- #1
hexag1
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I'm going through a calculus textbook in an attempt to learn it myself. So far so good, but I've been stuck on optimization problems.
I understand the concept. The maxima and minima of a function can be found by looking at where its derivative = 0.
I also see that a function that has no upper or lower bound can still have local maxima and minima.
I went through the problems in max/min section of my textbook just fine.
But I have run into trouble in the optimization section.
In these problems, one is asked to maximize or minimize some value (say the volume of a box, or the area of a window, or whatever).
First, one writes a function, almost always of more than one variable.
Secondly, using other information, one finds an expression to substitute into the function and write it in terms of a single variable.
Third, one differentiates the function, looks for the zeroes, and then evaluates the initial function at these values etc.
My problem is in step two. I can't find ways of writing functions in terms of a single variable.
Here's one that gives me trouble:
A right circular cylinder is inscribed in a sphere of radius R. What is the maximum surface area of such a cylinder?
So, the cylinder is inside the sphere, its edges touching the sphere.
We can call its radius c, and its height h.
Its area is:
A = 2 \pi c^2 + 2 \pi c h
Now we have to differentiate and find A' . But how do we find a substitute for h?
From the cross section of the cylinder, we can see that the cylinder's radius and height/2 form a right triangle with the radius R of the sphere. So:
R^2 = c^2 + (h/2)^2 Right?
solving for h, we have:
h = 2 \sqrt {R^2 - c^2}
We then substitute h into the area function, and differentiate. I get nothing like what my textbook gives, which is \pi R^2(1+ \sqrt{5})
I don't see how they got this.
I understand the concept. The maxima and minima of a function can be found by looking at where its derivative = 0.
I also see that a function that has no upper or lower bound can still have local maxima and minima.
I went through the problems in max/min section of my textbook just fine.
But I have run into trouble in the optimization section.
In these problems, one is asked to maximize or minimize some value (say the volume of a box, or the area of a window, or whatever).
First, one writes a function, almost always of more than one variable.
Secondly, using other information, one finds an expression to substitute into the function and write it in terms of a single variable.
Third, one differentiates the function, looks for the zeroes, and then evaluates the initial function at these values etc.
My problem is in step two. I can't find ways of writing functions in terms of a single variable.
Here's one that gives me trouble:
A right circular cylinder is inscribed in a sphere of radius R. What is the maximum surface area of such a cylinder?
So, the cylinder is inside the sphere, its edges touching the sphere.
We can call its radius c, and its height h.
Its area is:
A = 2 \pi c^2 + 2 \pi c h
Now we have to differentiate and find A' . But how do we find a substitute for h?
From the cross section of the cylinder, we can see that the cylinder's radius and height/2 form a right triangle with the radius R of the sphere. So:
R^2 = c^2 + (h/2)^2 Right?
solving for h, we have:
h = 2 \sqrt {R^2 - c^2}
We then substitute h into the area function, and differentiate. I get nothing like what my textbook gives, which is \pi R^2(1+ \sqrt{5})
I don't see how they got this.