Quick question about calculus (derivatives)

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SUMMARY

This discussion centers on the concept of differentiation in calculus, specifically regarding the visualization of functions and their graphs. The user expresses confusion about finding maximum and minimum values without a graph, emphasizing the importance of visual representation in understanding differentiation. The responses clarify that while graphs provide insight, applying differentiation rules is essential for obtaining exact values. The area function is highlighted as a key concept in relation to the problems presented.

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  • Ability to interpret area functions in relation to geometric shapes.
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a129
I thought Differentiation is all about understanding it in a graph. Every time I solve a question on differentiation I visualise it as a graph so it's more logical. After all, that IS what the whole topic is about, right? Or am I just wrong?

But when you look at these questions:
http://imgur.com/a/MdO6w

It's asking for the maximum/minimum value, and my question here is where is the graph? How do I find the maximum/minimum value without a graph? I can't imagine the graph

It's probably doable by just applying the formula, but seeing it in a graph would make sense.

If a question would be something like a person filling a cylinder of height x with water of volume 2cm^3 per second, then sure I can imagine the graph and I can find the rate of change..

So for these questions..how do the graphs look like? Or maybe there isn't any graph for it? I'm worried, I hope I didn't grasp on the wrong concept all this time..

Thanks! I really need some answers

EDIT: I am not asking for the answer. So I thought it's appropriate to post this on here instead of on the homework section
 
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a129 said:
It's probably doable by just applying the formula, but seeing it in a graph would make sense.
In this case, you need to have first a function ##f## which can be differentiated. The graph is then ##\{(x,f(x))\,\vert \,x \in D\}## with ##D## being the domain of allowed values for the variable ##x##.
 
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a129 said:
I thought Differentiation is all about understanding it in a graph. Every time I solve a question on differentiation I visualise it as a graph so it's more logical. After all, that IS what the whole topic is about, right? Or am I just wrong?

But when you look at these questions:
http://imgur.com/a/MdO6w

It's asking for the maximum/minimum value, and my question here is where is the graph? How do I find the maximum/minimum value without a graph? I can't imagine the graph
In the two problems you circled in the image you posted, the graph would be the area function. In the first problem, you have area as a function of r. In the second problem, area is a function of x.
a129 said:
It's probably doable by just applying the formula, but seeing it in a graph would make sense.
It's advantageous to have both a graph of the function and its formula. Seeing the graph would give you a rough idea where a maximum or minimum would be, but using differentiation rules would give you the exact value.
a129 said:
If a question would be something like a person filling a cylinder of height x with water of volume 2cm^3 per second, then sure I can imagine the graph and I can find the rate of change..

So for these questions..how do the graphs look like? Or maybe there isn't any graph for it? I'm worried, I hope I didn't grasp on the wrong concept all this time..

Thanks! I really need some answers

EDIT: I am not asking for the answer. So I thought it's appropriate to post this on here instead of on the homework section
 

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