You are here:

- Home
- Teens
- Homework/Study Tips
- Calculus
- Concavity of a curve

Advertisement

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) = x², 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.

- Add to this Answer
- Ask a Question

Rating(1-10) | Knowledgeability = 10 | Clarity of Response = 10 | Politeness = 10 |

Comment | Perfect explanation.Thanks! |

Calculus

Answers by Expert:

Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology (reproduction, insusion of chemicals into bloodstream).

Experience in the area: I have tutored students in all areas of mathematics since 1980.
Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors.
Awards and Honors: I have passed Actuarial tests 100, 110, and 135.
**Publications**

Maybe not a publication, but I have respond to well oveer 8,500 questions on the PC.
Well over 2,000 of them have been in calculus.
**Education/Credentials**

I aquired well over 40 hours of upper division courses. This was well over the number that were required.
I graduated with honors in both my BS and MS degree from Oregon State University.
I was allowed to jump into a few courses at college a year early.
**Awards and Honors**

I have been nominated as the expert of the month several times.
All of my scores right now are at least a 9.8 average (out of 10).
**Past/Present Clients**

My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.