Math Formula Calculus

Math Formulas Calculus with Examples

Introduction to Calculus

Calculus is a branch of mathematics that focuses on the study of change. It is divided into two main parts: differential calculus and integral calculus. Differential calculus deals with the concept of a derivative, which represents the rate of change of a function. Integral calculus, on the other hand, focuses on the concept of an integral, which represents the accumulation of quantities. 

Differential Calculus

Definition: Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as the limit:

\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]

Example:

Let's find the derivative of the function \( f(x) = x^2 \).

\[ f'(x) = \lim_{{h \to 0}} \frac{(x+h)^2 - x^2}{h} \]

\[ f'(x) = \lim_{{h \to 0}} \frac{x^2 + 2xh + h^2 - x^2}{h} \]

\[ f'(x) = \lim_{{h \to 0}} \frac{2xh + h^2}{h} \]

\[ f'(x) = \lim_{{h \to 0}} (2x + h) \]

\[ f'(x) = 2x \]

So, the derivative of \( f(x) = x^2 \) is \( f'(x) = 2x \).

Integral Calculus

Definition: Integral

The integral of a function \( f(x) \) over an interval \([a, b]\) is defined as:

\[ \int_a^b f(x) \, dx \]

The integral can be interpreted as the area under the curve of the function \( f(x) \) from \( x = a \) to \( x = b \).

Example:

Let's find the integral of the function \( f(x) = x^2 \) over the interval \([0, 1]\).

\[ \int_0^1 x^2 \, dx \]

To compute this, we use the antiderivative of \( x^2 \), which is \( \frac{x^3}{3} \):

\[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 \]

\[ = \frac{1^3}{3} - \frac{0^3}{3} \]

\[ = \frac{1}{3} - 0 \]

\[ = \frac{1}{3} \]

So, the integral of \( f(x) = x^2 \) from 0 to 1 is \( \frac{1}{3} \).

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concepts of differentiation and integration. It states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then:

\[ \int_a^b f(x) \, dx = F(b) - F(a) \]

This theorem shows that differentiation and integration are inverse processes.

Example:

Using the previous example, we know the antiderivative of \( f(x) = x^2 \) is \( F(x) = \frac{x^3}{3} \). According to the Fundamental Theorem of Calculus:

\[ \int_0^1 x^2 \, dx = F(1) - F(0) \]

\[ = \frac{1^3}{3} - \frac{0^3}{3} \]

\[ = \frac{1}{3} - 0 \]

\[ = \frac{1}{3} \]

This confirms our previous result.

Conclusion

Calculus is a powerful mathematical tool used to understand and describe change and accumulation. Differential calculus focuses on derivatives, representing rates of change, while integral calculus focuses on integrals, representing accumulated quantities. The Fundamental Theorem of Calculus elegantly connects these two branches, showing that they are essentially inverse operations.

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