Second Derivative: What Does it Represent? - James

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The second derivative of a function indicates its concavity, with positive values signifying upward curvature and negative values indicating downward curvature. In physics, the second derivative of a position function with respect to time represents acceleration, illustrating the rate of change of velocity. While the first derivative is often described as the gradient of the tangent line, the second derivative can similarly be viewed as the gradient of the tangent line of the first derivative's graph. The discussion emphasizes that while these interpretations are useful, functions can be understood in various contexts, leading to different meanings for the first and second derivatives. Overall, the second derivative plays a crucial role in understanding the behavior of functions and their applications.
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If the first derivative of a function represents the gradient of the tangent line...

What does the second derivative represent?

Thanks in advance
James
 
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the second derivative of some function f: R -> R represents concavity. positive values for the second derivative mean that the graph curves upward, negative values mean the graph curves downward. in physics the second derivative of position function with respect to time gives the acceleration.
 
Last edited:


rate of change of the rate of change.
like acceleration
 


It's the gradient of the tangent line of the derivative's graph.
 


I'm a little bit reluctant to say that the "first derivative of a function represents the gradient of the tangent line" or that the "second derivative of a function represents the curvature".

Certainly we can think of them that way, especially if we are focusing on graphs of functions. But functions have many different applications, we can think of functions in many different ways, and each gives a different interpretation of first and second derivative.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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