Why d2y dx2

Lorrine Breyer Pundit. How do you find the tangent line? Mack Zuviaguirre Pundit. What is the difference between first and second derivative test?

The biggest difference is that the firstderivative test always determines whether a function has alocal maximum, a local minimum, or neither; however, the secondderivative test fails to yield a conclusion when y'' is zero ata critical value. Chandra Appen Pundit. What does it mean if the first derivative is zero?

After some time, the slope flattened out to zero and the function had a local minimum. A positive derivativemeans that the function is increasing. A negative derivativemeans that the function is decreasing. A zero derivativemeans that the function has some special behaviour at the givenpoint.

Jelloul Sameth Pundit. What does the first derivative tell you? The first derivative of a function is anexpression which tells us the slope of a tangent line to thecurve at any instant. Because of this definition, the firstderivative of a function tells us much about thefunction. If is positive, then must be increasing. If is negative,then must be decreasing.

Yilian Torio Teacher. What is the symbol for derivative? Chrifa Ikatz Teacher. What happens if the second derivative is 0? A positive second derivative means concave up,negative means concave down. Well, an inflection point is when the concavity switches. So naturally the secondderivative has to equal zero at some point if our second derivative is going to switch signs. An inflectionpoint is the point where the concavity changes.

Marita Verriozar Teacher. What is a higher order derivative? Higher Order Derivatives. Cesar Ouchene Teacher. What does the 3rd derivative tell you? The third derivative is the derivative ofthe derivative of the derivative : the rate of changeof the rate of change of the rate of change.

On the other hand, ifA is position and B is time, then the derivative of A withrespect to B is velocity. The second derivative is the rateof change of velocity, or acceleration. Xinlei Reinholtz Reviewer. How does the second derivative show concavity? The sign of the second derivative gives usinformation about its concavity. Thus the derivative isincreasing! In other words, the graph of f is concaveup.

Raimundas Neant Beginner. What does the second derivative test tell you? The Second Derivative Test. The SecondDerivative Test relates the concepts of critical points,extreme values, and concavity to give a very useful tool fordetermining whether a critical point on the graph of a function isa relative minimum or maximum. Zhana Caridad Beginner. What is the second derivative test used for? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. However, we can still give rigorous meaning to these calculations without appealing to non-standard analysis by using the language of bilinear forms.

There is no possible way of understanding why Leibniz invented the notation he did unless you think about calculus the way Leibniz did, using infinitesimal numbers. Leibniz would have described it as the ratio of two infinitesimals. Nonstandard analysis shows that this idea can be made rigorous, but in any case limits didn't exist in Leibniz's time. The numerator is an infinitesimal number with units of meters.

The denominator is an infinitesimal with units of seconds. All of this does makes it harder for beginners to make sense of the notation. There are many, many more examples I could give of such 'lazy corruption'. I think of it as being like learning any foreign language, where there are always all sorts of quirks and customs that break a general rule. Once you understand the 'lazy corruption' in the context of its surroundings, the meaning is, more often than not, actually, perfectly clear.

In fact, if we wanted to derive the Leibniz notation for the second derivative in a more systematic way, we could use the quotient rule on the first derivative:. Now we have a notation that we can algebraically manipulate to derive identities to, e. To see this hands-on, attempt to prove 4 via algebraic manipulation.

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Create a free Team What is Teams? Learn more. Ask Question. Asked 10 years, 8 months ago. Active 1 month ago. Viewed 18k times. Martin Sleziak I am somewhat serious. Clark: Really? I mean, there are ways of making the notation rigorous I agree that the notation is sensible as evidenced by the answers below.

Really really the entire Leibniz notation, suggesting a ratio of differentials, is somewhat unfortunate. You could make it rigorous using e. Show 4 more comments.