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## Calculus - Differential Calculus - Problem Set IV - Outline of Contents:

Here's a quick outline of the kind of problems which will be solved in this tutorial

## A Quick look at some of the examples we cover in the tutorial document and first, some hints on how to handle such problems.

We will learn a lot about curves by taking a very detailed tour through problems such as the ones below :

• Find all the points (x, y) on the graph of x 2/3 + y 2/3 = 8 where lines tangent to the graph at (x, y) have slope −1.
First differentiate both sides to get the derivative. Then find the point of intersection of the tangent and the curve.

• Find the points of inﬂexion and determine the intervals of convexity and concavity of the curve y = e− (x ^ 2)
Find the expression for the first and second order derivatives. Then find the points where the second order derivative is zero. Find how the sign of the 2nd order derivatives changes while passing through specific points. FInd intervals of convexity and concavity and plot the curve.

• Show that the curve (a2 + x2 )y = a2 x has three points of inﬂexion.
Again, a similar approach. Find the expression for the first and second order derivatives. Then find the points where the second order derivative is zero. Find how the sign of the 2nd order derivatives changes while passing through specific points.

• Find the points of inﬂexion on the curve x = (log y)3 .
Again, a similar approach. Find the expression for the first and second order derivatives. Then find the points where the second order derivative is zero.

• Find the asymptotes of  curves like  y =  (x2 + 2x − 1) / x , y = e−x sin x + x
We discover both a vertical and an inclined asymptote. To determine the mutual positions of a curve and an asymptote, let us consider the difference of the ordinates of the curve and the asymptote for one and the same value of x. We find that his difference is negative for x > 0 and positive for x < 0; and so for x > 0, the curve lies below the asymptote, and for x < 0 it lies above. This is also something which helps us sketch the curve.

• Transforming the formula for radius of curvature of a curve, at a point, into the polar form (using the substitutions x = ρ cos θ and y = ρ sin θ)
We use parametric differentiation to replace (dy/dx) and (d2y/dx2) in terms of first and second order derivatives of ρ w.r.t θ

While going through these problems you will also be able to get an idea about :
• How to change the independent variable from x to t in a given diﬀerential equation .
• How to transform the formula for the radius of curvature, into polar co-ordinates .
• How to determine the mutual positions of a curve and an asymptote ,
• Investigate functions, finding their critical points and and constructing their graphs .
• Determining the domains of increase and decrease of the function .
• Determining the domains of convexity and concavity of the curve and the points of inﬂexion .
• Determining the maximum and minimum values of the function .
• Finding the domains of functions .
• Detecting the presence or absence of a vanishing point.
You will also gain an appreciation of why curve sketching is important