Fundamental Theorem of Calculus

Calculus is a branch of mathematics that helps us understand transformations between values that are related by a function. It is divided into two main branches: differential calculus and integral calculus. In grade 11 math, we focus on understanding the basic concepts of these two branches, with a special emphasis on integration. The fundamental theorem of calculus is a central idea that connects these branches.

The fundamental theorem of calculus is a bridge between differentiation and integration, and consists of two main parts. These parts make it possible to evaluate definite integrals and relate them to antiderivatives. This theorem allows us to understand how the derivative of a function and the area under its graph are related.

Part 1: The relationship between differentiation and integration

The first part of the fundamental theorem of calculus states that if we have a continuous function f(x) on the interval [a, b], and if F(x) is the antiderivative of f(x) on this interval, then:

            f(b) - f(a) = ∫[a to b] f(x) dx
  

This is a powerful result. It tells us that the integral of a function f(x) from a to b is equal to the change in the value of the antiderivative from a to b.

Let us understand this with a simple example:

Let f(x) = 3x^2. The antiderivative of f(x) can be F(x) = x^3, since the derivative of x^3 is 3x^2. Let's find the definite integral from 1 to 2.

According to the theorem, we calculate:

            f(2) - f(1) = (2^3) - (1^3) = 8 - 1 = 7
  

So the integral of 3x^2 from x = 1 to x = 2 is 7.

Now, let's visualize this process.

Part 2: Derivative of the integral

The second part of the fundamental theorem of calculus focuses on the derivative of the integral. It states that if you have a continuous function f(x) on the interval [a, b], and the integral of F(x) is defined by:

            F(x) = ∫[a to x] f(t) dt
  

Then, the derivative of F(x) with respect to x is f(x). In other words:

            f'(x) = f(x)
  

This means that if you accumulate the area under f(t) from a to x, and then take the derivative of this accumulated area, you get the original function f(x) back. This result is incredibly useful because it shows how integration and differentiation are inverse operations.

Let us demonstrate this with another example:

Suppose f(t) = cos(t). Then if we define:

            F(x) = ∫[0 to x] cos(t) dt
  

The derivative of F(x) is:

            f'(x) = cos(x)
  

So, by integrating the cosine function from 0 to x and then differentiating it with respect to x we get back to the original function.

Visual interpretation

Applications and examples

The fundamental theorem of calculus has wide applications in various fields such as physics, engineering, and economics. Understanding this theorem is important for solving real-world problems that involve rates of change and accumulation.

Example 1 - Finding the distance:

If the velocity of a car is given by the function v(t) = 4t, where t is the time in hours, we can find the total distance traveled from time t = 2 to t = 5 by integrating the velocity function:

            Distance = ∫[2 to 5] 4t dt = [2t^2][2 to 5]
                     = (2*(5^2)) - (2*(2^2))
                     = 50 - 8 
                     = 42
  

Thus, the car covered 42 units of distance during that time interval.

Example 2 - Average value of a function:

Consider a function f(x) = x^2 + 3 on the interval [2, 5]. To find the average value of the function, use the formula:

            Average value = (1 / (b - a)) * ∫[a to b] f(x) dx
                          = (1 / (5 - 2)) * ∫[2 to 5] (x^2 + 3) dx
                          = (1/3) * ([x^3/3 + 3x][2 to 5])
                          = (1/3) * (125/3 + 15 – 8/3 – 6)
                          = 17
  

Hence, the average value of the function over the specified interval is 17.

Conclusion

The fundamental theorem of calculus encompasses a deep connection between two fundamental concepts of calculus - differentiation and integration. By understanding this theorem, we can see how these seemingly different operations complement each other, providing powerful tools for analysis and problem-solving. Mastering this theorem not only enhances your mathematical skills, but it also prepares you for more advanced topics in mathematics and related subjects.

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