Core Theorems of Calculus

1. Intermediate Value Theorem (IVT)

• Definition: If f(x)f(x) is continuous on [a,b][a, b] and NN is any number between f(a)f(a) and f(b)f(b), then there exists at least one cc in (a,b)(a, b) such that f(c)=Nf(c) = N.

• Use: Guarantees a value exists (e.g., a root) between two points.

2. Extreme Value Theorem (EVT)

• Definition: If f(x)f(x) is continuous on [a,b][a, b], then f(x)f(x) has both a maximum and a minimum value on that interval.

• Use: Guarantees an absolute max and min exist on a closed interval.

3. Mean Value Theorem (MVT)

• Definition: If f(x)f(x) is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists at least one cc in (a,b)(a, b) such that

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}

• Use: Guarantees a point where the instantaneous rate of change equals the average rate of change.

4. Rolle’s Theorem

• Definition: If f(x)f(x) is continuous on [a,b][a, b], differentiable on (a,b)(a, b), and f(a)=f(b)f(a) = f(b), then there exists at least one cc in (a,b)(a, b) such that f′(c)=0f'(c) = 0.

• Use: Guarantees a horizontal tangent between two equal y-values.

5. Fundamental Theorem of Calculus – Part 1 (FTC1)

• Definition: If f(x)f(x) is continuous on [a,b][a, b] and

F(x)=∫axf(t) dt,F(x) = \int_a^x f(t) \, dt,

then F′(x)=f(x)F'(x) = f(x).

• Use: Differentiating an integral "undoes" the integral.

6. Fundamental Theorem of Calculus – Part 2 (FTC2)

• Definition: If f(x)f(x) is continuous on [a,b][a, b], then

∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)

where F′(x)=f(x)F'(x) = f(x).

• Use: Computes definite integrals using antiderivatives.

7. Net Change Theorem

• Definition: If F′(x)=f(x)F'(x) = f(x), then

∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)

• Use: The integral of a rate of change gives total change.