# Visual interpretation of Fundamental Theorem of Calculus

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• cask1
In summary, the Fundamental Theorem of Calculus provides a visual way of linking a function (F(x)) with its derivative (f(x)) through the concept of the "area function". This can be seen through the use of integrals and the difference between the values of the function at two points. While algebra may also be used to understand the theorem, visual representations such as graphs can provide a more intuitive understanding. Examples such as the sine and cosine waves and the concept of bank account values can help illustrate the relationship between a function and its derivative.
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.

cask1 said:
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

Thank you!

cask1 said:
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.

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.

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## 1. What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a fundamental concept in calculus that relates the concepts of differentiation and integration. It states that if a function is continuous on a closed interval, then the definite integral of the function over that interval can be calculated by finding the antiderivative of the function and evaluating it at the limits of integration.

## 2. How does visual interpretation help in understanding the fundamental theorem of calculus?

Visual interpretation can help in understanding the fundamental theorem of calculus by providing a graphical representation of the concepts involved. It can help visualize the relationship between the derivative and the integral, and how they are related through the fundamental theorem.

## 3. Can the fundamental theorem of calculus be applied to any function?

Yes, the fundamental theorem of calculus can be applied to any continuous function on a closed interval. However, the function must be continuous in order for the integral to be well-defined.

## 4. What is the difference between the first and second parts of the fundamental theorem of calculus?

The first part of the fundamental theorem of calculus states that the derivative of the definite integral of a function is equal to the original function. The second part states that the area under the curve of a function can be calculated by finding the antiderivative of the function and evaluating it at the limits of integration.

## 5. How is the fundamental theorem of calculus used in real-world applications?

The fundamental theorem of calculus has various applications in fields such as physics, engineering, economics, and more. It is used to solve problems involving rates of change, optimization, and calculating areas and volumes. It is also used in the development of other mathematical concepts and theories.

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