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>Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason I recently-ish helped my stepdaughter with an alge

by jpfed 10y ago

>Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason I recently-ish helped my stepdaughter with an algebra assignment where she was asked to find the maximum of some polynomial. I totally blanked on how to do this with only the tools that she was supposed to have. I didn't know how to do it or explain it without calculus.

mturmon 10y ago
If it was a quadratic, you could (in effect) complete the square and thereby get it to a form like a(x - b)^2 + c At that point, you get the answer by inspection. I.e., in this case, if a is positive, then the function is minimized when x = b, and the minimum value is c. In lucky cases, fourth-order polynomials could be put in such a form. Odd-degree polynomials will not have a unique maximum.
pdm55 10y ago
Use a tool like Symbolab. It shows you the steps for different methods. Very useful. Further, we should all remember to draw a graph as our first step. Symbolab, Desmos and Geogebra (my favourite) are all fantastic graphing software. Also, for a quadratic if you know the zeroes (roots), it's half-way between them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the mid-point (maximum) is at x=3. Your graphing will confirm this. Your stepdaughter might also have learnt that the mid-point of a quadratic ax^2 + bx + c = 0 is x=-b/2a. (This can be derived by setting the derivative equal to zero, but is generally just given to the students.) Her teacher may be expecting her to use that formula. Alternatively the students might be required to determine and plot various data points using, say, Excel, and find the maximum this way. The other point is that solutions often do not spring to mind immediately. That is why I ask students to send me their problems before our tutoring sessions. I often need time to think about them. The fun of math is this exploration to find an answer. So try to get problems and give yourselves time (days) to explore them together.
- jpfed 10y ago
  It was a cubic, but the point is that calculus so obviates the need for any degree-specific tricks that I don't have any particular memory for them.

by jpfed 10y ago

mturmon 10y ago

If it was a quadratic, you could (in effect) complete the square and thereby get it to a form like a(x - b)^2 + c At that point, you get the answer by inspection. I.e., in this case, if a is positive, then the function is minimized when x = b, and the minimum value is c. In lucky cases, fourth-order polynomials could be put in such a form. Odd-degree polynomials will not have a unique maximum.

pdm55 10y ago

Use a tool like Symbolab. It shows you the steps for different methods. Very useful. Further, we should all remember to draw a graph as our first step. Symbolab, Desmos and Geogebra (my favourite) are all fantastic graphing software. Also, for a quadratic if you know the zeroes (roots), it's half-way between them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the mid-point (maximum) is at x=3. Your graphing will confirm this. Your stepdaughter might also have learnt that the mid-point of a quadratic ax^2 + bx + c = 0 is x=-b/2a. (This can be derived by setting the derivative equal to zero, but is generally just given to the students.) Her teacher may be expecting her to use that formula. Alternatively the students might be required to determine and plot various data points using, say, Excel, and find the maximum this way. The other point is that solutions often do not spring to mind immediately. That is why I ask students to send me their problems before our tutoring sessions. I often need time to think about them. The fun of math is this exploration to find an answer. So try to get problems and give yourselves time (days) to explore them together.

jpfed 10y ago
It was a cubic, but the point is that calculus so obviates the need for any degree-specific tricks that I don't have any particular memory for them.