# 14 Additional Clickable Calculus Solutions Posted

October 03 2013
Maple

7

Fourteen Clickable Calculus examples have been added to the Teaching Concepts with Maple area of the Maplesoft web site. Four are sequence and series explorations taken from algebra/precalculus, four are applications of differentiation, four are applications of integration, and two are problems from the lines-and-planes section of multivariate calculus. By my count, this means some 111 Clickable Calculus examples have now been posted to the section.

Two of the applications of differentiation are related-rate problems: a sliding-ladder and a shadow problem. The sliding-ladder problem is a classic, in which the calculus enters after the correct expression to differentiate has been found. (Readers might also recall the The Sliding Ladder, a Classroom Tips and Techniques article that appeared in the June, 2013, Maple Reporter. In that article, the trajectory of an arbitrary point on the sliding ladder was found and graphed.)

The shadow problem asks for the rate at which a shadow, cast by a person walking away from a light, grows. A variant, not considered, can ask for the speed at which the head of the shadow moves along the ground. But again, what starts out as an illustration of the utility of the derivative often ends up as an algebraic challenge where the bottleneck is formulating the underlying model.

In the example "Minimize Area of Triangle", the area of the triangle whose vertices are the origin, and the intersection of a (movable) line with a parabola, is to be minimized. Finding the two intersections, and writing an expression for the area of a triangle in terms of its vertices, is algebra; the minimization step is the intended application of the calculus. The situation is a bit better for the example "Curvature of an Ellipse" in which the calculation of the curvature requires differential calculus to obtain the underlying model, and then continues using the calculus to find the points of maximum and minimum curvature along the ellipse.

The four applications of integration are more about evaluating definite integrals than about model-building. Calculating the average value of a function, verifying the Mean Value theorem for integrals, finding arc length and the surface area of a surface of revolution - these applications test integration skills more than they test algebraic and geometric training.

The two examples from the lines-and-planes section of the multivariate calculus course require the construction of a plane satisfying given conditions. Here, these planes are found with the standard vector and algebraic tools used in the typical calculus text. However, New Tools for Lines and Planes, the Classroom Tips and Techniques article in the March, 2013, Maple Reporter, shows how these, and other exercises from this section of the course can be solved with a set of tools new to Maple 17. Constructors for a line or plane object, and manipulators for these objects are available through the Context Menu system so that the standard exercises in this section of the multivariate calculus course can be solve with syntax-free calculations.

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