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Calculus/Concavity of a curve


Hi.Could you tell me why concavity of a curve should be found by using its second derivative instead of its first derivative?I think using its first derivative only is enough for the operation!

The concavity of a curve determines which way the curve is changing.
The curve is like the first derivative, so to find the way it is changing,
take the derivative of that.

For example, if we have f(x) = x, then f'(x) = 2x.
As is expected, f'(x)<0 when x<0 and f'(x)>0 when x>0.

The second derivative f"(x) = 2, and this is always positive.
That means the curve is always curved in the upwards direction.
That is what is referred to as concave up.

If the tangent line to the curve at some point is always locally below the curve,
the curve is concave up at that point.  If the tangent line to the curve at some point
is always locally above the curve, then the curve is concave down at that point.

For f(x) = sin(x), f'(x) = cos(x), and f"(x) = -sin(x).
This says that whenever sin(x) is positive, the curve is opening down (concave down) and
whenever sin(x) is negative, the curve is opening up (concave up).

Knowing what the sin() curve looks like confirms this fact.

Another thing the second derivative tells us is whenever f"(x) = 0,
that is a point on the curve of that is called an inflection point.

For f(x)= sin(x), it was just found that f"(x) = -sin(x).
That is known to be 0 at x = nπ, where n is any integer.
As can be seen on the curve, that is every time the sin() curve crosses the axis.

At points such as this, drawing a parallel line to the curve puts one side of the line below the axis and the other side above the axis.  


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