Calculus Q & A

Question:

An email from a Calculus student:

Hi Tom,

First, I want to sincerely thank you for the support of your wonderful programs. You’ve inspired a knowledge of calculus that my prof cannot.

I’ve got a problem to find the extremas of an exponential function over a given closed interval but don’t know which program to use. Here’s the problems:

f(x) = (3x-1)e^(-x), on the interval [0, 2]

and

f(x) = (ln (x+1))/(x+1), on the interval [0, 2]

any advice would be greatly appreciated!

Thanks!

Answer:

In short you enter the function into my “graph by hand” menu program.

Calculus to me is like teaching you how to multiply through the 9’s and then make a student take three semesters siting a million areas of the usage of multiplication.

Calculus finds two things, the derivative finds the “slope of a line”, that’s it!!! and the integral finds the “area” under a smooth curve (x^2,x^3,x^4,x^5 etc and sin(x) and cos(x) (called a sine wave) as you graph those on an x y graph, and only finds the area, if you give that integral a range, called a “definite integral” an indefinite integral finds nothing.

To me calculus is the Sudoku of math, the study of cross word problems. Something to do while waiting for a plane to Phoenix. In calculus they don’t say “rose” (ya know the flower) they say “hibiscus mutabilis” they constantly make easy things into extremely hard things. Linear approximation is a prime example, as well as related rates.

That said, “extrema” “max and mins” of a function is the absolute or maximum high points or low points as you look at a graph of a function. If you graph -x^2 (minus) this is a smooth mountain, extrema is standing at the top. The opposite is true of x^2 (positive) this is a smooth valley, if you graph it, and you are under that valley, touching it with your finger at the lowest point.

That said, it just so happens that when the slope of line is “horizontal” it has a slope of zero, and if you set that horizontal line on top of the mountain it will touch at only one point (tangent) and that point will be the highest possible point on that mountain. So calculus take the first derivative of a function (slope of a line), sets it equal to zero and then solves for the x values. It then puts those x values back into the origininal function, and when solved, finds the “y” value. That point (x,y) is the maximum or extrema of that graphed function. Those x values are called “critical numbers” because they lie on the x axis. They become “critical points” when you plug them into the original function and solve for “y”.

Now to me in your first example, given what I’ve just taught you, They say “over an interval” and then give you the inteval [0,2] (Notice the brackets which indicate an interval where parenthesis would indicate a point (x,y) in math — There is only one critical point (no interval) where the hoizontal line would touch the graph (tangent), so I guess this “over the interval” is to throw you off, or check whether your understanding is as good as mine.

My graph program will do all this for you, but in the first example if written properly (get used to doing this) would look like this: (3x-1)e^(-x). Notice the times sign in front of the “e” that tells you product rule to find the derivative. In the second example quotient rule would be used to find the derivative.