Calculus  Differential Calculus  Problem Set V
Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE ALevel; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE/AIEEE Anyone else who needs this Tutorial as a reference!
Calculus  Differential Calculus  Problem Set V  Outline of Contents:
More examples of investigating and sketching curves, parametric representation of curves.
Why curve sketching is important.
Curve sketching is important to help students visualize what they're doing. If they understand the basis of how a curve is sketch, it also set the ground for them to understand optimization and the concepts of maxima and minima. Free hand sketches of curves are not made for accuracy, they are made for the purpose of understanding general features of the curves such as zeroes, poles, points of discontinuity, inflexion points, aymptotes, tangents, loops, etc. You sketch curves in order to understand the properties of the function. Calculus helps you to determine those properties (such as. gradients and turning points)
Here's a quick look at the curves we will study in this tutorial :
Astroid Curve :
We will Investigate the curve given by the equation x = a cos^{3} t and y = a sin^{3} t ( a > 0)
The quantities x and y are deﬁned for all values of t. But since the functions cos^{3} t and sin3 t are periodic, of a period 2π, it is suﬃcient to consider the variation of the parameter t in the range from 0 to 2π; here the interval [−a, a] is the range of x and the interval [−a, a] is the range of y. We try to find the points at which the tangents to the curve are vertical and horizontal. We also show how this curve has no asymptotes. What we end up getting, is the astroid curve. The tutorial will explain how we go about plotting this curve.
A Loop Shaped Curve :
Construct the curve given by the following equations x = 3at / (1 + t^{3}) and y = 3at^{2 }/ (1 + t^{3}) for a > 0. We go through various steps such as determining the value of 't' for which the curve is defined. We try to find "critial" values of parameter t. We try to find the points at which the tangents to the curve are vertical and horizontal and where it intercepts the axes. We also try to compute the asymptotes to the curve. This tutorial will explain to you, how we can figure out the shape of this curve, identify its aymptotes and zeroes, etc.
