∫ Calculus Study Guide

📚 High School / AP Calculus AB & BC ⏱ Study time: ~60 min 🎯 Limits · Derivatives · Integration · Applications 📋 Common Core · AP Curriculum 🎯 SAT / AP Exam Essential

Before You Begin

Functions — calculus studies rates of change of functions
Limits concept — values approaching but not reaching a point
Algebra & trigonometry — used throughout differentiation and integration

Concept Path Functions › Trigonometry › Calculus › Differential Equations › Engineering Math

📋

Prerequisite Check

Before this unit, make sure you can work with: Algebra (functions & equations) · Fractions & rationals · Basic trigonometry concepts

📌 By the end of this unit you will be able to

Evaluate limits algebraically, including indeterminate forms
Apply the power, product, quotient, and chain rules to differentiate functions
Find critical points and determine local maxima/minima using the first derivative test
Evaluate indefinite and definite integrals and apply the Fundamental Theorem of Calculus

🌱 Why learn Calculus?

Calculus is the mathematics of change. Derivatives answer "how fast is this changing right now?" — your speedometer reads a derivative. Integrals answer "how much has accumulated?" — fuel consumed on a trip is an integral. Newton invented calculus to describe planetary orbits; today it powers AI, climate models, structural engineering, and medical imaging. Understanding calculus means you can model any continuous process in the real world.

⚡ 30-Second Summary

Limits: The value f(x) approaches as x → a. Left limit = right limit → limit exists
Derivatives: f′(x) = instantaneous rate of change (slope of tangent). Power rule: (xⁿ)′ = nxⁿ⁻¹
Chain / Product / Quotient rules for composite functions
Antiderivatives: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Fundamental Theorem: ∫[a→b] f(x)dx = F(b) − F(a) — connects differentiation and integration
Applications: Critical points (f′=0), area between curves, related rates

🏛 Origin of the Concept

Calculus was independently invented by Isaac Newton (1666, called "method of fluxions") and Gottfried Leibniz (1675, publishing first in 1684). Their bitter priority dispute consumed both men and split European mathematics for decades. Every symbol you use today — dy/dx, the integral sign ∫, and variable notation — comes from Leibniz's notation, while Newton's physics intuition gave us the laws of motion. Calculus unified both.

1. Limits

A limit describes the value a function approaches as the input approaches some value. Limits are the foundation of all of calculus.

\[ \lim_{x \to a} f(x) = L \]

Read: "the limit of f(x) as x approaches a equals L"

Limit Laws (a and b are real numbers):

Sum: \(\lim[f(x) + g(x)] = \lim f(x) + \lim g(x)\)
Product: \(\lim[f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x)\)
Quotient: \(\lim\dfrac{f(x)}{g(x)} = \dfrac{\lim f(x)}{\lim g(x)}\) (if denominator ≠ 0)

Indeterminate Forms — L'Hôpital's Rule

When direct substitution gives \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\):

\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \]

Important Limits

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1, \qquad \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \]

2. Derivatives

🧠 Intuition First (no symbols)

Derivative = instant snapshot. The speedometer in your car shows the derivative of your position — how fast you're changing position right now. Integration = accumulator. Adding up all those speed readings over time gives you total distance traveled. That's the Fundamental Theorem: differentiation and integration are opposites.

The derivative measures the instantaneous rate of change of a function. Geometrically, it is the slope of the tangent line.

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Differentiation Rules AP Exam

Constant: \(\dfrac{d}{dx}[c] = 0\)
Power Rule: \(\dfrac{d}{dx}[x^n] = nx^{n-1}\)
Constant Multiple: \(\dfrac{d}{dx}[cf] = cf'\)
Sum/Difference: \(\dfrac{d}{dx}[f \pm g] = f' \pm g'\)
Product Rule: \(\dfrac{d}{dx}[fg] = f'g + fg'\)
Quotient Rule: \(\dfrac{d}{dx}\!\left[\dfrac{f}{g}\right] = \dfrac{f'g - fg'}{g^2}\)
Chain Rule: \(\dfrac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

Derivatives of Common Functions

\(\dfrac{d}{dx}[\sin x] = \cos x\), \(\dfrac{d}{dx}[\cos x] = -\sin x\), \(\dfrac{d}{dx}[\tan x] = \sec^2 x\)
\(\dfrac{d}{dx}[e^x] = e^x\), \(\dfrac{d}{dx}[\ln x] = \dfrac{1}{x}\)
\(\dfrac{d}{dx}[a^x] = a^x \ln a\)

Example — Chain Rule: Find \(f'(x)\) for \(f(x) = (3x^2 + 1)^5\)

\(f'(x) = 5(3x^2+1)^4 \cdot 6x = 30x(3x^2+1)^4\)

Applications of Derivatives

Critical points: where \(f'(x) = 0\) or \(f'(x)\) is undefined
Increasing: where \(f'(x) > 0\)
Decreasing: where \(f'(x) < 0\)
Concave up: where \(f''(x) > 0\)
Concave down: where \(f''(x) < 0\)
Inflection point: where concavity changes (\(f''(x) = 0\) and sign changes)

3. Integration

Integration is the reverse of differentiation. The indefinite integral (antiderivative) of \(f(x)\) is \(F(x)\) where \(F'(x) = f(x)\).

Basic Integration Rules

Power Rule: \(\displaystyle\int x^n\, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)\)
Constant: \(\displaystyle\int c\, dx = cx + C\)
\(\displaystyle\int e^x\, dx = e^x + C\), \(\displaystyle\int \frac{1}{x}\, dx = \ln|x| + C\)
\(\displaystyle\int \sin x\, dx = -\cos x + C\), \(\displaystyle\int \cos x\, dx = \sin x + C\)

The Fundamental Theorem of Calculus AP Exam

\[ \int_a^b f(x)\, dx = F(b) - F(a) \]

where F is any antiderivative of f. The definite integral gives the net area under the curve from x = a to x = b.

Example: Evaluate \(\displaystyle\int_1^3 (2x + 1)\, dx\)

\(F(x) = x^2 + x\)
\(F(3) - F(1) = (9+3) - (1+1) = 12 - 2 = \mathbf{10}\)

U-Substitution

When the integrand contains a composite function, let \(u = g(x)\), so \(du = g'(x)\, dx\).

Example: \(\displaystyle\int 2x(x^2+1)^3\, dx\)

Let \(u = x^2 + 1\), \(du = 2x\, dx\)
\(= \displaystyle\int u^3\, du = \dfrac{u^4}{4} + C = \dfrac{(x^2+1)^4}{4} + C\)

Power Rule memory tip"bring the exponent down, then subtract 1" — \((x^5)' = 5x^4\), \((x^3)' = 3x^2\). The coefficient multiplies the front.

4. Practice Problems

⭐ Basic Find \(\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\).
Show answer

Factor: \(\dfrac{(x+2)(x-2)}{x-2} = x+2 \to \mathbf{4}\)
⭐ Basic Differentiate \(f(x) = 4x^3 - 3x^2 + 7x - 2\).
Show answer

\(f'(x) = 12x^2 - 6x + 7\)
⭐⭐ Standard Differentiate \(g(x) = x^2 \sin x\) (use product rule).
Show answer

\(g'(x) = 2x\sin x + x^2\cos x\)
⭐⭐ Standard Evaluate \(\displaystyle\int (6x^2 - 4x + 3)\, dx\).
Show answer

\(2x^3 - 2x^2 + 3x + C\)
⭐⭐ Standard Evaluate \(\displaystyle\int_0^2 (3x^2 + 1)\, dx\).
Show answer

\([x^3 + x]_0^2 = (8+2) - 0 = \mathbf{10}\)
⭐⭐⭐ Challenge Find all critical points of \(f(x) = x^3 - 3x + 2\) and classify each.
Show answer

\(f'(x) = 3x^2-3 = 0 \Rightarrow x=\pm1\) · \(x=-1\): local max (f' + → −) · \(x=1\): local min (f' − → +)

🔓 Master This to Unlock

Trig Derivatives — d/dx(sin x) = cos x Physics Mechanics — velocity and acceleration as derivatives

Calculus is the language of every STEM field. Now that you understand limits and derivatives, trigonometric differentiation and physics equations of motion all speak the same language you just learned.

Calculus complete!

You've mastered differentiation and integration. Head to Trigonometry next to see how sin and cos extend every formula you just learned.

Start Trigonometry →

Math series

11 / 12

NEXT UNIT

Trigonometry

Unit circle · sin · cos · tan, addition formulas and graph transformations

Advanced ⏱ ~30 min

Pre-Test Checklist