MATH 1501 Calculus I:
Maple Explorations

Here you will find a collection of Maple worksheets designed to help you in your studies. The following online resourses were written with this course in mind.

• An introductory tutorial on using Maple in this course. This tutorial covers the use of a few simple Maple commands that you will find very useful in doing your homework for this course. It was written expressly for this course.
• Slopes, Tangent Lines, and Derivatives: This worksheet gives animated visualization for the computation and role of a tangent line. The worksheet is appropriate at an early stage in a study of differential calculus.
• Reflections on Differentiation: This worksheet is designed to introduce how to differentiate both functions and expressions. The problem of finding extreme values of simple functions is illustrated by using Fermat's Principle to find the path of reflected light. Some fancier plotting capabilities and also the equation solving ability of Maple are also introduced.
• Solving Equations: This worksheet is about continuity--the intermediate value property of continuous functions, etc. Equations are solved using "bisection," regula falsi, and Newton's method. This assignment help you understand the Maple procedures.
• Solving f(a) = b: The Method of Bisection: The Method of Bisection for finding roots of an equation is not conceptually hard. It has two especially good features: first, it is easily understood, and second, it can provide initial guesses for methods which might converge faster -- such as Newton's Method. We illustrate the method and provide an example to show that a little care must be taken.
• Maple example for a Max-Min problem. This example was contributed by the student Roger Lang.
• Maximum Heart-Rate for a Runner: The heart's pulse-rate for a well-trained runner will increase at the beginning of a run, and then decrease as the runner experiences a "second wind." There after, the pulse-rate will gradually increase as the runner switches from metabolizing glycogen to converting fat into an energy source. This toy model mimics this phenomena. It asks when the maximum heart rate will occur. The answer is obtained with the standard tool: set the derivative equal to zero and solve. It may be that the "solving" is the most interesting part from the perspective of computing in this worksheet.
• Important geometric relationships exist between the graphs of function f, the derivative of f and the integral of f. This worksheet compares their graphs and asks the user to identify which is which in an overlay of the three graphs.
• Area by Chance: The Riemann integral is usually defined as the limit of a collection of approximating sums. There after, the fundamental theorem of integral calculus provides methods for evaluating integrals without computing limits of sums. As an alternative idea, this worksheet introduces a random number generator and what is usually called "Monte-Carlo Techniques" to evaluate integrals. "Area by Chance" is a good worksheet to examine early in the introduction of the integral.
• Techniques of Integration: Substitution and Integration by Parts: It used to be that a calculus class would study the techniques of integra tion so well that the student could work out how to integrate functions such as x arcsin(x). The computer can be used to help with the calculus when methods suc h as substitution or integration-by-parts are correct choices for integration techniques.
• A Distribution of Weights : Most of us have experiences with balancing a series of weights on a line segment . The balance point is located at a point which may be computed as a combination of sums and products of weights and distances from the balance point. For the p urposes of the sciences, we often consider a solid as being concentrated at a si ngle point. The location of the center of mass of a continuously distributed wei ght is computed in this worksheet.
• Rotations : When drawing a figure of revolution, there are several ideas that ar ise: What will the figure look like? What is the area of the resulting surface? What is the volume of the enclosed solid? All three questions are addressed in this worksheet.
• The Theorem of Pappus : Suppose that the plane region R is revolved about the line L in the xy-plane. S uppose also that the line does not intersect the region. The volume of the solid generated is equal to the product of the area of R and the length of the circum ference of the circle traced by the centroid of R. This result of Pappus is illustrated in this worksheet.
• The Year of the Fastest Growth for the US Population: The U. S. Constitution requires a census every ten years. We provide a "Logistic Fit" for that data and graph the fit superimposed with the data. This worksheet then asks the following question: In what year does this logistic fit suggest the fastest growth in the U. S. census data?
• Growth, Decay, and Exponential Functions: The real reason we learn about the exponential function in this course is that it has a lot of important applications. In this worksheet we discuss some of their applications to archaeology (carbon dating), atmospheric radiation, and predictions of populations.

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Last Modified: . School of Mathematics, Georgia Institute of Technology.