Find the real number which exceeds its square by the greatest amount.

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Solution:

As with any optimization problem (finding maximum or minimum values), our first job is to find a function which represents what we are trying to optimize. In this example, we want to find the greatest difference, so we use this function: f(x) = x - x², since this function represents a number, x, less its square. To maximize this function, we find its derivative, set it equal to zero, and solve (don't forget we also need to find where this derivative is undefined, but in this case, there is no such place).

Now we have f '(x) = 1 - 2x. Set it equal to zero and solve to get x = 1/2. We can prove this is a maximum with a wiggle graph or the Second Derivative Test. Try to do both to review!

Calculus II (BC): A Mystery Integration Problem

Try this integration problem on for size:

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Difficulty Rating: 

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Solution:

Since we have a radical function, it's looking like a trigonometric substitution problem. However, it doesn't seem to match any of our formats. Let's complete the square to force it into a format. Now, it's clear that we are in the form for tangent trigonometric substitution. So, carry on to get the final answer.

I'll leave the final computation to you as review ... I know powers of secant and tangent are not overly fun. You'll need to try Parts to integrate (dv = sec²x and u = secx). Then, you'll need to substitute in using a Pythagorean trig substitution to get another sec³x in the problem to be able to finish. Enjoy ...

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