58
1.6. THE SECOND DERIVATIVE
decreasing, and we say that the function is decreasing at a decreasing rate.
This leads us to introduce the notion of concavity which provides simpler language to
describe some of these behaviors. Informally, when a curve opens up on a given interval,
like the upright parabola y = x 2 or the exponential growth function y = e x , we say that
the curve is concave up on that interval. Likewise, when a curve opens down, such as the
parabola y = −x 2 or the opposite of the exponential function y = −e x , we say that the
function is concave down. This behavior is linked to both the first and second derivatives
of the function.
In Figure 1.31, we see two functions along with a sequence of tangent lines to each.
On the lefthand plot where the function is concave up, observe that the tangent lines to
the curve always lie below the curve itself and that, as we move from left to right, the
slope of the tangent line is increasing. Said differently, the function f is concave up on the
interval shown because its derivative, f ′ , is increasing on that interval. Similarly, on the
righthand plot in Figure 1.31, where the function shown is concave down, there we see that
the tangent lines alway lie above the curve and that the value of the slope of the tangent
line is decreasing as we move from left to right. Hence, what makes f concave down on
the interval is the fact that its derivative, f ′ , is decreasing.
Figure 1.31: At left, a function that is concave up; at right, one that is concave down.
We state these most recent observations formally as the definitions of the terms concave
up and concave down.
Definition 1.6. Let f be a differentiable function on an interval (a, b). Then f is concave
up on (a, b) if and only if f ′ is increasing on (a, b); f is concave down on (a, b) if and only
if f ′ is decreasing on (a, b).
The following activities lead us to further explore how the first and second derivatives
of a function determine the behavior and shape of its graph. We begin by revisiting
1.6. THE SECOND DERIVATIVE
decreasing, and we say that the function is decreasing at a decreasing rate.
This leads us to introduce the notion of concavity which provides simpler language to
describe some of these behaviors. Informally, when a curve opens up on a given interval,
like the upright parabola y = x 2 or the exponential growth function y = e x , we say that
the curve is concave up on that interval. Likewise, when a curve opens down, such as the
parabola y = −x 2 or the opposite of the exponential function y = −e x , we say that the
function is concave down. This behavior is linked to both the first and second derivatives
of the function.
In Figure 1.31, we see two functions along with a sequence of tangent lines to each.
On the lefthand plot where the function is concave up, observe that the tangent lines to
the curve always lie below the curve itself and that, as we move from left to right, the
slope of the tangent line is increasing. Said differently, the function f is concave up on the
interval shown because its derivative, f ′ , is increasing on that interval. Similarly, on the
righthand plot in Figure 1.31, where the function shown is concave down, there we see that
the tangent lines alway lie above the curve and that the value of the slope of the tangent
line is decreasing as we move from left to right. Hence, what makes f concave down on
the interval is the fact that its derivative, f ′ , is decreasing.
Figure 1.31: At left, a function that is concave up; at right, one that is concave down.
We state these most recent observations formally as the definitions of the terms concave
up and concave down.
Definition 1.6. Let f be a differentiable function on an interval (a, b). Then f is concave
up on (a, b) if and only if f ′ is increasing on (a, b); f is concave down on (a, b) if and only
if f ′ is decreasing on (a, b).
The following activities lead us to further explore how the first and second derivatives
of a function determine the behavior and shape of its graph. We begin by revisiting