Calculus Fundamentals
(Level 2)
Algebra tells you what an equation equals right now. Calculus tells you how fast it's changing and where it's going next. Derivatives give you instantaneous rates—velocity from position, acceleration from velocity, slope at a single point. Integrals do the reverse: they add up infinitely many tiny pieces to get totals—distance from velocity, area under curves, accumulated work. This is where engineering stops being static and starts modeling reality.
1. Limits and Continuity
Derivatives and integrals are built on limits. A limit asks: "What value does this function approach as x gets closer and closer to some number?" Not what it equals at that point—what it approaches. Sometimes they're the same. Sometimes they're not. That difference is what makes calculus more powerful than algebra.
Continuity means no jumps, no breaks, no holes. If you can draw a function without lifting your pen, it's probably continuous. Mathematically, a function is continuous at a point if the limit equals the actual function value there. Most engineering functions are continuous almost everywhere—that's what makes calculus useful instead of pathological.
Just plug in 2. The function is continuous, so the limit equals the value.
Can't plug in 0 (you'd get 0/0). This is a famous limit—memorize it.
This defines the derivative of eˣ. Limits aren't just theory—they create the rules you'll use constantly.
Why This Matters Later
Every derivative, every integral, every differential equation—all defined using limits. If you don't understand limits, calculus is just symbol-pushing. With limits, it's a rigorous tool for analyzing change.
2. Differentiation
The derivative is the slope of a curve at a single point. Not average slope over an interval—instantaneous slope at one exact spot. If position is a function of time, the derivative is velocity. If velocity is a function of time, its derivative is acceleration. The pattern repeats: derivatives measure how things change, instantly.
You don't compute derivatives from the limit definition every time (though that's how they're defined). You learn rules: power rule, product rule, chain rule. These let you differentiate almost anything you'll encounter in engineering without going back to first principles. But knowing the limit definition exists keeps you honest—it's not magic, it's a well-defined operation.
Power rule: bring down the exponent, subtract 1 from the exponent.
Power rule on each term. Constants disappear, linear terms become constants.
Chain rule: derivative of outer function times derivative of inner. The 2 from 2x comes along for the ride.
Why This Matters Later
Velocity, acceleration, slope, rate of reaction, marginal cost—all derivatives. Differential equations model how systems change over time. Numerical optimization finds maxima and minima. None of it works without differentiation.
3. Applications of Differentiation
Once you can take derivatives, you can find where functions peak, valley, or change curvature. Set the derivative equal to zero, solve for x, and you've found critical points—potential maxima or minima. Check the second derivative to confirm: positive means concave up (minimum), negative means concave down (maximum).
This isn't abstract. Minimizing material cost? Maximizing strength-to-weight ratio? Finding the angle that maximizes range? All optimization problems, all solved by setting derivatives to zero. Engineers don't guess-and-check—they differentiate.
Why This Matters Later
Every time you design something—minimize weight, maximize efficiency, find the optimal angle—you're solving an optimization problem. Derivatives tell you where the answer is.
4. Integration
Integration is the reverse of differentiation. If derivatives break things down into rates, integrals build them back up into totals. You integrate velocity to get displacement. You integrate force over distance to get work. You integrate a function over an interval to get the area under the curve.
An indefinite integral (antiderivative) gives you a family of functions: ∫ 2x dx = x² + C. That C matters—without boundary conditions, you don't know which member of the family you need. A definite integral gives you a number: ∫₀² 2x dx = 4. That's the exact area under y = 2x from 0 to 2.
Reverse the power rule: increase exponent by 1, divide by the new exponent.
Integrate term by term. Don't forget +C for indefinite integrals.
Reverse chain rule: divide by the inner derivative (2 from 2x).
Why This Matters Later
Work, energy, mass, charge, probability—all computed by integration. Differential equations give you rates; integrals give you totals. You need both.
5. The Fundamental Theorem of Calculus
This theorem says: differentiation and integration are inverse operations. If you differentiate a function and then integrate the result, you get back where you started (plus a constant). If you integrate a derivative over an interval, you get the net change in the original function.
Why does this matter? Because it means you can evaluate integrals without summing infinitely many rectangles. You just find an antiderivative, plug in the endpoints, and subtract. That's it. This single insight makes calculus computationally feasible.
Plug in the upper limit, subtract the lower limit. That's the theorem at work.
Why This Matters Later
This isn't just a trick for calculating integrals—it's the bridge between derivatives and integrals, between rates and totals. Every application of calculus depends on this connection.
6. Applications of Integration
The most direct use of integration: finding areas. The area under f(x) from a to b is ∫ᵃᵇ f(x) dx. That's not approximate—it's exact, assuming you can find the antiderivative. Later you'll extend this to volumes (revolving curves around axes), arc lengths, and surface areas. For now, the key insight is that integration lets you calculate geometric quantities that have no simple formula.
Exact area. No approximation, no rectangles—just calculus.
Why This Matters Later
Volumes of revolution, centers of mass, fluid pressure, arc length—all extensions of the same idea. Integration takes continuous quantities and adds them up exactly. That's power.
Ready for the Next Level?
Once you understand derivatives and integrals, you're ready to explore multivariable calculus and vector analysis.
Continue to Level 3: Multivariable and Linear Algebra →