What is an Antiderivative? Slope Fields

Overview:In this lesson, students will learn the definition of an antiderivative. The main part of the lesson concerns slope fields or direction fields. The students learn what slope fields are, what their purpose is, and how to make them.

Grade Level/Subject:This lesson is for 12th graders in AP Calculus.

Time:1-50 minute class period

Purpose:This lesson will teach students yet another application of derivatives. We will return briefly to the car problem that was discussed in section 6.1 and learn another way of solving the same problem.

Prerequisite Knowledge:

Students should:

-Have a very good understanding of derivatives

-Know how to graph slopes

Objectives:

  1. Students will develop an understanding of antiderivatives and how they are used.
  2. Students will learn slope fields and become comfortable in creating them.

Standards:

  1. Algebra: Students will have to recall their algebra in order to graph the slopes in these problems.
  2. Problem Solving: Students will look at the same problem they looked at in section 6.1 and solve it in a new and different way.

Resources/Materials Needed:

  1. Graph Paper
  2. Calculus Book
  3. Dry Erase Board and Dry Erase Markers

Activities/Procedures:

  1. Give quiz over 6.1 and 6.2.
  2. Give the definition of an antiderivative. Antidifferentiation is obviously the reverse process of differentiation. Given a function f, we say that F is an antiderivative of f provided that F’ = f.
  3. In order to find antiderivatives, slope fields are used. Begin with the simplest case, a function whose graph has a constant slope. f(x) = y0 + m(x - x0). Yet, a function like this could have infinitely many solutions corresponding to all the parallel lines with slope m. Therefore, we need a specific point on the graph
    (x0, y0) known as the initial condition. Draw a graph that represents this type of slope field.
  4. Move on to a more complicated example. f(x) = x2 + 1. Draw the graph of the family of curves. Explain once again that an initial condition is needed.
  5. Introduce slope fields. The easiest way to do this is to se a concrete example. Suppose you have the following function values: f(-4.5) = -5, f(-3) = -2,
    f(-1.5) = -1/3, f(0) = 2, f(1.5) = 3, f(3) = 3, f(4.5) = 2.
    These given function values tell us the slope of any antiderivative’s graph at the same input x.
  6. Draw the slope field for these function values. Next, draw in some of the family of curves.
  7. Return to the example of f(x) = x2 + 1. Give the initial condition y(1.5) = 2.5. Draw a slope field for f(x) for all integer coordinates between -4 < x < 4. Then draw in the graph that satisfies the initial condition.
  8. Return to the car problem from 6.1. First assume that the car travels at a constant speed of 50 mph. The graph of this using speed vs. time is simply a horizontal line. The vertical height of the horizontal line represents the slope of the original graph, distance vs. time. If we assume an initial position as t = 0, d = 0, then we can recover the original graph. Draw the graphs to show how this is done.
  9. Show how this graph will change if the initial conditions are changed.
  10. Now return to a car that travels at varying velocities. Use the values from their trips they took in 6.1. Break it down into hour by hour velocities. Reproduce the original graph using these values.

Homework: Read Section 6.4 and take notes. Complete attached worksheet.

Name:______

Generate a slope field for each function over the indicated interval and sketch the graphs of three antiderivatives of the function over this interval. Use the attached graph paper.

1)f(x) = 1/x, [1, 5]

2)f(x) = tan(x),

3)f(x) = sec(x),

4)f(x) = arctan(x), [0, 4]

5)f(x) = arcsin(x), [-1,1]

6)f(x) = arcsec(x), [1, 2]

7)f(x) = arccsc(x), [1, 2]

8)f(x) = log4(x), [0.5, 2]