MHB Volume Rate of Change: Understanding Math

Click For Summary
The discussion focuses on understanding the volume rate of change in a mathematical context. It confirms that the results for part a) are correct but highlights an error in evaluating $$\frac{dV}{dt}$$, where $$\frac{dr}{dt}$$ should be expressed as $$3\,\frac{\text{in}}{\text{min}}$$. Additionally, it clarifies that $$\frac{dV}{dt}$$ is proportional to the square of the radius $$r$$, indicating a quadratic change rather than an exponential one. This distinction is crucial for accurately solving the problem. Understanding these relationships is essential for mastering the concept of volume rate of change.
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
View attachment 1561

on b) I think this is the point they wanted us to get
hope answers are correct its an even problem # so no ans in bk.
 
Physics news on Phys.org
Re: volume rate of change

Your results for part a) are correct. :D

In your working, where you evaluate $$\frac{dV}{dt}$$ for the given values of $r$, you should have replaced $$\frac{dr}{dt}$$ with $$3\,\frac{\text{in}}{\text{min}}$$ instead of $$3\text{ in}$$. Also, I think what they are looking for in part b) is that $$\frac{dV}{dt}$$ is proportional to the square of $r$. Thus it changes quadratically, not exponentially.
 

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)}...
View full post »

Similar threads

Undergrad Limits confusion -- What is meant by "instantaneous rate of change"?
  • · Replies 18 ·
Replies
18
Views
2K
Undergrad Can I always think of derivatives this way?
  • · Replies 2 ·
Replies
2
Views
2K
MHB How to use change of variables technique here?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Gradient Vector- largest possible rate of change?
  • · Replies 1 ·
Replies
1
Views
2K
MHB *Volume of a cone change of rate of volume with respect to h and r
  • · Replies 8 ·
Replies
8
Views
35K
MHB 205.23 related rates sphere volume
  • · Replies 6 ·
Replies
6
Views
2K
MHB Gradients of curves to find average growth rate between two points
  • · Replies 2 ·
Replies
2
Views
2K
MHB Partial Derivatives of Functions
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Calculating the change of the volume of a sphere using this integral
  • · Replies 3 ·
Replies
3
Views
3K
MHB Am I doing this related rates right?
  • · Replies 2 ·
Replies
2
Views
2K