Calculus I

The limit describes the value a function approaches as its input approaches some point, without necessarily ever reaching it. Limits are the rigorous foundation underneath every other idea in calculus: the derivative is defined as a limit, and the integral is defined as a limit of sums.

A function is continuous at a point if you can draw its graph through that point without lifting your pen — no jumps, holes, or asymptotes. Formally, continuity means the limit of the function as x approaches a point equals the function's actual value at that point.

The derivative measures how a function's output changes in response to an infinitesimally small change in its input — its instantaneous rate of change. Geometrically, it's the slope of the line tangent to the function's graph at a point. The derivative is the central object of differential calculus and underlies physics (velocity, acceleration), economics (marginal cost), and machine learning (gradient descent).

The chain rule tells you how to differentiate a composition of functions — a function inside another function. It's the single most-used differentiation rule in practice, because most real functions you'll differentiate are compositions (sin(x²), e^(3x), √(x+1), etc.).

Curve sketching uses the first and second derivatives of a function to determine its overall shape — where it increases or decreases, where it's concave up or down, and where it has local extrema or inflection points — without needing to plot many individual points. It systematizes graphing into a checklist derived directly from calculus.

The natural exponential function e^x has the remarkable property of being its own derivative. This single fact, combined with the chain rule and change-of-base formulas, generates the derivatives of every exponential and logarithmic function.

The six trigonometric functions each have derivatives that can be derived from the limit definition using two key limits: lim(h→0) sin(h)/h = 1 and lim(h→0) (cos(h)-1)/h = 0. Once sin and cos are established, the derivatives of tan, cot, sec, and csc follow from the quotient rule.

The Extreme Value Theorem (EVT) guarantees that a continuous function on a closed, bounded interval [a,b] attains both a maximum and a minimum value somewhere on that interval. It's an existence theorem: it doesn't say how to find the extrema, only that they're guaranteed to exist, which is what makes optimization on closed intervals well-posed.

Implicit differentiation is a technique for finding dy/dx when y is defined implicitly by an equation relating x and y (e.g. x² + y² = 25), rather than explicitly as y = f(x). The method differentiates both sides of the equation with respect to x, treating y as a function of x and applying the chain rule whenever y appears, then solves algebraically for dy/dx.

L'Hôpital's Rule provides a way to evaluate limits that produce indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator separately (not using the quotient rule) and taking the limit of the resulting ratio. It converts a difficult limit problem into a (hopefully simpler) derivative problem.

The Mean Value Theorem (MVT) is one of the most important results in differential calculus: it guarantees that for a function continuous on [a,b] and differentiable on (a,b), there exists at least one point c in (a,b) where the instantaneous rate of change (the derivative) equals the average rate of change over the whole interval. Geometrically, some tangent line inside the interval is parallel to the secant line connecting the endpoints. The MVT is the theoretical bridge that connects local information about derivatives to global information about a function's behavior — it underlies the proofs that a positive derivative implies an increasing function, that antiderivatives differing by a constant, and much of the rest of calculus's logical structure.

Optimization problems ask for the maximum or minimum value of some quantity subject to constraints — the classic 'applied max/min' problems of calculus. The general method: express the quantity to be optimized as a function of a single variable (using constraints to eliminate extra variables), then use derivatives to find and classify its critical points.

The product rule gives the derivative of a product of two functions. It is not simply the product of the derivatives — a common error — but rather a sum of two terms, each holding one factor fixed while differentiating the other.

The quotient rule gives the derivative of a ratio of two functions. It can be derived from the product rule combined with the chain rule, and it's essential whenever a function is written as f(x)/g(x) rather than a product.

Related rates problems involve finding how fast one quantity changes by relating it to the rate of change of another quantity, using an equation that connects them. The technique differentiates that connecting equation with respect to time using implicit differentiation, then substitutes known values to solve for the unknown rate.

Rolle's Theorem is the special case of the Mean Value Theorem where the function takes equal values at both endpoints. It guarantees that between two points where a function has equal height, there must be at least one point where the tangent line is horizontal — a critical point.