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Section 5.1 Constructing Accurate Graphs of Antiderivatives

A recurring theme in our discussion of differential calculus has been the question “Given information about the derivative of an unknown function \(f\text{,}\) how much information can we obtain about \(f\) itself?” In Activity 1.8.3, the graph of \(y = f'(x)\) was known (along with the value of \(f\) at a single point) and we endeavored to sketch a possible graph of \(f\) near the known point. In Example 3.1.7 — we investigated how the first derivative test enables us to use information about \(f'\) to determine where the original function \(f\) is increasing and decreasing, as well as where \(f\) has relative extreme values. If we know a formula or graph of \(f'\text{,}\) by computing \(f''\) we can find where the original function \(f\) is concave up and concave down. Thus, knowing \(f'\) and \(f''\) enables us to understand the shape of the graph of \(f\text{.}\)

We returned to this question in even more detail in Section 4.1. In that setting, we knew the instantaneous velocity of a moving object and worked to determine as much as possible about the object's position function. We found connections between the net signed area under the velocity function and the corresponding change in position of the function, and the Total Change Theorem further illuminated these connections between \(f'\) and \(f\text{,}\) showing that the total change in the value of \(f\) over an interval \([a,b]\) is determined by the net signed area bounded by \(f'\) and the \(x\)-axis on the same interval.

In what follows, we explore the situation where we possess an accurate graph of the derivative function along with a single value of the function \(f\text{.}\) From that information, we'd like to determine a graph of \(f\) that shows where \(f\) is increasing, decreasing, concave up, and concave down, and also provides an accurate function value at any point.

Preview Activity 5.1.1.

Suppose that the following information is known about a function \(f\text{:}\) the graph of its derivative, \(y = f'(x)\text{,}\) is given in Figure 5.1.1. Further, assume that \(f'\) is piecewise linear (as pictured) and that for \(x \le 0\) and \(x \ge 6\text{,}\) \(f'(x) = 0\text{.}\) Finally, it is given that \(f(0) = 1\text{.}\)

Figure 5.1.1. At left, the graph of \(y = f'(x)\text{;}\) at right, axes for plotting \(y = f(x)\text{.}\)
  1. On what interval(s) is \(f\) an increasing function? On what intervals is \(f\) decreasing?

  2. On what interval(s) is \(f\) concave up? concave down?

  3. At what point(s) does \(f\) have a relative minimum? a relative maximum?

  4. Recall that the Total Change Theorem tells us that

    \begin{equation*} f(1) - f(0) = \int_0^1 f'(x) \, dx\text{.} \end{equation*}

    What is the exact value of \(f(1)\text{?}\)

  5. Use the given information and similar reasoning to that in (d) to determine the exact value of \(f(2)\text{,}\) \(f(3)\text{,}\) \(f(4)\text{,}\) \(f(5)\text{,}\) and \(f(6)\text{.}\)

  6. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of \(y = f(x)\) on the axes provided, being sure to indicate the behavior of \(f\) for \(x \lt 0\) and \(x \gt 6\text{.}\)

Subsection 5.1.1 Constructing the graph of an antiderivative

Preview Activity 5.1.1 demonstrates that when we can find the exact area under the graph of a function on any given interval, it is possible to construct a graph of the function's antiderivative. That is, we can find a function whose derivative is given. We can now determine not only the overall shape of the antiderivative graph, but also the actual height of the graph at any point of interest.

This is a consequence of the Fundamental Theorem of Calculus: if we know a function \(f\) and the value of the antiderivative \(F\) at some starting point \(a\text{,}\) we can determine the value of \(F(b)\) via the definite integral. Since \(F(b) - F(a) = \int_a^b f(x) \, dx\text{,}\) it follows that

\begin{equation} F(b) = F(a) + \int_a^b f(x) \, dx\text{.}\label{AWG}\tag{5.1.1} \end{equation}

We can also interpret the equation \(F(b) - F(a) = \int_a^b f(x) \, dx\) in terms of the graphs of \(f\) and \(F\) as follows. On an interval \([a,b]\text{,}\)

differences in heights on the graph of the antiderivative given by \(F(b) - F(a)\) correspond to the net signed area bounded by the original function on the interval \([a,b]\text{,}\) which is given by \(\int_a^b f(x) \, dx\text{.}\)

Activity 5.1.2.

Suppose that the function \(y = f(x)\) is given by the graph shown in Figure 5.1.2, and that the pieces of \(f\) are either portions of lines or portions of circles. In addition, let \(F\) be an antiderivative of \(f\) and say that \(F(0) = -1\text{.}\) Finally, assume that for \(x \le 0\) and \(x \ge 7\text{,}\) \(f(x) = 0\text{.}\)

Figure 5.1.2. At left, the graph of \(y = f(x)\text{.}\)
  1. On what interval(s) is \(F\) an increasing function? On what intervals is \(F\) decreasing?

  2. On what interval(s) is \(F\) concave up? concave down? neither?

  3. At what point(s) does \(F\) have a relative minimum? a relative maximum?

  4. Use the given information to determine the exact value of \(F(x)\) for \(x = 1, 2, \ldots, 7\text{.}\) In addition, what are the values of \(F(-1)\) and \(F(8)\text{?}\)

  5. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of \(y = F(x)\) on the axes provided, being sure to indicate the behavior of \(F\) for \(x \lt 0\) and \(x \gt 7\text{.}\) Clearly indicate the scale on the vertical and horizontal axes of your graph.

  6. What happens if we change one key piece of information: in particular, say that \(G\) is an antiderivative of \(f\) and \(G(0) = 0\text{.}\) How (if at all) would your answers to the preceding questions change? Sketch a graph of \(G\) on the same axes as the graph of \(F\) you constructed in (e).

Subsection 5.1.2 Multiple antiderivatives of a single function

In the final question of Activity 5.1.2, we encountered a very important idea: a function \(f\) has more than one antiderivative. Each antiderivative of \(f\) is determined uniquely by its value at a single point. For example, suppose that \(f\) is the function given at left in Figure 5.1.3, and suppose further that \(F\) is an antiderivative of \(f\) that satisfies \(F(0) = 1\text{.}\)

Figure 5.1.3. At left, the graph of \(y = f(x)\text{.}\) At right, three different antiderivatives of \(f\text{.}\)

Then, using Equation (5.1.1), we can compute

\begin{align*} F(1) &= F(0) + \int_0^1 f(x) \, dx\\ &= 1 + 0.5\\ &= 1.5\text{.} \end{align*}

Similarly, \(F(2) = 1.5\text{,}\) \(F(3) = -0.5\text{,}\) \(F(4) = -2\text{,}\) \(F(5) = -0.5\text{,}\) and \(F(6) = 1\text{.}\) In addition, we can use the fact that \(F' = f\) to ascertain where \(F\) is increasing and decreasing, concave up and concave down, and has relative extremes and inflection points. We ultimately find that the graph of \(F\) is the one given in blue in Figure 5.1.3.

If we want an antiderivative \(G\) for which \(G(0) = 3\text{,}\) then \(G\) will have the exact same shape as \(F\) (since both share the derivative \(f\)), but \(G\) will be shifted vertically from the graph of \(F\text{,}\) as pictured in red in Figure 5.1.3. Note that \(G(1) - G(0) = \int_0^1 f(x) \, dx = 0.5\text{,}\) just as \(F(1) - F(0) = 0.5\text{,}\) but since \(G(0) = 3\text{,}\) \(G(1) = G(0) + 0.5 = 3.5\text{,}\) whereas \(F(1) = 1.5\text{.}\) In the same way, if we assigned a different initial value to the antiderivative, say \(H(0) = -1\text{,}\) we would get still another antiderivative, as shown in magenta in Figure 5.1.3.

This example demonstrates an important fact that holds more generally:

If \(G\) and \(H\) are both antiderivatives of a function \(f\text{,}\) then the function \(G - H\) must be constant.

To see why this result holds, observe that if \(G\) and \(H\) are both antiderivatives of \(f\text{,}\) then \(G' = f\) and \(H' = f\text{.}\) Hence,

\begin{equation*} \frac{d}{dx}[ G(x) - H(x) ] = G'(x) - H'(x) = f(x) - f(x) = 0\text{.} \end{equation*}

Since the only way a function can have derivative zero is by being a constant function, it follows that the function \(G - H\) must be constant.

We now see that if a function has at least one antiderivative, it must have infinitely many: we can add any constant of our choice to the antiderivative and get another antiderivative. For this reason, we sometimes refer to the general antiderivative of a function \(f\text{.}\)

To identify a particular antiderivative of \(f\text{,}\) we must know a single value of the antiderivative \(F\) (this value is often called an initial condition). For example, if \(f(x) = x^2\text{,}\) its general antiderivative is \(F(x) = \frac{1}{3}x^3 + C\text{,}\) where we include the “\(+C\)” to indicate that \(F\) includes all of the possible antiderivatives of \(f\text{.}\) If we know that \(F(2) = 3\text{,}\) we substitute 2 for \(x\) in \(F(x) = \frac{1}{3}x^3 + C\text{,}\) and find that

\begin{equation*} 3 = \frac{1}{3}(2)^3 + C\text{,} \end{equation*}

or \(C = 3 - \frac{8}{3} = \frac{1}{3}\text{.}\) Therefore, the particular antiderivative in this case is \(F(x) = \frac{1}{3}x^3 + \frac{1}{3}\text{.}\)

Activity 5.1.3.

For each of the following functions, sketch an accurate graph of the antiderivative that satisfies the given initial condition. In addition, sketch the graph of two additional antiderivatives of the given function, and state the corresponding initial conditions that each of them satisfy. If possible, find an algebraic formula for the antiderivative that satisfies the initial condition.

  1. original function: \(g(x) = \left| x \right| - 1\text{;}\) initial condition: \(G(-1) = 0\text{;}\) interval for sketch: \([-2,2]\)

  2. original function: \(h(x) = \sin(x)\text{;}\) initial condition: \(H(0) = 1\text{;}\) interval for sketch: \([0,4\pi]\)

  3. original function: \(p(x) = \begin{cases}x^2, \amp \text{ if } 0 \lt x \lt 1 \\ -(x-2)^2, \amp \text{ if } 1 \lt x \lt 2 \\ 0 \amp \text{ otherwise } \end{cases}\text{;}\) initial condition: \(P(0) = 1\text{;}\) interval for sketch: \([-1,3]\)

Subsection 5.1.3 Functions defined by integrals

Equation (5.1.1) allows us to compute the value of the antiderivative \(F\) at a point \(b\text{,}\) provided that we know \(F(a)\) and can evaluate the definite integral from \(a\) to \(b\) of \(f\text{.}\) That is,

\begin{equation*} F(b) = F(a) + \int_a^b f(x) \, dx\text{.} \end{equation*}

In several situations, we have used this formula to compute \(F(b)\) for several different values of \(b\text{,}\) and then plotted the points \((b,F(b))\) to help us draw an accurate graph of \(F\text{.}\) This suggests that we may want to think of \(b\text{,}\) the upper limit of integration, as a variable itself. To that end, we introduce the idea of an integral function, a function whose formula involves a definite integral.

Definition 5.1.4.

If \(f\) is a continuous function, we define the corresponding integral function \(A\) according to the rule

\begin{equation} A(x) = \int_a^x f(t) \, dt\text{.}\label{FGz}\tag{5.1.2} \end{equation}

Note that because \(x\) is the independent variable in the function \(A\text{,}\) and determines the endpoint of the interval of integration, we need to use a different variable as the variable of integration. A standard choice is \(t\text{,}\) but any variable other than \(x\) is acceptable.

One way to think of the function \(A\) is as the “net signed area from \(a\) up to \(x\)” function, where we consider the region bounded by \(y = f(t)\text{.}\) For example, in Figure 5.1.5, we see a function \(f\) pictured at left, and its corresponding area function (choosing \(a = 0\)), \(A(x) = \int_0^x f(t) \, dt\) shown at right.

Figure 5.1.5. At left, the graph of the given function \(f\text{.}\) At right, the area function \(A(x) = \int_0^x f(t) \ dt\text{.}\)

The function \(A\) measures the net signed area from \(t = 0\) to \(t = x\) bounded by the curve \(y = f(t)\text{;}\) this value is then reported as the corresponding height on the graph of \(y = A(x)\text{.}\) At http://gvsu.edu/s/cz, we find a java applet 1  that brings the static picture in Figure 5.1.5 to life. There, the user can move the red point on the function \(f\) and see how the corresponding height changes at the light blue point on the graph of \(A\text{.}\)

David Austin, Grand Valley State University

The choice of \(a\) is somewhat arbitrary. In the activity that follows, we explore how the value of \(a\) affects the graph of the integral function.

Activity 5.1.4.

Suppose that \(g\) is given by the graph at left in Figure 5.1.6 and that \(A\) is the corresponding integral function defined by \(A(x) = \int_1^x g(t) \, dt\text{.}\)

Figure 5.1.6. At left, the graph of \(y = g(t)\text{;}\) at right, axes for plotting \(y = A(x)\text{,}\) where \(A\) is defined by the formula \(A(x) = \int_1^x g(t) \ dt\text{.}\)
  1. On what interval(s) is \(A\) an increasing function? On what intervals is \(A\) decreasing? Why?

  2. On what interval(s) do you think \(A\) is concave up? concave down? Why?

  3. At what point(s) does \(A\) have a relative minimum? a relative maximum?

  4. Use the given information to determine the exact values of \(A(0)\text{,}\) \(A(1)\text{,}\) \(A(2)\text{,}\) \(A(3)\text{,}\) \(A(4)\text{,}\) \(A(5)\text{,}\) and \(A(6)\text{.}\)

  5. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of \(y = A(x)\) on the axes provided, being sure to indicate the behavior of \(A\) for \(x \lt 0\) and \(x \gt 6\text{.}\)

  6. How does the graph of \(B\) compare to \(A\) if \(B\) is instead defined by \(B(x) = \int_0^x g(t) \, dt\text{?}\)

Subsection 5.1.4 Summary

  • Given the graph of a function \(f\text{,}\) we can construct the graph of its antiderivative \(F\) provided that (a) we know a starting value of \(F\text{,}\) say \(F(a)\text{,}\) and (b) we can evaluate the integral \(\int_a^b f(x) \, dx\) exactly for relevant choices of \(a\) and \(b\text{.}\) For instance, if we wish to know \(F(3)\text{,}\) we can compute \(F(3) = F(a) + \int_a^3 f(x) \, dx\text{.}\) When we combine this information about the function values of \(F\) together with our understanding of how the behavior of \(F' = f\) affects the overall shape of \(F\text{,}\) we can develop a completely accurate graph of the antiderivative \(F\text{.}\)

  • Because the derivative of a constant is zero, if \(F\) is an antiderivative of \(f\text{,}\) it follows that \(G(x) = F(x) + C\) will also be an antiderivative of \(f\text{.}\) Moreover, any two antiderivatives of a function \(f\) differ precisely by a constant. Thus, any function with at least one antiderivative in fact has infinitely many, and the graphs of any two antiderivatives will differ only by a vertical translation.

  • Given a function \(f\text{,}\) the rule \(A(x) = \int_a^x f(t) \, dt\) defines a new function \(A\) that measures the net-signed area bounded by \(f\) on the interval \([a,x]\text{.}\) We call the function \(A\) the integral function corresponding to \(f\text{.}\)

Exercises 5.1.5 Exercises

1. Definite integral of a piecewise linear function.
2. A smooth function that starts out at 0.
3. A piecewise constant function.
4. Another piecewise linear function.
5.

A moving particle has its velocity given by the quadratic function \(v\) pictured in Figure 5.1.7. In addition, it is given that \(A_1 = \frac{7}{6}\) and \(A_2 = \frac{8}{3}\text{,}\) as well as that for the corresponding position function \(s\text{,}\) \(s(0) = 0.5\text{.}\)

  1. Use the given information to determine \(s(1)\text{,}\) \(s(3)\text{,}\) \(s(5)\text{,}\) and \(s(6)\text{.}\)

  2. On what interval(s) is \(s\) increasing? On what interval(s) is \(s\) decreasing?

  3. On what interval(s) is \(s\) concave up? On what interval(s) is \(s\) concave down?

  4. Sketch an accurate, labeled graph of \(s\) on the axes at right in Figure 5.1.7.

  5. Note that \(v(t) = -2 + \frac{1}{2}(t-3)^2\text{.}\) Find a formula for \(s\text{.}\)

Figure 5.1.7. At left, the given graph of \(v\text{.}\) At right, axes for plotting \(s\text{.}\)
6.

A person exercising on a treadmill experiences different levels of resistance and thus burns calories at different rates, depending on the treadmill's setting. In a particular workout, the rate at which a person is burning calories is given by the piecewise constant function \(c\) pictured in Figure 5.1.8. Note that the units on \(c\) are “calories per minute.”

Figure 5.1.8. At left, the given graph of \(c\text{.}\) At right, axes for plotting \(C\text{.}\)
  1. Let \(C\) be an antiderivative of \(c\text{.}\) What does the function \(C\) measure? What are its units?

  2. Assume that \(C(0) = 0\text{.}\) Determine the exact value of \(C(t)\) at the values \(t = 5, 10, 15, 20, 25, 30\text{.}\)

  3. Sketch an accurate graph of \(C\) on the axes provided at right in Figure 5.1.8. Be certain to label the scale on the vertical axis.

  4. Determine a formula for \(C\) that does not involve an integral and is valid for \(5 \le t \le 10\text{.}\)

7.

Consider the piecewise linear function \(f\) given in Figure 5.1.9. Let the functions \(A\text{,}\) \(B\text{,}\) and \(C\) be defined by the rules \(A(x) = \int_{-1}^{x} f(t) \, dt\text{,}\) \(B(x) = \int_{0}^{x} f(t) \, dt\text{,}\) and \(C(x) = \int_{1}^{x} f(t) \, dt\text{.}\)

  1. For the values \(x = -1, 0, 1, \ldots, 6\text{,}\) make a table that lists corresponding values of \(A(x)\text{,}\) \(B(x)\text{,}\) and \(C(x)\text{.}\)

  2. On the axes provided in Figure 5.1.9, sketch the graphs of \(A\text{,}\) \(B\text{,}\) and \(C\text{.}\)

  3. How are the graphs of \(A\text{,}\) \(B\text{,}\) and \(C\) related?

  4. How would you best describe the relationship between the function \(A\) and the function \(f\text{?}\)

Figure 5.1.9. At left, the given graph of \(f\text{.}\) At right, axes for plotting \(A\text{,}\) \(B\text{,}\) and \(C\text{.}\)