📖 What is Calculus?
Calculus is the mathematical study of continuous change. It has two major branches: differential calculus (concerning rates of change and slopes) and integral calculus (concerning accumulation and areas). Together, they're connected by the Fundamental Theorem of Calculus.
Why Learn Calculus?
- Physics & Engineering: Describes motion, forces, electricity, and waves
- Economics: Optimization, marginal analysis, growth modeling
- Medicine: Drug dosage models, spread of diseases
- Computer Science: Machine learning, graphics, optimization algorithms
- Critical Thinking: Develops rigorous logical reasoning skills
🎯 Key Concepts
1. Limits
A limit describes what value a function approaches as the input approaches some value. Limits are the foundation of calculus.
"As $x$ approaches $a$, $f(x)$ approaches $L$"
Limit Laws
- 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: $\lim\frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$ (if denominator ≠ 0)
- Constant: $\lim[c \cdot f(x)] = c \cdot \lim f(x)$
- Power: $\lim[f(x)]^n = [\lim f(x)]^n$
Special Limits to Memorize
- $\lim_{x \to 0} \frac{\sin x}{x} = 1$
- $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$
- $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$
- $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
2. Derivatives
The derivative measures the instantaneous rate of change of a function. Geometrically, it's the slope of the tangent line at a point.
What Derivatives Tell Us
- $f'(x) > 0$: Function is increasing
- $f'(x) < 0$: Function is decreasing
- $f'(x) = 0$: Potential maximum, minimum, or inflection point
- $f''(x) > 0$: Concave up (holds water)
- $f''(x) < 0$: Concave down (spills water)
3. Differentiation Rules
Basic Rules
- Constant: $\frac{d}{dx}[c] = 0$
- Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$
- Sum Rule: $\frac{d}{dx}[f + g] = f' + g'$
- Constant Multiple: $\frac{d}{dx}[cf] = cf'$
- Product Rule: $\frac{d}{dx}[fg] = f'g + fg'$
- Quotient Rule: $\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$
- Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
4. Integrals
Integration is the reverse of differentiation. It calculates accumulated quantities like area under curves, total distance, or accumulated change.
Where $F'(x) = f(x)$ and $C$ is constant of integration
Calculates the net area from $x = a$ to $x = b$
Differentiation and integration are inverse operations
5. Applications
Common Applications of Derivatives
- Velocity & Acceleration: $v = \frac{ds}{dt}$, $a = \frac{dv}{dt}$
- Optimization: Find max/min by setting $f'(x) = 0$
- Related Rates: Connect changing quantities using chain rule
- Linear Approximation: $f(x) \approx f(a) + f'(a)(x-a)$
Common Applications of Integrals
- Area Under Curve: $A = \int_a^b f(x)\,dx$
- Area Between Curves: $A = \int_a^b [f(x) - g(x)]\,dx$
- Average Value: $\bar{f} = \frac{1}{b-a}\int_a^b f(x)\,dx$
- Volume (Disk Method): $V = \pi\int_a^b [r(x)]^2\,dx$
📝 Essential Formulas
Derivative Formulas
| Function $f(x)$ | Derivative $f'(x)$ |
|---|---|
| $x^n$ | $nx^{n-1}$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\log_a x$ | $\frac{1}{x \ln a}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
| $\cot x$ | $-\csc^2 x$ |
Integral Formulas
| Function | Integral |
|---|---|
| $x^n$ $(n \neq -1)$ | $\frac{x^{n+1}}{n+1} + C$ |
| $\frac{1}{x}$ | $\ln|x| + C$ |
| $e^x$ | $e^x + C$ |
| $a^x$ | $\frac{a^x}{\ln a} + C$ |
| $\sin x$ | $-\cos x + C$ |
| $\cos x$ | $\sin x + C$ |
| $\sec^2 x$ | $\tan x + C$ |
| $\csc^2 x$ | $-\cot x + C$ |
| $\sec x \tan x$ | $\sec x + C$ |
| $\frac{1}{\sqrt{1-x^2}}$ | $\arcsin x + C$ |
| $\frac{1}{1+x^2}$ | $\arctan x + C$ |
Integration Techniques
Let $u = g(x)$, then $du = g'(x)\,dx$
LIATE: Logarithmic, Inverse trig, Algebraic, Trig, Exponential (priority for $u$)
✏️ Practice Problems
$\frac{d}{dx}[3x^4] = 3 \cdot 4x^3 = 12x^3$
Answer: $12x^3 - 4x + 5$
$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
Answer: $15(3x+1)^4$
$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$
Answer: $2x^3 + 2x^2 - 3x + C$
$F(x) = \frac{x^3}{3} + x$
Answer: $\frac{14}{3}$ or approximately $4.67$
Answer: $\frac{(x^2+1)^5}{5} + C$
💡 Tips & Tricks
"Bring down the power, reduce by one": $x^n \to nx^{n-1}$. For integration, do the opposite: "Increase power by one, divide by new power."
"Outer, inner, times derivative of inner." For $(stuff)^n$, bring down $n$, keep stuff to $(n-1)$ power, multiply by derivative of stuff.
Look for a function and its derivative appearing together. If you see $f(g(x)) \cdot g'(x)$, let $u = g(x)$.
Choose $u$ in this priority order: Log, Inverse trig, Algebraic, Trig, Exponential. The other part becomes $dv$.
For indefinite integrals, differentiate your answer—you should get back the original function. This catches most errors!
1) Draw picture & identify variables
2) Write function to optimize
3) Find domain/constraints
4) Take derivative, set = 0
5) Test critical points & endpoints
⚠️ Common Mistakes to Avoid
For indefinite integrals, ALWAYS add "+ C"
$\int x^2\,dx = \frac{x^3}{3} + C$ (not just $\frac{x^3}{3}$)
Wrong: $\frac{d}{dx}[\sin(3x)] = \cos(3x)$
Right: $\frac{d}{dx}[\sin(3x)] = 3\cos(3x)$
Don't forget to multiply by derivative of inner function!
Wrong: $\frac{d}{dx}[x \cdot \sin x] = 1 \cdot \cos x$
Right: $\frac{d}{dx}[x \cdot \sin x] = \sin x + x\cos x$
Product rule: $f'g + fg'$, not $f' \cdot g'$
$\frac{d}{dx}[\sin x] = \cos x$ (positive)
$\frac{d}{dx}[\cos x] = -\sin x$ (negative!)
The "co" functions get a negative sign.
Area below x-axis gives negative values. If you want total area (not net area), use:
$\int_a^b |f(x)|\,dx$
Master Calculus with Smart Planning
Use Centauri to schedule your calculus study sessions, create spaced repetition for formulas, and track your progress.
Get Early Access