What is a Curve

Open curve is defined as a curve whose ends do not meet. Example is parabola, hyperbola.

Closed curves are curves whose ends are joined. Closed curves do not have end points. Examples of closed curves are ellipse and circle.

When we discuss about a curve, let us see points L, M and N on the curve given below.

N is outside the curve, so it is present in the exterior of the curve.

M is on the boundary of the curve, while L lies inside the curve that is interior of the curve.

• The x axis, if its equation remains the same when y is replaced by - y
• The y axis, if its equation is not altered when x is replaced by - x
• The origin, if it is not altered when x is replaced by - x and y is replaced by - y simultaneously.
• The line y = x, if its equation remains unchanged when x and y are replaced by y and x.
• The line y = -x, if its equation is unchanged when x and y are replaced by - y and - x.

When we sketch a curve, we have to see for x intercept, y intercept, local minima, local maxima, and points of inflection are taken into consideration. There are some steps to be followed to find all these.

• To find the x intercept, we plug in y = 0.
• To find the y intercept, we plug in x = 0
• Local minima occur when the first derivative`dy/dx` is 0 and the sign changes from positive to negative.
• Local maxima occur when the first derivative `dy/dx` is 0 and sign changes from negative to positive.
• Point of inflection is the point where the second derivative `(d^(2)y)/dx^2` is 0 and second derivative changes sign.
• The shape of the curve at any point is determined by the second derivative test.

If `(d^(2)y)/dx^2` > 0, the curve is concave up.

If `(d^(2)y)/dx^2` < 0, the curve is concave down.