Basic Calculus Concepts

Why This Matters

Calculus is the mathematical language of change—and Contemporary Mathematics tests whether you can apply this language to real-world problems. You're not just being asked to memorize formulas; you're being tested on your ability to recognize when a situation involves rates of change (derivatives) versus when it involves accumulation (integrals). These two big ideas—along with the functions that model them—form the backbone of everything from economics to biology to engineering.

The concepts here build on each other deliberately. Functions describe relationships, limits let us handle the infinitely small, derivatives capture instantaneous change, and integrals sum up infinite pieces. Master the connections between these ideas, and you'll handle any application problem thrown at you. Don't just memorize the power rule—know why derivatives give you slopes and when to reach for integration instead.

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The Foundation: Functions and Their Behavior

Before calculus can describe change, you need to understand what's changing. Functions are the objects we study, and their properties determine which calculus tools apply.

Functions and Their Graphs

• A function assigns exactly one output to each input—this one-to-one mapping is what makes calculus possible; if inputs could have multiple outputs, rates of change would be meaningless
• Graph features reveal function behavior—intercepts show where outputs equal zero, slopes indicate steepness, and asymptotes mark boundaries the function approaches but never reaches
• Domain and range restrictions matter for real-world applications; you can't take the square root of negative numbers or divide by zero in most contexts

Exponential and Logarithmic Functions

• Exponential functions have the form $f(x) = a^x$ and model situations where growth rate is proportional to current size—the bigger it gets, the faster it grows
• Logarithmic functions are inverses of exponentials, written $\log_a(x)$, and are essential for solving equations where the variable is in the exponent
• Natural exponential $e \approx 2.718$ appears constantly in calculus because its derivative equals itself: $\frac{d}{dx}e^x = e^x$

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Approaching Infinity: Limits and Continuity

Limits are the conceptual engine of calculus. They let us talk precisely about what happens as we get infinitely close to something without ever quite reaching it.

Limits and Continuity

• A limit describes the value a function approaches as input approaches some value; written $\lim_{x \to a} f(x) = L$, this doesn't require the function to actually equal $L$ at $x = a$
• Continuity means no breaks, jumps, or holes—formally, a function is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$; the limit exists and equals the actual function value
• Limits enable derivatives and integrals by letting us calculate instantaneous rates (not just average rates) and exact areas (not just approximations)

Sequences and Series

• A sequence is an ordered list like $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8},...$ while a series is the sum of sequence terms: $1 + \frac{1}{2} + \frac{1}{4} + ...$
• Convergence vs. divergence determines whether an infinite series adds up to a finite number or grows without bound—this is a limit question in disguise
• Geometric series with ratio $|r| < 1$ converge to $\frac{a}{1-r}$, making them useful for modeling diminishing returns in economics and repeated processes

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Measuring Change: Derivatives and Differentiation

Derivatives answer the question: How fast is something changing right now? This instantaneous rate of change is the slope of the tangent line at any point.

Derivatives and Differentiation Rules

• The derivative $f'(x)$ represents instantaneous rate of change—defined as $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, this limit captures the slope at a single point rather than between two points
• Power rule: $\frac{d}{dx}x^n = nx^{n-1}$—this handles polynomials; combine with the product rule $\frac{d}{dx}[fg] = f'g + fg'$ and quotient rule for more complex functions
• Chain rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$—essential for composite functions; differentiate the outside, then multiply by the derivative of the inside

Applications of Derivatives

• Optimization finds maximum and minimum values by setting $f'(x) = 0$ and solving for critical points—these are where the function "levels off" before changing direction
• Second derivative test determines whether critical points are maxima ($f''(x) < 0$, concave down) or minima ($f''(x) > 0$, concave up)
• Related rates problems use the chain rule to connect how different quantities change together—if you know how fast one thing changes, you can find how fast a related thing changes

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Measuring Accumulation: Integrals and Integration

If derivatives break things into infinitely small pieces to find rates, integrals do the reverse—they add up infinitely many infinitely small pieces to find totals.

Integrals and Basic Integration Techniques

• Definite integrals $\int_a^b f(x)\,dx$ calculate accumulated quantities—geometrically, this represents the signed area between the curve and the x-axis from $a$ to $b$
• Indefinite integrals yield antiderivatives plus a constant: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$; the $+C$ accounts for the fact that many functions share the same derivative
• Substitution (u-substitution) reverses the chain rule—set $u = g(x)$, find $du = g'(x)\,dx$, and rewrite the integral in terms of $u$

Fundamental Theorem of Calculus

• Part 1 connects derivatives and integrals as inverse operations—if $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$; differentiation undoes integration
• Part 2 provides the evaluation shortcut: $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$—find an antiderivative, plug in the bounds, subtract
• This theorem is why calculus works as a unified subject; without it, derivatives and integrals would be separate tools with no deep connection

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Real-World Applications

Calculus becomes powerful when applied to model dynamic systems and solve practical problems. These applications show up frequently on exams.

Applications of Integrals

• Area between curves uses $\int_a^b [f(x) - g(x)]\,dx$ where $f(x)$ is the upper curve and $g(x)$ is the lower curve—always subtract lower from upper
• Volumes of revolution use disk method $V = \pi \int_a^b [r(x)]^2\,dx$ when rotating around an axis; washer method adds a hole in the middle
• Total change from rate of change—if $f'(t)$ gives velocity, then $\int_a^b f'(t)\,dt$ gives total displacement; this pattern applies to any rate-to-total conversion

Basic Differential Equations

• Differential equations relate functions to their derivatives—$\frac{dy}{dx} = ky$ says "the rate of change is proportional to the current amount," modeling exponential growth/decay
• Separable equations can be solved by getting all $y$ terms on one side and all $x$ terms on the other, then integrating both sides
• Initial conditions determine the specific solution from a family of solutions—without them, you'll have that $+C$ constant still floating around

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Quick Reference Table

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Self-Check Questions

1. Both the derivative and the definite integral involve limits in their definitions. What conceptual question does each one answer, and how does the Fundamental Theorem connect them?

2. You're given a function and asked to find where it reaches its maximum value on a closed interval. What's your step-by-step process, and why must you check endpoints too?

3. Compare and contrast: When would you use the product rule versus the chain rule? Give an example function for each.

4. A problem gives you a rate function $r(t)$ and asks for the total accumulated quantity between $t = 0$ and $t = 5$. What calculus operation do you use, and what does your answer represent geometrically?

5. The differential equation $\frac{dy}{dt} = 0.03y$ models a bank account with continuous interest. What type of function is the solution, and how would you find the specific solution if you know the initial deposit was $1000?