This is a Test of Mathematics Solution Subjective 124 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Sketch on plain paper, the graph of
There are several steps to find graph of a function. We will use calculus to analyze the function. Here y=f(x)
Domain: The function is defined at all real numbers except x =1 and x = -1 which makes the denominator 0.
Even/Odd: Clearly f(x) = f(-x). Hence it is sufficient to investigate the function for positive values of x and then reflect it about y axis.
Critical Points: Next we investigate the critical points. Critical Points are those values of x for which the first derivative of f(x) is either 0 or undefined. Since , then . This implies Hence critical points are x =0 , 1, -1
Monotonicity: The first derivative is negative for all positive values of x (note that we are only investigating for positive x values, since we can then reflect the picture about y axis as previously found). Hence the function is ‘decreasing’ for all positive value of x.
Second Derivative: We compute the second derivative to understand a couple things:
convexity/concavity of the function
examine whether the critical points are maxima, minima, inflection points.
Clearly implying at x = 0 we have local maxima. Since f(0) = – 1, we have (0, -1) as a local maxima. Also the second derivative is negative from x = 0 to x = 1 and positive after x = 1. Hence the curve is under-tangent (concave) from x = 0 to x = 1, and above-tangent (convex) from x =1 onward.
Vertical Asymptote: We next examine what happens near x = 1. We want to know what happens when we approach x=1 from left and from right. To that end we compute the following limits:
(since the denominator gets infinitesimally small with a negative sign, and numerator is about 2)
(since the denominator gets infinitesimally small with a positive sign, and numerator is about 2)
Horizontal Asymptote: Finally we examine what happens when x approaches . To that end we compute the following:
Drawing the graph:
Local Maxima at (0,-1)
Even function hence we draw for positive x values and reflect about y axis
Vertical asymptote at x =1
From x = 0 to 1, the function decreasing to negative infinity, staying under tangent all the time.
From x = 1 to positive infinity, the function decreases from positive infinity to 1 staying above tangent all the time.
Horizontal Asymptote at y= 1
What is this topic: Graph Sketching using Calculus
What are some of the associated concept: Maxima, Minima, Derivative, Convexity, Concavity, Asymptotes