Core Theorems of Calculus

Intermediate Value Theorem (IVT)

If $f(x)$ is continuous on $[a,b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one $c$ in $(a,b)$ such that $f(c) = N$. Guarantees a value exists between two points.

Extreme Value Theorem (EVT)

If $f(x)$ is continuous on $[a,b]$, then $f(x)$ has both a maximum and a minimum value on that interval. Guarantees an absolute max and min exist on a closed interval.

Mean Value Theorem (MVT)

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one $c$ in $(a,b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. Guarantees a point where the instantaneous rate of change equals the average rate of change.

Rolle’s Theorem

If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists at least one $c$ in $(a,b)$ such that $f'(c) = 0$. Guarantees a horizontal tangent between two equal y-values.

Fundamental Theorem of Calculus – Part 1 (FTC1)

If $f(x)$ is continuous on $[a,b]$ and $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$. Differentiating an integral "undoes" the integral.

Fundamental Theorem of Calculus – Part 2 (FTC2)

If $f(x)$ is continuous on $[a,b]$, then $\int_a^b f(x) \, dx = F(b) - F(a)$ where $F'(x) = f(x)$. Computes definite integrals using antiderivatives.

Net Change Theorem

If $F'(x) = f(x)$, then $\int_a^b f(x) \, dx = F(b) - F(a)$. The integral of a rate of change gives total change.