1.3 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, showing they're inverse operations. It proves that finding the area under a curve and calculating rates of change are related, revolutionizing how we solve math problems.

This theorem has two parts. The first shows how to find antiderivatives, while the second gives us a way to evaluate definite integrals. Together, they provide powerful tools for solving real-world problems in physics, economics, and engineering.

The Fundamental Theorem of Calculus

Mean Value Theorem for Integrals

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• States for a continuous function $f$ on $[a, b]$, there exists a point $c$ in $[a, b]$ such that $\int_a^b f(x) [dx](https://www.fiveableKeyTerm:dx) = f(c)(b - a)$
• Geometrically means the area under the curve [object Object],[object Object] over $[a, b]$ equals the area of a rectangle with base $b - a$ and height $f(c)$ (average value of $f$ on $[a, b]$)
• Guarantees the existence of an average value for a continuous function over a closed interval (mean value)
• Provides a connection between the definite integral and the average value of a function (area and average height)
• Relies on the intermediate value theorem to ensure the existence of point $c$

Fundamental Theorem of Calculus

• Part 1 (Existence of antiderivatives) states if $f$ is continuous on $[a, b]$, then the function $F(x) = \int_a^x f(t) dt$ is an antiderivative of $f$ on $[a, b]$, meaning $F'(x) = f(x)$
• Part 2 (Evaluating definite integrals) states if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$, providing a method for evaluating definite integrals using antiderivatives
• Connects the concepts of differentiation and integration, showing they are inverse operations (derivative of integral is original function, integral of derivative is original function up to a constant)
• Demonstrates the definite integral of a function's derivative over an interval equals the change in the function's value over that interval (rate of change and accumulated change)
• Also known as the ###Newton-Leibniz_Formula_0###

Derivatives of integrals

• To find the derivative of a definite integral with a variable upper limit, use $\frac{d}{dx} \int_a^x f(t) dt = f(x)$, where $a$ is a constant and $f$ is continuous on $[a, x]$ (rate of change of accumulated area)
• To find the derivative of a definite integral with a variable lower limit, use $\frac{d}{dx} \int_x^b f(t) dt = -f(x)$, where $b$ is a constant and $f$ is continuous on $[x, b]$ (negative rate of change of accumulated area)
• Allows for finding the rate of change of a quantity defined by an integral (marginal cost, marginal revenue)
• Can be applied to accumulation functions to analyze their behavior

Computing definite integrals

• To evaluate a definite integral $\int_a^b f(x) dx$:
  1. Find an antiderivative $F(x)$ of $f(x)$
  2. Compute $F(b) - F(a)$ (fundamental theorem of calculus part 2)
• Often easier than using Riemann sums or other approximation techniques (trapezoidal rule, Simpson's rule)
• Useful for finding areas, volumes, and other accumulated quantities (work, average value)

Differentiation vs integration

• Differentiation finds the rate of change or slope of a function at a point (instantaneous rate of change)
• Integration finds the accumulated value or area under a curve over an interval (total change)
• The fundamental theorem of calculus connects these two concepts, showing they are inverse operations
  • The derivative of the integral of a function is the original function (FTC part 1)
  • The definite integral of a function's derivative over an interval equals the total change in the function's value over that interval (FTC part 2)
• Understanding the relationship between differentiation and integration is crucial for solving various problems in calculus and its applications (optimization, differential equations)

Continuity and Limits in the Fundamental Theorem

• Continuity of the function is a key requirement for the Fundamental Theorem of Calculus
• The theorem relies on the properties of continuous functions, including the ability to take limits
• The chain rule is often used in conjunction with the Fundamental Theorem when dealing with composite functions