5.2. THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
281
Example 5.1. Investigate the behavior of the integral function
E(x) =
x
0
e
−t 2 dt.
Solution. E is closely related to the well known error function 2 , a function that is
particularly important in probability and statistics. It turns out that the function e −t 2
does not have an elementary antiderivative that we can express without integrals. That is,
whereas a function such as f (t) = 4 − 2t has elementary antiderivative F(t) = 4t − t 2 , we
are unable to find a simple formula for an antiderivative of e −t 2 that does not involve a
definite integral. We will learn more about finding (complicated) algebraic formulas for
antiderivatives without definite integrals in the chapter on infinite series.
Returning our attention to the function E, while we cannot evaluate E exactly for any
value other than x = 0, we still can gain a tremendous amount of information about the
function E. To begin, applying the rule in Equation (5.4) to E, it follows that
E
′ (x) =
d
dx
x
0
e
−t 2 dt
= e
−x 2
,
so we know a formula for the derivative of E. Moreover, we know that E(0) = 0. This
information is precisely the type we were given in problems such as the one in Activity 3.1
and others in Section 3.1, where we were given information about the derivative of a
function, but lacked a formula for the function itself.
Here, using the first and second derivatives of E, along with the fact that E(0) = 0,
we can determine more information about the behavior of E. First, with E ′ (x) = e −x 2 , we
note that for all real numbers x, e −x 2 > 0, and thus E ′ (x) > 0 for all x. Thus E is an
always increasing function. Further, we note that as x → ∞, E ′ (x) = e −x 2
→ 0, hence the
slope of the function E tends to zero as x → ∞ (and similarly as x → −∞). Indeed, it
turns out (due to some more sophisticated analysis) that E has horizontal asymptotes as x
increases or decreases without bound.
In addition, we can observe that E ′′ (x) = −2xe −x 2 , and that E ′′ (0) = 0, while
E ′′ (x) < 0 for x > 0 and E ′′ (x) > 0 for x < 0. This information tells us that E is concave
up for x < 0 and concave down for x > 0 with a point of inflection at x = 0.
The only thing we lack at this point is a sense of how big E can get as x increases.
If we use a midpoint Riemann sum with 10 subintervals to estimate E(2), we see that
E(2) ≈ 0.8822; a similar calculation to estimate E(3) shows little change (E(3) ≈ 0.8862),
so it appears that as x increases without bound, E approaches a value just larger than
2 The error function is defined by the rule erf(x) =
2
√
π
x
0
e −t
2 dt and has the key property that 0 ≤
erf(x) < 1 for all x ≥ 0 and moreover that lim
x→∞
erf(x) = 1.
281
Example 5.1. Investigate the behavior of the integral function
E(x) =
x
0
e
−t 2 dt.
Solution. E is closely related to the well known error function 2 , a function that is
particularly important in probability and statistics. It turns out that the function e −t 2
does not have an elementary antiderivative that we can express without integrals. That is,
whereas a function such as f (t) = 4 − 2t has elementary antiderivative F(t) = 4t − t 2 , we
are unable to find a simple formula for an antiderivative of e −t 2 that does not involve a
definite integral. We will learn more about finding (complicated) algebraic formulas for
antiderivatives without definite integrals in the chapter on infinite series.
Returning our attention to the function E, while we cannot evaluate E exactly for any
value other than x = 0, we still can gain a tremendous amount of information about the
function E. To begin, applying the rule in Equation (5.4) to E, it follows that
E
′ (x) =
d
dx
x
0
e
−t 2 dt
= e
−x 2
,
so we know a formula for the derivative of E. Moreover, we know that E(0) = 0. This
information is precisely the type we were given in problems such as the one in Activity 3.1
and others in Section 3.1, where we were given information about the derivative of a
function, but lacked a formula for the function itself.
Here, using the first and second derivatives of E, along with the fact that E(0) = 0,
we can determine more information about the behavior of E. First, with E ′ (x) = e −x 2 , we
note that for all real numbers x, e −x 2 > 0, and thus E ′ (x) > 0 for all x. Thus E is an
always increasing function. Further, we note that as x → ∞, E ′ (x) = e −x 2
→ 0, hence the
slope of the function E tends to zero as x → ∞ (and similarly as x → −∞). Indeed, it
turns out (due to some more sophisticated analysis) that E has horizontal asymptotes as x
increases or decreases without bound.
In addition, we can observe that E ′′ (x) = −2xe −x 2 , and that E ′′ (0) = 0, while
E ′′ (x) < 0 for x > 0 and E ′′ (x) > 0 for x < 0. This information tells us that E is concave
up for x < 0 and concave down for x > 0 with a point of inflection at x = 0.
The only thing we lack at this point is a sense of how big E can get as x increases.
If we use a midpoint Riemann sum with 10 subintervals to estimate E(2), we see that
E(2) ≈ 0.8822; a similar calculation to estimate E(3) shows little change (E(3) ≈ 0.8862),
so it appears that as x increases without bound, E approaches a value just larger than
2 The error function is defined by the rule erf(x) =
2
√
π
x
0
e −t
2 dt and has the key property that 0 ≤
erf(x) < 1 for all x ≥ 0 and moreover that lim
x→∞
erf(x) = 1.