Curves with Extrema?

We spend a lot of time in calculus studying curves. We look for maximums, minimums, asymptotes, end behavior, and on and on, but what about in “real life”? For some time I’ve been trying to find a real situation determined or modeled by a non-trigonometric curve with more than one extreme value. I’ve not been…

Soda Cans

A typical calculus optimization question asks you to find the dimensions of a cylindrical soda can with a fixed volume that has a minimum surface area (and therefore is cheaper to manufacture). Let r be the radius of the cylinder and h be its height. The volume, V, is constant and . The surface area…

The Marble and the Vase

A fairly common max/min problem asks the student to find the point on the parabola that is closest to the point .  The solution is not too difficult. The distance, L(x), between A and the point  on the parabola  is given by And the minimum distance can be found when This occurs when . The local…

Why Radians?

Calculus is always done in radian measure. Degree (a right angle is 90 degrees) and gradian measure (a right angle is 100 grads) have their uses. Outside of the calculus they may be easier to use than radians. However, they are somewhat arbitrary. Why 90 or 100 for a right angles? Why not 10 or…

Related Rate Problems I

Related rate problems provide an early opportunity for students to use calculus in a, more or less, real context and practice implicit differentiation. One of the problems students have with these problems is that almost all of them involve writing the model or starting equation based on some geometric situation. Students have to switch from…