MHB Approximating Position with Riemann Sums

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The discussion revolves around using a Riemann sum to approximate position based on given time and velocity values. The user calculated a sum of 58.5 feet but is unclear about its significance. It is noted that this sum represents the change in position, as velocity indicates the rate of change of position over time. Clarification is requested for the complete problem to enhance understanding. Overall, the conversation highlights the application of Riemann sums in estimating positional changes from velocity data.
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The question provides a table of values for time and velocity.

Part c of the question asks to use a Riemann sum to approximate (not specifying which one). Part d asks what the answer to part c represents and to explain my reasoning. The answer that I got for the sum is 58.5 feet, but I do not know what this really means.
 
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As velocity is the time rate of change of position, then such a summation of velocities will approximate the change in position. It would be helpful though if you posted the problem in its entirety. :D
 

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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