Fundamental Theorem of Calculus, Part 1

Definition

The Fundamental Theorem of Calculus, Part 1 states that if a function is continuous on an interval $[a, b]$, then the function defined by the integral of this function from $a$ to $x$ is differentiable and its derivative is the original function. This theorem bridges the concept of differentiation and integration.

5 Must Know Facts For Your Next Test

1. The theorem applies to functions that are continuous on a closed interval $[a, b]$.
2. If $F(x)$ is defined as $\int_a^x f(t) \, dt$, then $F'(x) = f(x)$ for all $x$ in $(a, b)$.
3. The theorem shows that integration can be reversed by differentiation.
4. This part of the theorem constructs an antiderivative of a given function.
5. It provides a practical way to evaluate definite integrals by finding antiderivatives.

Review Questions

• What is the derivative of the integral function $F(x) = \int_a^x f(t) \, dt$?
• Under what conditions does the Fundamental Theorem of Calculus, Part 1 hold true?
• Explain how the Fundamental Theorem of Calculus, Part 1 links differentiation and integration.

Related terms

Continuous Function: A function without breaks, jumps, or discontinuities over its domain.

Antiderivative: A function whose derivative is the given function.

Definite Integral: $\int_a^b f(x) \, dx$, which represents the area under a curve from point $a$ to point $b$.