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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

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

They are:

  • Open curve

  • Closed curve

Open Curve

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

open curve curve

Closed Curve

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

closed curve Types of Curves

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

General Curve

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

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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 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.

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