David M. Bressoud Macalester College, St. Paul, Minnesota AP National Conference, Houston, TX July 17, 2005

This PowerPoint presentation is available at 1.What does the FTC really say: 2004 AB3. 2.Three problems from the 2005 AB exam that use the FTC. 3.A brief history of the FTC.

2004 AB3(d) A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by v(t) = 1 – tan –1 (e t ). At time t = 0, the particle is at y = –1. Find the position of the particle at time t = 2. y '(t) = v(t) = 1 – tan –1 (e t ) y(t) = ? Copyright © 2004 by the College Entrance Examination Board. All rights reserved.

Velocity  Time = Distance time velocity distance

Areas represent distance moved (positive when v > 0, negative when v < 0).

This is the total accumulated distance from time t = 0 to t = 2.

Change in y-value equals We now use the fact that y(0) = –1:

The Fundamental Theorem of Calculus (evaluation part): If then

If we know an anti-derivative, we can use it to find the value of the definite integral. If we know the value of the definite integral, we can use it to find the change in the value of the anti-derivative. The Fundamental Theorem of Calculus (evaluation part):

If then If we know an anti-derivative, we can use it to find the value of the definite integral. If we know the value of the definite integral, we can use it to find the change in the value of the anti-derivative. The Fundamental Theorem of Calculus (evaluation part):

The Fundamental Theorem of Calculus (antiderivative part): In other words, is a perfectly acceptable antiderivative of v(t).

2005 AB3/BC3 Distance x (cm) Temperature T(x) (°C) A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature T(x), in degrees Celsius (°C), of the wire x cm from the heated end. (c) Find, and indicate units of measure. Explain the meaning of in terms of the temperature of the wire. Copyright © 2005 by the College Entrance Examination Board. All rights reserved.

2005 AB3/BC3 Distance x (cm) Temperature T(x) (°C) A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature T(x), in degrees Celsius (°C), of the wire x cm from the heated end. (c) Find, and indicate units of measure. Explain the meaning of in terms of the temperature of the wire. Evaluation part of FTC: This integral represents the change in temperature (in °C) from the heated end of the wire to the other end. Copyright © 2005 by the College Entrance Examination Board. All rights reserved.

2005 AB5/BC5 A car is traveling on a straight road. For 0 ≤ t ≤ 24 seconds, the car’s velocity v(t), in meters per second, is modeled by the piecewise-linear function defined by the graph above. (a) Find Using correct units, explain the meaning of Copyright © 2005 by the College Entrance Examination Board. All rights reserved.

2005 AB5/BC5 A car is traveling on a straight road. For 0 ≤ t ≤ 24 seconds, the car’s velocity v(t), in meters per second, is modeled by the piecewise-linear function defined by the graph above. (a) Find Using correct units, explain the meaning of Interpret integral as area (trapezoid): Copyright © 2005 by the College Entrance Examination Board. All rights reserved.

2005 AB5/BC5 A car is traveling on a straight road. For 0 ≤ t ≤ 24 seconds, the car’s velocity v(t), in meters per second, is modeled by the piecewise-linear function defined by the graph above. (a) Find Using correct units, explain the meaning of Interpret integral as area (trapezoid): Evaluation part of FTC, interpret integral as change in the antiderivative: Copyright © 2005 by the College Entrance Examination Board. All rights reserved.

2005 AB4 The values of f and its first and second derivatives are specified at x = 0, 1, 2, and 3. The sign of f and its derivatives is given for each of the intervals (0,1), (1,2), (2,3), and (3,4). (c) Let g be the function defined by on the open interval (0,4). For 0 < x <4, find all values of x at which g has a relative extremum. Determine whether g has a relative maximum or a relative minimum at each of these values. Justify your answer. Copyright © 2005 by the College Entrance Examination Board. All rights reserved.

2005 AB4 The values of f and its first and second derivatives are specified at x = 0, 1, 2, and 3. The sign of f and its derivatives is given for each of the intervals (0,1), (1,2), (2,3), and (3,4). (c) Let g be the function defined by on the open interval (0,4). For 0 < x <4, find all values of x at which g has a relative extremum. Determine whether g has a relative maximum or a relative minimum at each of these values. Justify your answer. Antiderivative part of FTC: Read information about derivatives of g from values and signs of f and its derivative. Copyright © 2005 by the College Entrance Examination Board. All rights reserved.

There will be many more examples of the Fundamental Theorem of Calculus in the new College Board Professional Development package which should be available this year.

Richard Courant, Differential and Integral Calculus (1931), first calculus textbook to state and designate the Fundamental Theorem of Calculus in its present form.

FTC in present form does not appear in most commonly used calculus texts until George Thomas’s Calculus in the 1950s.

1630’s Descartes, Fermat, and others discover general rule for slope of tangent to a polynomial. René Descartes (1596–1650)Pierre de Fermat (1601–1665)

1630’s Descartes, Fermat, and others discover general rule for slope of tangent to a polynomial. Amazingly, these are inverse to the long known formulas of quadrature (finding areas): Archimedes (287–212 BC) al-Haytham (965–1039) Levi ben Gerson (1288–1344) Johannes Kepler (1571–1630)

1630’s Descartes, Fermat, and others discover general rule for slope of tangent to a polynomial. 1639, Descartes describes reciprocity in letter to DeBeaune

Hints of the reciprocity result in studies of integration by Wallis (1658), Neile (1659), and Gregory (1668) John Wallis (1616–1703)James Gregory (1638–1675)

First published proof by Barrow (1670) Isaac Barrow (1630–1677)

Discovered by Newton (1666, unpublished); and by Leibniz (1673) Isaac Newton (1643–1727)Gottfried Leibniz (1646–1716)

Wm. A. Granville, Pennsylvania (Gettysburg) College, Elements of the Differential and Integral Calculus, 1904, 1911, 1941 “The problems of the Integral Calculus depend on the inverse operation, namely: To find a function f (x) whose derivative is given.”

Wm. A. Granville, Pennsylvania (Gettysburg) College, Elements of the Differential and Integral Calculus, 1904, 1911, 1941 “Theorem. The difference of the values of for x = a and x = b gives the area bounded by the curve whose ordinate is y, the axis of X, and the ordinates corresponding to x = a and x = b. This difference is represented by the symbol and is read ‘the integral from a to b of ydx.’ ”

The Problem: This is how most students think of integration, as the inverse of differentiation. They think of a definite integral as the difference of the antiderivative evaluated at the two endpoints. The FTC does not look like a theorem to them. It looks like a definition. The Solution: Emphasize the usefulness of the limit definition of the definite integral (e.g. as what graphing calculators use to approximate integrals). Emphasize the connection that FTC establishes between antidifferentiation and area or total change.

What if a function is not the derivative of some identifiable function?

Joseph Fourier (1807): Put the emphasis on definite integrals (he invented the notation ) and defined them in terms of area between graph and x-axis. What if a function is not the derivative of some identifiable function?

Joseph Fourier (1807): Put the emphasis on definite integrals (he invented the notation ) and defined them in terms of area between graph and x-axis. How do you define area? What if a function is not the derivative of some identifiable function?

A.-L. Cauchy: First to define the integral as the limit of the summation Also the first (1823) to explicitly state and prove the antiderivative part of the FTC when f is continuous:

Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series Defined as limit of

Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series Defined as limit of When is a function integrable? Does the Fundamental Theorem of Calculus always hold?

The Fundamental Theorem of Calculus (antiderivative part): Riemann found an example of a function f that is integrable over any interval but whose antiderivative is not differentiable at x if x is a rational number with an even denominator.

The Fundamental Theorem of Calculus (evaluation part): If then

Vito Volterra, 1881, found a bounded function f with an anti-derivative F so that F'(x) = f(x) for all x, but there is no interval over which the definite integral of f(x) exists. The Fundamental Theorem of Calculus (evaluation part): If then

Henri Lebesgue, 1901, came up with a totally different way of defining integrals that is the same as the Riemann integral for nice functions, but that avoids the problems with the Fundamental Theorem of Calculus.

Antiderivative part of FTC (Lebesgue 1904): If f is integrable (in the Lebesgue sense) on [a, b], then almost everywhere on [a, b].

Antiderivative part of FTC (Lebesgue 1904): If f is integrable (in the Lebesgue sense) on [a, b], then almost everywhere on [a, b]. The set of x for which this does not hold can be contained in a countable union of intervals, the sum of whose lengths is as small as desired

Evaluation part of FTC (Lebesgue 1904): If f is absolutely continuous on [a, b], then f is differentiable almost everywhere on [a, b], the derivative of f is integrable (in the Lebesgue sense), and for all x in [a, b].

Evaluation part of FTC (Lebesgue 1904): If f is absolutely continuous on [a, b], then f is differentiable almost everywhere on [a, b], the derivative of f is integrable (in the Lebesgue sense), and for all x in [a, b]. Given any  > 0, there exists a response  so that for any collection of pairwise disjoint intervals the sum of whose lengths is less the , the sum of the changes in the values of functions over these intervals will be less than  :

This PowerPoint presentation is available at Now working on a new text, The Fundamental Theorem of Calculus, for the MAA.