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# What is a Curve

In our day to day life, a curve is defined as a line which is not straight. But, in math, a curve can also be straight or it is also called continuous line. Curve has certain properties. In this topic you could see types of curve and region of a curve, which is mentioned below.

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## Types of Curves

Based on these properties, curves are classified into two types.

They are:

• Open curve

• Closed curve

## Open Curve

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

## Closed Curve

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.

## Region of a Curve

The boundary along with the interior portion of a curve is called as the region of a curve. Two dimensional curves are algebraically represented as polynomials in variable x and y. We can plot the curve on a graph paper. The curve is symmetrical about:

• 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 derivativedy/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.

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