- The range of a projectile is , where v 0 is its initial velocity, g is the acceleration due
to gravity and is a constant, and θ is its firing angle. Find the angle that maximizes the projectile’srange.9.A box with a square base, vertical sides, and an open top contains 500 ft^3 of material. Find the
dimensions of the base that minimize the surface area.10.Where on the curve y = does the tangent line have the greatest slope?CURVE SKETCHING
Another topic on which students spend a lot of time in calculus is curve sketching. In the old days, whole
courses (called “Analytic Geometry”) were devoted to the subject, and students had to master a wide
variety of techniques to learn how to sketch a curve accurately.
Fortunately (or unfortunately, depending on your point of view), students no longer need to be as good at
analytic geometry. There are two reasons for this: (1) The AP Exam tests only a few types of curves; and
(2) you can use a graphing calculator. Because of the calculator, you can get an idea of the shape of the
curve, and all you need to do is find important points to label the graph. We use calculus to find some of
these points.
When it’s time to sketch a curve, we’ll show you a four-part analysis that’ll give you all the information
you need.
Step 1: Test the Function
Find where f(x) = 0. This tells you the function’s x-intercepts (or roots). By setting x = 0, we can
determine the y-intercepts. Then find any horizontal and/or vertical asymptotes.
Step 2: Test the First Derivative
Find where f ́(x) = 0. This tells you the critical points. We can determine whether the curve is rising or
falling, as well as where the maxima and minima are. It’s also possible to determine if the curve has any
points where it’s nondifferentiable.
Step 3: Test the Second Derivative
Find where f ́ ́(x) = 0. This shows you where any points of inflection are. (These are points where the
graph of a function changes concavity.) Then we can determine where the graph curves upward and where