Key Calculus Concepts

Why This Matters

Calculus is the mathematical language of change and accumulation—two ideas that show up everywhere from physics to economics to biology. In this course, you're being tested on your ability to understand how functions behave, why rates of change matter, and how accumulation connects back to those rates. The concepts here aren't isolated facts; they form an interconnected system where limits enable derivatives, derivatives reverse into integrals, and the Fundamental Theorem ties everything together.

Don't just memorize formulas and rules. Every concept in this guide demonstrates a deeper principle: limits capture the idea of approaching behavior, derivatives quantify instantaneous change, and integrals measure total accumulation. When you study each item, ask yourself what mathematical principle it illustrates and how it connects to the others. That's what separates students who struggle from those who truly understand calculus.

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The Foundation: Limits and Continuity

Before you can talk about rates of change or accumulation, you need to understand what happens as inputs get arbitrarily close to specific values. Limits formalize the intuitive idea of "approaching" and make the rest of calculus rigorous.

Limits

• A limit describes the value a function approaches as the input gets arbitrarily close to a specific point—written as $\lim_{x \to a} f(x) = L$
• Limits can be finite, infinite, or fail to exist—understanding one-sided limits ($\lim_{x \to a^+}$ and $\lim_{x \to a^-}$) helps diagnose function behavior at tricky points
• Every major calculus concept depends on limits—derivatives are defined as limits of difference quotients, and integrals arise from limits of Riemann sums

Continuity

• A function is continuous at a point when three conditions hold—the limit exists, the function is defined there, and $\lim_{x \to a} f(x) = f(a)$
• Discontinuities come in types—removable (holes), jump, and infinite (vertical asymptotes), each revealing different function behaviors
• Continuous functions guarantee key theorems apply—the Intermediate Value Theorem and Extreme Value Theorem require continuity on closed intervals

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Measuring Change: Derivatives and Their Rules

The derivative answers a fundamental question: how fast is something changing right now? It transforms the geometric idea of slope into a precise tool for analyzing instantaneous rates of change.

Derivatives

• The derivative measures instantaneous rate of change—defined formally as $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Geometrically, it gives the slope of the tangent line—at any point $(a, f(a))$, the tangent line has slope $f'(a)$
• Differentiability implies continuity—if $f'(a)$ exists, then $f$ must be continuous at $a$ (but not vice versa)

Differentiation Rules

• The power rule handles polynomial terms—for $f(x) = x^n$, we have $f'(x) = nx^{n-1}$, the most frequently used rule
• Product and quotient rules manage combinations—$(fg)' = f'g + fg'$ and $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
• The chain rule unlocks composite functions—if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$, essential for nested expressions

Applications of Derivatives

• Critical points occur where $f'(x) = 0$ or is undefined—these are candidates for local maxima, minima, or neither
• The second derivative reveals concavity—$f''(x) > 0$ means concave up, $f''(x) < 0$ means concave down, and sign changes indicate inflection points
• Related rates problems connect multiple changing quantities—use implicit differentiation with respect to time to link rates like $\frac{dr}{dt}$ and $\frac{dV}{dt}$

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Measuring Accumulation: Integrals and Techniques

While derivatives break things into instantaneous snapshots, integrals reassemble those pieces into totals. Integration answers the question: if I know the rate, what's the total amount?

Integrals

• Definite integrals compute net accumulation—$\int_a^b f(x)\,dx$ represents the signed area between the curve and the x-axis from $a$ to $b$
• Indefinite integrals find antiderivatives—$\int f(x)\,dx = F(x) + C$ where $F'(x) = f(x)$ and $C$ is the constant of integration
• Riemann sums provide the foundational definition—integrals are limits of sums $\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x$

Integration Techniques

• Substitution reverses the chain rule—set $u = g(x)$, then $du = g'(x)\,dx$ to simplify composite integrands
• Integration by parts reverses the product rule—$\int u\,dv = uv - \int v\,du$, useful when the integrand is a product
• Partial fractions decompose rational functions—split $\frac{P(x)}{Q(x)}$ into simpler fractions when the denominator factors

Applications of Integrals

• Area between curves uses $\int_a^b [f(x) - g(x)]\,dx$—where $f(x) \geq g(x)$ on the interval
• Volumes of revolution apply disk or shell methods—disks use $V = \pi \int_a^b [r(x)]^2\,dx$, shells use $V = 2\pi \int_a^b x \cdot f(x)\,dx$
• Physical applications include work and average value—work done by a variable force is $W = \int_a^b F(x)\,dx$

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The Bridge: Connecting Derivatives and Integrals

The Fundamental Theorem of Calculus is the most important result in the course—it reveals that differentiation and integration aren't just related, they're inverse operations.

Fundamental Theorem of Calculus

• Part 1 says differentiation undoes integration—if $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$, provided $f$ is continuous
• Part 2 provides the evaluation shortcut—$\int_a^b f(x)\,dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$
• This theorem justifies all antiderivative-based integration—without it, we'd have to compute limits of Riemann sums every time

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Extending the Framework: Sequences and Series

When functions are too complex to work with directly, we can approximate them using infinite sums. Sequences and series extend calculus tools to handle infinite processes and function approximations.

Sequences and Series

• A sequence is an ordered list $\{a_n\}$; a series is the sum $\sum_{n=1}^{\infty} a_n$—convergence means the partial sums approach a finite limit
• Key convergence tests include ratio, comparison, and integral tests—each applies to different series structures
• Taylor and Maclaurin series represent functions as infinite polynomials—$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ near $x = a$

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Quick Reference Table

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Self-Check Questions

1. What three conditions must be satisfied for a function to be continuous at a point, and which condition typically fails for a function with a "hole"?

2. Both the chain rule and substitution involve composite functions—explain how one is used for differentiation and the other for integration.

3. If $f'(x) > 0$ and $f''(x) < 0$ on an interval, what can you conclude about the function's behavior? Which concepts are you applying?

4. Compare and contrast the two parts of the Fundamental Theorem of Calculus: what question does each part answer, and when would you use each?

5. A problem gives you a rate function $r(t)$ and asks for the total accumulated quantity over $[0, 5]$. What operation do you perform, and how does this connect to the relationship between derivatives and integrals?