Fundamental Theorem of Calculus, Part 1

Definition

The Fundamental Theorem of Calculus, Part 1 states that if $F$ is an antiderivative of $f$ on an interval $[a, b]$, then the integral of $f$ from $a$ to any point $x$ in that interval is equal to $F(x) - F(a)$. It links the process of differentiation and integration.

5 Must Know Facts For Your Next Test

1. The theorem provides a way to evaluate definite integrals using antiderivatives.
2. If $f$ is continuous on $[a, b]$, then there exists a function $F$ such that $F'(x) = f(x)$ for all $x$ in $(a, b)$.
3. The integral $\int_a^b f(t) \, dt = F(b) - F(a)$ where $F$ is any antiderivative of $f$.
4. This theorem is fundamental because it shows that differentiation and integration are inverse processes.
5. The Fundamental Theorem of Calculus connects the concept of an area under a curve with antiderivatives.

Review Questions

• What does the Fundamental Theorem of Calculus, Part 1 state about the relationship between integration and differentiation?
• How can you use an antiderivative to evaluate a definite integral?
• What conditions must be satisfied for the Fundamental Theorem of Calculus, Part 1 to hold?

Related terms

Antiderivative:

A function whose derivative is the given function. If \( F'(x) = f(x) \), then \( F \) is an antiderivative of \( f \).

Definite Integral:

The integral of a function over a specific interval. It gives the net area under the curve between two points.

Continuous Function:

A function without any breaks or jumps. For every point in its domain, small changes in input lead to small changes in output.