Looking for a math problem and solution as a transition between Algebra II and Calculus I

Factual Questions
1

Hi

I’ve been searching for an ideal math problem and solution that encompasses both Algebra II and Calculus I. I look forward to your feedback.

2

How about the problem in this thread Math Question - Factual Questions - Straight Dope Message Board

3

I’m not sure what you’re looking for, or why. My first thought was something like “Find the maximum (largest possible) value of the function f(x) = -2x[sup]2[/sup] + 28x - 87” which could be done using algebra (complete the square), graphing (find the vertex of the parabola), or calculus (find where the derivative = 0).

But I’m not at all sure if this is the kind of thing you’re looking for. Plus, as stated, it’s very dry and abstract and lacks any context about why you’d want to find such a thing.

4

This may not be at all what you’re looking for, but there are some classic problems which might mark such a “transition.” Regiomontanus’ famous angle maximization problem was posed and solved in the 15th century; the Wikipedia article shows solutions with geometry, calculus and algebra. Here’s a pdf write-up of the same problem from Mathematics Teacher in which the goal is changed from viewing a work of art to kicking a rugby goal.

5

Thanks septimus. Precisely what I’m looking for. Thank you all.

6

Here’s a problem my Algebra II teacher gave that is just plain ol’ algebra,
yet foreshadows Calculus-think:

Let p(x) be a polynomial in x. (Choose a specific polynomial to work with, e.g.,
x[sup]2[/sup] + 4x - 7

(Try working this with several different specific polynomials.)

Let h represent some arbitrary constant.

Expand and simplify:


p(x + h) - p(x)
---------------
       h

Think about what happens when h is zero. Note, in the given expression (in the box above),
h cannot be zero. But expand and simplify the above without worrying about that. Once you
do that, do you get a simplified expression in which h can be zero? Okay, let h be zero
in the resulting expression and then what expression do you have?

You get: 2x + 4. Your algebra student doesn’t know the earth-shaking significance
of this (yet), but this is the derivative p’(x) = 2x + 4, and this is how you develop
the derivative of polynomials. Derivatives of other kinds of functions are a bit more
complicated, but follow the same general logic of expanding and simplifying
[ f(x + h) - f(x) ] / h and then seeing if you can get some meaningful result when
h is zero.

7

Here you go.

8

I used to give a problem to my classes that could be solved using everything from guess and check to calculus. Basically, it involved a farmer with a field that had a certain number of trees (say, 600). The trees produced an average number of fruit per tree (say 30). The farmer wishes to increase the total number of fruit produced, so he plans to add trees to the orchard. However, he knows from experience that, for every five trees added, the average fruit produced per tree will drop by one. Determine the number of trees he should plant to maximize his total yield.

I forget the actual numbers I used to use; they were calculated to produce an answer that could be determined relatively quickly through both guess-and-check and tabular approaches. However, not shockingly, the problem reduces to an equation for which a local maximum can be determined using calculus (or pre-calculus, or algebra, etc.).

9

The question of the largest rectangle with a given perimeter is readily solved using calculus. With a bit of cleverness it can be solved by pure algebra using the equation
xy = P^2/16 - (x-y)^2/4, which is clearly maximized when x = y.