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Mathematics → Calculus

Calculus

Calculus is the mathematics of continuous change. Developed independently by Newton and Leibniz in the 17th century, it provides the language for physics, engineering, economics, and every quantitative science.

Limits

The limit is the foundation of calculus. It captures what value a function approaches as its input approaches some point.

Definition — Limit (informal)

We write lim⁡x→af(x)=L\lim_{x \to a} f(x) = L if f(x)f(x) can be made arbitrarily close to LL by taking xx sufficiently close to aa (but not equal to aa).
lim⁡x→2(3x+1)=7lim⁡x→0sin⁡xx=1lim⁡x→∞1x=0\lim_{x \to 2}(3x + 1) = 7 \qquad \lim_{x \to 0}\frac{\sin x}{x} = 1 \qquad \lim_{x \to \infty}\frac{1}{x} = 0

The first and third limits follow by direct substitution. The middle one — sin⁡(x)/x\sin(x)/x as x→0x \to 0 — is a different story entirely. Direct substitution gives 0/00/0, L'Hôpital's rule is circular here, and the proof requires a geometric argument about areas on a unit circle. See the full proof →

Limit laws

If lim⁡x→af(x)=L\lim_{x\to a} f(x) = L and lim⁡x→ag(x)=M\lim_{x\to a} g(x) = M, then:

lim⁡x→a[f(x)+g(x)]=L+Mlim⁡x→a[f(x)⋅g(x)]=LMlim⁡x→af(x)g(x)=LM  (M≠0)\lim_{x\to a}[f(x) + g(x)] = L + M \qquad \lim_{x\to a}[f(x)\cdot g(x)] = LM \qquad \lim_{x\to a}\frac{f(x)}{g(x)} = \frac{L}{M}\;(M\neq 0)

Continuity

Definition — Continuity

A function ff is continuous at aa if: (1) f(a)f(a) is defined, (2) lim⁡x→af(x)\lim_{x\to a} f(x) exists, and (3) lim⁡x→af(x)=f(a)\lim_{x\to a} f(x) = f(a).

Intuitively: the graph has no holes, jumps, or vertical asymptotes at aa. Polynomials, exponentials, and trigonometric functions are continuous everywhere on their domains.

The Intermediate Value Theorem states that if ff is continuous on [a,b][a,b] and kk is between f(a)f(a) and f(b)f(b), then there exists c∈(a,b)c \in (a,b) with f(c)=kf(c) = k. This guarantees roots exist — it's why every polynomial of odd degree has at least one real root.

The Derivative

Definition — Derivative

The derivative of ff at aa is the limit of the difference quotient:
f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
If this limit exists, ff is differentiable at aa.

Geometrically, f′(a)f'(a) is the slope of the tangent line to the graph of ff at the point (a,f(a))(a, f(a)). Physically, if f(t)f(t) is position, then f′(t)f'(t) is velocity and f′′(t)f''(t) is acceleration.

The equation of the tangent line to ff at x=ax = a is: y=f(a)+f′(a)(x−a)y = f(a) + f'(a)(x - a)

Differentiation Rules

Rather than computing limits every time, these rules let us differentiate any elementary function:

Power Rule

ddxxn=nxn−1\dfrac{d}{dx}x^n = nx^{n-1}

Sum Rule

(f+g)′=f′+g′(f+g)\prime = f\prime + g\prime

Product Rule

(fg)′=f′g+fg′(fg)\prime = f\prime g + fg\prime

Quotient Rule

(fg)′=f′g−fg′g2\left(\dfrac{f}{g}\right)\prime = \dfrac{f\prime g - fg\prime}{g^2}

Chain Rule

(f∘g)′(x)=f′(g(x))⋅g′(x)(f \circ g)\prime(x) = f\prime(g(x))\cdot g\prime(x)

Exponential

ddxex=ex\dfrac{d}{dx}e^x = e^x

Common derivatives

ddxsin⁡x=cos⁡xddxcos⁡x=−sin⁡xddxln⁡x=1xddxax=axln⁡a\frac{d}{dx}\sin x = \cos x \qquad \frac{d}{dx}\cos x = -\sin x \qquad \frac{d}{dx}\ln x = \frac{1}{x} \qquad \frac{d}{dx}a^x = a^x \ln a

The Integral

Definition — Definite Integral

The definite integral of ff from aa to bb is the signed area between the graph and the xx-axis:
∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗) Δx\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\,\Delta x
where Δx=(b−a)/n\Delta x = (b-a)/n and xi∗x_i^* is any sample point in the ii-th subinterval.

The indefinite integral (antiderivative) ∫f(x) dx\int f(x)\,dx is a family of functions F(x)+CF(x) + C such that F′(x)=f(x)F'(x) = f(x). The constant CC reflects the fact that derivatives of constants vanish.

Common antiderivatives

∫xn dx=xn+1n+1+C(n≠−1)∫ex dx=ex+C∫1x dx=ln⁡∣x∣+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n\neq -1) \qquad \int e^x\,dx = e^x + C \qquad \int \frac{1}{x}\,dx = \ln|x| + C
∫sin⁡x dx=−cos⁡x+C∫cos⁡x dx=sin⁡x+C\int \sin x\,dx = -\cos x + C \qquad \int \cos x\,dx = \sin x + C

Fundamental Theorem of Calculus

The Fundamental Theorem unifies differentiation and integration — revealing that they are inverse operations.

Part I — Differentiation of an Integral

If ff is continuous on [a,b][a,b] and g(x)=∫axf(t) dtg(x) = \int_a^x f(t)\,dt, then gg is differentiable and:
g′(x)=f(x)g'(x) = f(x)

Part II — Evaluation Theorem

If FF is any antiderivative of ff on [a,b][a,b], then:
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

Part II is the engine of calculus: to compute the area under a curve, find an antiderivative and evaluate at the endpoints. What seemed to require an infinite sum reduces to two function evaluations.

Example: ∫0πsin⁡x dx=[−cos⁡x]0π=−cos⁡π+cos⁡0=1+1=2\int_0^{\pi} \sin x\,dx = [-\cos x]_0^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2

Next: Linear Algebra →

On this page

Limits

Continuity

The Derivative

Differentiation Rules

The Integral

Fundamental Theorem