Visual interpretation of Fundamental Theorem of Calculus

[cask1]

Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If you graph x^2 and 2x, they look nothing alike, and there’s no clue as to how they are related, but the area from 1 to 2 under the curve y=2x is always equal to (2)^2 – (1)^2. The units work out also.

 

[dkotschessaa]

Try playing around with the presentation here: http://www.matematicasvisuales.com/english/html/analysis/ftc/ftc1.html

I haven't scrutinized it, but it looks to deliver a good explanation visually.

-Dave K

 

[Ssnow]

Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))?

The link is the ''area function''. If we permit $x$ to varies in an intervall $[a,b]$ then the area under $f(x)$ depends by $x$ and is a function in one variable $\mathcal{A}(x)$ given by:
$\mathcal{A}(x)=\int_{a}^{x}f(s)ds= F(x)-F(a)=\text{Area under} \ \ f \ \ \text{between} \ \ a \ \ \text{and} \ \ x$

so the link is the Area that you can write in integral from $\int_{a}^{x}f(s)ds$ or as the difference $F(x)-F(a)$ (where $F'(x)=f(x)$ and we assume $f$ continuous on $[a,b]$). As @dkotschessaa said I suggest the same link where this can be visualize very well...

Ssnow

 

[cask1]

Thank you!

 

[Stephen Tashi]

Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))?

My comment is that there are instances where algebra is a better way of understanding theorems than pictures. The Calculus of Finite Differences makes the fundamental theorem of calculus seem very natural.

 

[newjerseyrunner]

The derivative is simply the rate of change. The first graphic here is a good example, where the wavy line is f(x) and the red bars represent the derivative at each point. http://m.sparknotes.com/math/calcab/applicationsofthederivative/section5.rhtml

The easiest way to think about how derivatives work is by thinking of the sine wave and costume wave. Why are they derivatives of each other? Visually, it becomes quite obvious when you put them on top of each other. When the sine wave crosses the y axis, it's going up with a slope of exactly 1, so where sine crosses the y-axis from beneath, its derivative is 1, which is the cosines of the same x. When the sine wave is at a value of 1, what's it doing? It's at the top of its period and headed back down, so it's not going up or down at all, giving it a derivative of zero.

Oh, and if you look carefully, you can tell why 2x is the derivative of x^2. Look at how the graph changes on x^2. What is the slope of the line at any given x alone that line? It's a curve so you know it has to be changing. How's it changing? 2x.

A better example with something concrete: your bank account. Your bank account value is f(x). So today u have 50, tomorrow you have 75... so f(1) = 50, f(2) = 75... So from your real values, what was the rate of change? 25. That's the first derivative of your bank account. So next week, you have 100 in your account for f(3), the rate of change f'(x) was again 25. If you take it one step further, you'll notice that the account went up 25 each time. So what was the rate at which the rate itself changed? Well the rate didn't change at all, it was 25 both times, so 0. That's the second derivative. That's essentially the same as position, velocity, acceleration. Derivatives tell you how much a function above it changes.