CALCULUSCURVE SKETCHING

Sarvajanik College of Engg. &Technology

GROUP MEMBERS

RAJ CHAUHAN 66

VISHAL BAJAJ 69

SAGAR SAKPAL 72

PIYUSH JAIN 13

ABDUL MOTORWALA 56

KUNJAN RANA 55

CONTENTS

1. Monotonicity of a curve2. Concavity of a curve3. Tracing of Cartesian curves4. Asymptotes6. Region Of Existence7. Strophoid8. Folium of Descrates9. Polar Co-ordinates & Polar Curves10. Tracing Of Polar Curves11. Cycloid12. Inverted Cycloid

MONOTONICITY

Increasing or Decreasing Nature of a Function

Increasing f(x) if x1 < x2 and f(x1) < f(x2).

Decreasing f(x) if x1 < x2 and f(x1) > f(x2).

MONOTONICITY

Let the function f be continuous on [a,b] and differentiable on (a,b)1. If f’(x)>0 for every value of x € (a,b), then f is increasing on [a,b].2. If f’(x)<0 for every value of x € (a,b), then f is decreasing on [a,b].3. If f’(x)=0 for every value of x € (a,b), then f is constant on [a,b].

Any value of x in the domain of f’(x)=0 or at which f is not differentiable is called a critical point.

CONCAVITY

Cases where curves concave upward:

Cases where curves concave downward:

CONCAVITY

f is said to be concave up on (a, b) if f is increasing on (a, b).

f is said to be concave down or convex on (a, b) if f is decreasing on (a, b).

f has an inflection point at a if it is continuous at a and f changes concavity at a.

Criteria for ConcavityIf f’’(x) > 0, f is concave up on (a, b).

If f”(x) < 0, f is concave down on (a, b).

TRACING OF CARTESIAN CURVES

#Symmetry-

(a)Symmetry about X-axis:

The equation of curve remains unaltered on replacing y by –y, i.e.only even powers of y occurs in equation. E.g: y2 =4ax.

CONTD…

(b)Symmetry about y-axis:

The equation of curve remains unaltered when x is replaced by –x, i.e.only even powers of x occur in the equation.

E.g.: x2 =4ay.

CONTD…

(c)Symmetry about both x & y axes:

The equation is such that the powers of x & y both are even everywhere.

E.g.: x2 +y2=a2

CONTD…

(d)Symmetry in opposite quadrants:

The equation of curve remains unaltered when x is replaced by –x and y is replaced by –y simultaneously.

E.g.: xy=c2

CONTD…

(e)Symmetry about line y=x:

The equation of curve remains unaltered when x & y are interchanged.

E.g.: xy=c2

#Origin-

(i) Curves through origin:

The equation of curve doesn’t contain any constant term and passes through origin if it is satisfied by point (0,0).

Eg: y2=4ax

CONTD…

(ii) Tangents at origin:If the curve passes through origin and then tangents to the curve at origin can be obtained by the lowest degree term of equation to zero[0].Eg:in equation y2=4ax,

-4ax=0, x=0.

So,y-axis is only tangent at origin.

CONTD…

#Special points on the curves

-A point is called a DOUBLE POINT if two branches of curve passes through it.

(1)Node:

If the tangents are real and distinct ,the double point is called a “Node”.

CONTD…

(2)Cusps:

If the tangents are real and coincident, the double point is called a “Cusp”.

(3)A double point is called a Conjugate/Isolated point if it is neither a node nor a cusp, i.e.the two tangents at that point are all imaginary.

CONTD…

#Point of intersection with the co-ordinate axes

The points of intersection(if any),with x & y-axis can be respectively obtained by putting y=0 and x=0.

E.g.:-in circle X2+y2=a2 meets x-axis where y=0, i.e. X2=a2; x=+/-a

meets y-axis where x=0, i.e. y2=a2; y=+/-a.

Hence, the point of intersection is(+/-a,0)& (0,+/-a).

#Points where tangents are parallel to co-ordinate axes

This is given by dy/dx=0 in case tangents are parallel to x-axis but is given by dy/dx=+/-∞

ASYMPTOTES

An asymptote to a curve is a straight line which is a tangent to the curve at infinity.

There are three types of asymptotes :1. Parallel to x-axis: Asymptotes parallel to x-axis are obtained by equating to zero

the coefficient of the Highest power of x, provided the coefficient is merely a constant.

2. Parallel to y-axis: Asymptotes parallel to y-axis are obtained by equating to zero the coefficient of the Highest power of y, provided the coefficient is not merely a constant

3. Oblique asymptotes: To obtain an oblique asymptotes y=mx + c in the equation. Simplify the equation and equate the coefficients of highest power of x, next highest power of x,and so on. Solve these equations for m and c. Put the values of m and c in y = mx+c which is the required equation of an oblique asymptote.

CONTD…

#Trace the curve y²(2a-x)=x³ Asymptotes :

a) Since coefficient of highest power of is constant, there is no parallel asymptote to x- axis.

b) Equating the coefficient of highest degree term of y to zero, we get

2a-x= 0

=>x= 2a is the asymptote parallel to y-axis. x= 2a

REGION OF EXISTANCE

This region is obtained by expressing one variable in terms of other, i.e., y=f(x)[or x=f(y)] and then finding the values of x (or y) at which y(or x) becomes imaginary. The curve does not exist in the region which lies between these values of x (or y).

E.g. : Trace the curve y²(2a-x)=x³ Region

We can write the equation of curve like y²= x³/ (2a- x)

The value of y becomes imaginary when x<0 or x>2a.

Therefore, the curve exist in the region 0<x<2a.

x= 2a

STROPHOID

#

STROPHOID

Symmetry: The curve is symmetric about

Origin: The curve passes through origin.

Point of intersection: The curve meets the coordinates axes at origin and point .

Region of Existence: The curve lies in region .

Asymptotes: a. Curve does not have Asymptotes parallel to .b. Asymptotes parallel to y-axis is

FOLIUM OF DESCRATES

FOLIUM OF DESCRATES

POLAR CO-ORDINATE & POLAR CURVES

# Pole We choose a point in the plane that is called the pole (or origin) and is labeled O.#Polar axis Then, we draw a ray (half-line) starting at O called the polar axis. This axis is usually drawn

horizontally to the right corresponding to the positive x-axis in Cartesian coordinate.#Another Point If P is any other point in the plane, let:

r be the distance from O to P. θ be the angle (in radians) between the polar axis and the line OP. P is represented by the ordered pair (r, θ).

CONTD…

We use the convention that an angle is:

If P = O, then r = 0, and we agree that (0, θ) represents the pole for any value of θ.

sign directionPositive(+) Counter-clockwiseNegative(-) Clockwise

CONTD…

We extend the meaning of polar coordinates (r, θ) to the case in which r is negative—as follows:

We agree that, as shown, the points (–r, θ) and (r, θ) lie on the same line through O and at the same distance | r | from O, but on opposite sides of O.

If r > 0, the point (r, θ) lies in the same quadrant as θ.

If r < 0, it lies in the quadrant on the opposite side of

the pole.

Notice that (–r, θ) represents the same point as (r, θ + π).

POLAR CURVES

The graph of a polar equation r = f(θ) [or, more generally, F(r, θ) = 0] consists of all points that have at least one polar representation (r, θ), whose coordinates satisfy the equation.

Substituting value of θ we get values of r.

Plot the points where r is maximum and minimum and find values of θ when curve passes through origin ,if at all it does

TRACING OF POLAR CURVES

If the equation can’t be trace in Cartesian form we have to observe following points to trace a curve whose equation is in polar coordinates.

#Symmetry

To Find whether the curve is symmetrical about any line or pole.

(a) Curve is symmetrical about x-axis if equation remains unaltered when θ is replaced by – θ.

Example : r2=a2 + cos2θ

r = a + bcosθ

(b) If the equation is unchanged when r is

replaced by –r, or when θ is replaced by θ + π, the curve is symmetric about the pole.

CONTD…

(c) If the equation is unchanged when θ is replaced by π – θ, the curve is symmetric about the vertical line θ = π/2.

#Region or Extent (a) Find the greatest numerical value of r, so that the curve lies within the circle

of radius r. (b) Find the value of θ for which r is imaginary.This gives us an idea on the

region into which the curve doesn’t extend.

CONTD…

# Asymptotes

Find the asymptotes of the curve if any .

#Special points

Give different values to θ.Find the corresponding values of r and determine the points coincide with radius vector or perpendicular to it .i.e. where

tanφ= r(dθ/dr)=0 or ∞

CONTD…

E.g. Sketch the curve r=a(1-cosθ) (Cardioid with its axis along negative x-axis)

Sol.

1. Symmetry : Replace θ by –θ .The equation remains unaltered .Hence symmetry is about x-axis.

2. Pole : Put r=0, cosθ=1 or θ=0 .Hence θ=0 is tangent to the curve.

3. Table of values : As θ increases from 0 to π ,r also increases from 0 to 2 as is corresponding values of r and θ given.

CONTD…

E.g. Sketch the curve r=a(1-cosθ) (Cardioid with its axis along negative x-axis)

Sol.

1. Symmetry : Replace θ by –θ .The equation remains unaltered .Hence symmetry is about x-axis.

2. Pole : Put r=0, cosθ=1 or θ=0 .Hence θ=0 is tangent to the curve.

3. Table of values : As θ increases from 0 to π ,r also increases from 0 to 2 as is corresponding values of r and θ given.

CONTD…

4. Plotting these points and joining the points by means of a smooth curve ,we get a curve as shown in figure.

θ =0 π/3 π/2 2π/3 πr =0 a/2 a 3a/2 2a

CYCLOID

The cycloid is the locus of a point on the rim of a circle rolling along a straight line.

CYCLOID

#Parametric Equation of CycloidWe choose as parameter the angle of rotation of the circle ( = 0 when P is at the

origin). Suppose the circle has rotated through radians.

Because the circle has been in contact with the line, we see from Figure 14 that the distance it has rolled from the origin is |OT | = arc PT = r

CYCLOID

Therefore the center of the circle is C (r, r). Let the coordinates of P be (x, y). Then from Figure we see that

x = |OT | – |PQ| = r – r sin = r ( – sin )

y = |TC| – |QC| = r – r cos = r (1 – cos )

Therefore parametric equations of the cycloid are

x = r ( – sin ) y = r (1 – cos ) R

INVERTED CYCLOID

INVERTED CYCLOID

#Parametric Equation of Inverted Cycloid

When you multiply the parameter , which controls the angle, by (-1) you will change the direction of the rotation of the circle generating the cycloid

x = r ((- ) – sin (- )) R

y = r (1 – cos (- )) R

INVERTED CYCLOID

#Parametric Equation of Inverted Cycloid

When you multiply the parameter , which controls the angle, by (-1) you will change the direction of the rotation of the circle generating the cycloid

x = r ((- ) – sin (- )) R

y = r (1 – cos (- )) R