is a differentiable function for which the following information
is known: f (2) = −3, f ′ (2) = 1.5, f ′′ (2) = −0.25.
(a) Is f increasing or decreasing at x = 2? Is f concave up or concave down at
x = 2?
(b) Do you expect f (2.1) to be greater than −3, equal to −3, or less than −3? Why?
(c) Do you expect f ′ (2.1) to be greater than 1.5, equal to 1.5, or less than 1.5?
Why?
(d) Sketch a graph of y = f (x) near (2, f (2)) and include a graph of the tangent
line.
1.6. THE SECOND DERIVATIVE
and f ′′ is
on the interval as well
but of course with the blanks filled in. Throughout, view the scale of the grid for the
graph of f as being 1 × 1, and assume the horizontal scale of the grid for the graph of
f ′ is identical to that for f . If you need to adjust the vertical scale on the axes for the
graph of f ′ or f ′′ , you should label that accordingly.
⊳
Summary
In this section, we encountered the following important ideas:
• A differentiable function f is increasing at a point or on an interval whenever its first
derivative is positive, and decreasing whenever its first derivative is negative.
• By taking the derivative of the derivative of a function f , we arrive at the second
derivative, f ′′ . The second derivative measures the instantaneous rate of change of the
first derivative, and thus the sign of the second derivative tells us whether or not the
slope of the tangent line to f is increasing or decreasing.
• A differentiable function is concave up whenever its first derivative is increasing (or
equivalently whenever its second derivative is positive), and concave down whenever its
first derivative is decreasing (or equivalently whenever its second derivative is negative).
Examples of functions that are everywhere concave up are y = x 2 and y = e x ; examples
of functions that are everywhere concave down are y = −x 2 and y = −e x .
• The units on the second derivative are “units of output per unit of input per unit of
input.” They tell us how the value of the derivative function is changing in response to
changes in the input. In other words, the second derivative tells us the rate of change
of the rate of change of the original function.
Exercises
1. Suppose that y = f (x) is a differentiable function for which the following information
is known: f (2) = −3, f ′ (2) = 1.5, f ′′ (2) = −0.25.
(a) Is f increasing or decreasing at x = 2? Is f concave up or concave down at
x = 2?
(b) Do you expect f (2.1) to be greater than −3, equal to −3, or less than −3? Why?
(c) Do you expect f ′ (2.1) to be greater than 1.5, equal to 1.5, or less than 1.5?
Why?
(d) Sketch a graph of y = f (x) near (2, f (2)) and include a graph of the tangent
line.