MHB Rate of Change in Cone: $$\frac{\text{in}^3}{min}$$

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The discussion centers on whether the constant $$\pi$$ should be included in the rate of change calculation expressed in $$\frac{\text{in}^3}{min}$$. It is clarified that since $$\pi$$ is a dimensionless constant, its inclusion does not alter the units of measurement. Participants agree that unless a decimal approximation is specifically requested, it is preferable to present answers in terms of $$\pi$$. This approach maintains mathematical precision and clarity in the results. The conversation emphasizes the importance of unit consistency in mathematical expressions.
karush
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if correct? wasn't sure if $$\pi$$ should be in answer since answer is in$$\frac{\text{in}^3}{min}$$
 
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Looks perfect...since $\pi$ is a dimensionless constant, it does not affect your units. Personally, unless asked for a decimal approximation, I would leave the answers in terms of $\pi$. :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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