Thread: Finding Formulas for Families of Functions?

1. Finding Formulas for Families of Functions?

Hello all. I have two problems here that I am not sure how to answer. Both involve finding formulas for families of functions. Here are the problems:

Find a formula for the following functions:

- A function of the form y= at +b/t with a local minimum (3,12) and a local maximum at (-3,-12).
-A function of the form y= be^(-(x-a)^2)/2) with its maximum at the point (0,3).

I think the second problem is meant to be a variant of the bell-shaped curve (y= e^(-(x-a)^2)/b)If anyone can answer these problems, I would be very grateful!

2. Have you learnt finding derivatives and setting them to zero to locate critical values? This is the same, just treating a and b as constant values whilst differentiating.

In the first problem, setting the derivative to zero enables you to express either a or b in terms of the other, and restate the given function so it mentions only a or b instead of both. Then think, I'm assuming we're at the point (3, 12), so t and y are specifically the values 3 and 12...

Second problem, setting the derivative to zero enables you to reason that a must be zero. Again, refine the given formula and see what follows from (0, 3) being a point on the curve.

The derivatives aren't hard, but here's a couple of pics...

Spoiler:

... and...

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

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