Curve Tracing : Curve Tracing by
Dr Indrani Pramod Kelkar
Assocaite Professor
Vignan’s Institiute Of Information Technology, Visakhapatnam
Geometric behavior of curves : Geometric behavior of curves Study geometric behavior of curves is required in many applications of Integral Calculus like-
Evaluation of lengths
Volumes of revolution of plane curves
Areas of regions bounded by plane curves
Analytical Method : Analytical Method The analytical method of obtaining approximate shape of a curve
without the labor of
plotting a large number of points
is called
Curve Tracing.
Steps for Tracing a curve : Steps for Tracing a curve Study of symmetry
Study behavior of curve near origin
Points of intersection with axes
Study of asymptotes
Finding extent of the curve
Study of special points Cartesian Curve y=f(x), x = f(y), f(x,y)=0
Slide 5: Step 1 : Study of Symmetry
- Check for equation f(x,y)= 0 is
Symmetric about When equation is
X-axis unaffected if y is changed to –y
i.e. equation involves only even powers of y.
Y-axis unaffected if x is changed to –x
i.e. equation involves only even powers of x.
y=x unaffected if x & y are interchanged
y=-x unaffected if x & -y are interchanged
Oppo. unchanged if replace x by –x & y by –y
Quadrants
Slide 7: Step 2 : Study behavior of curve near origin
If equation has no constant term then curve passes through origin and then find the tangent at origin by equating the lowest degree terms in the equation to zero.
Decide whether O is a node, cusp or an isolated point
When there are two or more tangents at a point it is called a double point or multiple point.
If the tangents at a double point are
real and distinct then it is called a node.
real and coincident then it is called a cusp.
imaginary then it is an isolated points,
which can not be traced in the graph.
Slide 8: Step 3 : Find points of intersection with the axes
Find point of intersection with x-axis (if any) by putting y=0 in the equation of the curve.
Shift origin to this point and then using step 2 find whether this point is a node, cusp or an isolated point.
If the curve is symmetric about the line y = ± x then find the point of intersection of the curve and the line y = ± x . Shift origin to this point and then using step 2 find whether this point is a node, cusp or an isolated point.
Slide 9: Step 4 : Study of asymptotes
If p(x,y) is in continuopus motion along y = f(x) when one of its coordinates approaches infinity and at the same time distance of the point from the straight line tends to zero then this straight line is an asymptote or a tangent at infinity.
(note : Closed and bounded curves and curves without an infinite branch have no asymptotes)
Types of asymptotes : Types of asymptotes Vertical : put coefficient of highest power of y = 0
Horizontal : put coefficient of highest power of x = 0
Oblique : When the curve is a rational polynomial with degree of numerator one greater than denominator.
Substitute y = mx + c and then equate coefficient of first two highest powers of x to find values of m and c.
Slide 11: Step 5: Find horizontal extent or region
The horizontal extent is defined by those values of x for which y is defined. Find it from expression y = f(x).
Similarly using x = g(y) find vertical extent of the curve.
Eg. If y2 is negative for x > a then the curve does not lie to the right of ordinate of x = a.
Slide 12: Step 6 : Study of special points
Tangents parallel to axes
Solve y’=0 or x’= 0 to get critical points
Solve y’=∞ at which curve changes character
Points of extremum
Maximum if y’ changes from + to – at x=c
Minimum if y’ changes from - to + at x=c
No extremum if y’ does not change in sign
Points of inflection and concavity
Concave up y” > 0
Concave down y” < 0
Slide 13: Points of extremum