Fundamental Calculus Theorems

Why This Matters

These theorems aren't just abstract mathematical statements—they're the logical backbone of everything you'll do in calculus. You're being tested on your ability to recognize when a theorem applies, what conditions must be satisfied, and what conclusions you can draw. The theorems here connect three major ideas: continuity, differentiability, and integrability. Understanding how these concepts relate to each other is what separates students who struggle with proofs and applications from those who breeze through them.

Every FRQ that asks you to justify why a value exists, why a function has a maximum, or why a derivative equals zero somewhere is really asking: which theorem applies here? Don't just memorize the statements—know the hypotheses (what you need to check) and the conclusions (what you get to claim). When you see "continuous on a closed interval," your brain should immediately flag several theorems. That pattern recognition is what earns full credit.

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Existence Theorems for Continuous Functions

These theorems guarantee that certain values or points exist based solely on continuity. Continuity on a closed interval is the key hypothesis—it ensures no gaps or jumps that could let values "slip through."

Intermediate Value Theorem

• If $f$ is continuous on $[a, b]$, then $f$ takes on every value between $f(a)$ and $f(b)$—no skipping allowed
• Guarantees existence of solutions—if $f(a) < 0$ and $f(b) > 0$, there must be some $c$ where $f(c) = 0$
• Used to prove roots exist in an interval without actually finding them—perfect for justifying that equations have solutions

Extreme Value Theorem

• Continuous functions on closed intervals attain their bounds—both a maximum and minimum value must occur somewhere on $[a, b]$
• Closed interval is essential—open intervals or discontinuous functions can approach but never reach extreme values
• Foundation for optimization—this theorem justifies why optimization problems on closed intervals always have solutions

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Derivative Existence Theorems

These theorems tell you that under certain conditions, a derivative with specific properties must exist at some point. The key pattern: continuity plus differentiability yields guaranteed derivative behavior.

Rolle's Theorem

• When $f(a) = f(b)$, and $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists some $c$ where $f'(c) = 0$
• Guarantees a horizontal tangent—the function must "turn around" somewhere if it starts and ends at the same height
• Building block for MVT—Rolle's is the special case that proves the more general Mean Value Theorem

Mean Value Theorem

• Connects average and instantaneous rates—there exists $c$ in $(a, b)$ where $f'(c) = \frac{f(b) - f(a)}{b - a}$
• Requires both continuity on $[a, b]$ and differentiability on $(a, b)$—check both conditions before applying
• Powerful for proofs—if you know bounds on $f'$, MVT lets you bound how much $f$ can change

Fermat's Theorem (Stationary Point Theorem)

• Local extrema occur at critical points—if $f$ has a local max or min at $c$ and $f'(c)$ exists, then $f'(c) = 0$
• Does NOT work in reverse—$f'(c) = 0$ doesn't guarantee an extremum (think of $f(x) = x^3$ at $x = 0$)
• First step in optimization—find where $f' = 0$ or $f'$ doesn't exist, then test those candidates

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The Calculus Bridge: Connecting Derivatives and Integrals

This is the theorem that makes calculus calculus—it reveals that differentiation and integration are inverse operations, transforming two separate tools into one unified theory.

Fundamental Theorem of Calculus

• Part 1: Differentiation undoes integration—if $F(x) = \int_a^x f(t)\, dt$ and $f$ is continuous, then $F'(x) = f(x)$
• Part 2: Antiderivatives evaluate definite integrals—$\int_a^b f(x)\, dx = F(b) - F(a)$ where $F' = f$
• The practical payoff—instead of computing limits of Riemann sums, find any antiderivative and subtract endpoint values

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Limit Evaluation Techniques

When direct substitution fails, these theorems provide alternative strategies for finding limits.

L'Hôpital's Rule

• Resolves indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$—replace $\lim \frac{f(x)}{g(x)}$ with $\lim \frac{f'(x)}{g'(x)}$
• Can be applied repeatedly until you escape the indeterminate form—but verify the form each time
• Only works for quotients—convert products, differences, and exponentials to quotient form first (e.g., $0 \cdot \infty$ becomes $\frac{0}{1/\infty}$)

Squeeze Theorem

• Traps a function between two others—if $f(x) \leq g(x) \leq h(x)$ and $\lim f(x) = \lim h(x) = L$, then $\lim g(x) = L$
• Essential for oscillating functions—classic example: $\lim_{x \to 0} x^2 \sin(1/x) = 0$ by squeezing between $-x^2$ and $x^2$
• Requires bounding inequalities—you must establish the inequalities before applying the theorem

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Approximation and Implicit Relationships

These theorems extend calculus to more complex situations: approximating functions with polynomials and handling relationships where $y$ isn't explicitly solved.

Taylor's Theorem

• Any smooth function equals its Taylor polynomial plus an error term—$f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k + R_n(x)$
• Higher degree means better approximation near the center point $a$—but accuracy degrades as you move away
• Error bounds are testable—the Lagrange remainder $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ lets you quantify approximation accuracy

Implicit Function Theorem

• Determines when $F(x, y) = 0$ defines $y$ as a function of $x$—requires $\frac{\partial F}{\partial y} \neq 0$ at the point
• Justifies implicit differentiation—the derivative $\frac{dy}{dx} = -\frac{F_x}{F_y}$ exists when conditions are met
• Extends to higher dimensions—foundational for multivariable calculus and constrained optimization

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Quick Reference Table

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Self-Check Questions

1. Both the Intermediate Value Theorem and the Extreme Value Theorem require continuity on a closed interval. What different conclusions do they guarantee, and when would you use each?

2. A function $f$ is continuous on $[0, 5]$, differentiable on $(0, 5)$, and satisfies $f(0) = f(5) = 3$. Which theorem guarantees the existence of a horizontal tangent, and what can you conclude?

3. Compare the hypotheses of the Mean Value Theorem and L'Hôpital's Rule. Why does MVT require continuity on a closed interval while L'Hôpital's does not?

4. You need to prove that $x^3 + x - 1 = 0$ has a solution in $(0, 1)$. Which theorem applies, and what must you verify?

5. An FRQ gives you $g(x) = \int_1^x \frac{\sin t}{t}\, dt$ and asks for $g'(x)$. Which part of the Fundamental Theorem of Calculus applies, and what is the answer?