curve tracing

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