Module 1: Limits Module 2: Derivatives Module 3: Advanced Derivatives Module 4: Exp/Log/Inverse Module 5: Applications Module 6: Optimization Module 7: Integration

Calculus 1 Ultimate Cheat Sheet

Functions • Limits • Derivatives • Applications • Integration

Module 1: Functions, Limits & Continuity

Function Types

Piecewise Functions

f(x) = { x² if x < 0 { 2x if x ≥ 0

Composite Functions

(f ∘ g)(x) = f(g(x))
Domain: x where g(x) is in domain of f

Limits

lim[x→a] f(x) = L means f(x) approaches L as x approaches a
Direct Substitution
If f is continuous at a:
lim[x→a] f(x) = f(a)
Factoring
lim[x→a] (x²-4)/(x-2)
= lim[x→2] (x+2)(x-2)/(x-2)
= lim[x→2] (x+2) = 4
Rationalization
Multiply by conjugate:
(√a + √b)(√a - √b) = a - b
Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) and
lim g(x) = lim h(x) = L
then lim f(x) = L

Limits at Infinity

For rational functions f(x) = P(x)/Q(x):
• If deg(P) < deg(Q): lim = 0
• If deg(P) = deg(Q): lim = leading coefficient ratio
• If deg(P) > deg(Q): lim = ±∞

Continuity

f is continuous at x = a if:
1. f(a) exists
2. lim[x→a] f(x) exists
3. lim[x→a] f(x) = f(a)

Module 2: Basic Derivatives

Definition of Derivative

f'(x) = lim[h→0] [f(x+h) - f(x)]/h

Geometric: slope of tangent line
Physical: instantaneous rate of change

Basic Differentiation Rules

Power Rule

d/dx [x^n] = nx^(n-1)

d/dx [c] = 0 (constant)
d/dx [x] = 1

Sum/Difference

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

d/dx [cf(x)] = cf'(x)

Product Rule

d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

"First times derivative of second plus second times derivative of first"

Quotient Rule

d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²

"Low d-high minus high d-low, all over low squared"

Chain Rule

d/dx [f(g(x))] = f'(g(x)) · g'(x)

"Derivative of outside times derivative of inside"
Chain Rule Steps:
  1. Identify outer function f and inner function g
  2. Find f'(x) and g'(x)
  3. Evaluate f'(g(x))
  4. Multiply: f'(g(x)) · g'(x)

Module 3: Advanced Derivatives

Implicit Differentiation

When y is defined implicitly by equation F(x,y) = 0:
1. Differentiate both sides with respect to x
2. Remember d/dx [y] = dy/dx = y'
3. Solve for dy/dx
Example: x² + y² = 25
d/dx [x² + y²] = d/dx [25]
2x + 2y(dy/dx) = 0
dy/dx = -x/y

Related Rates

Related Rates Strategy:
  1. Draw diagram and label variables
  2. Write equation relating variables
  3. Differentiate implicitly with respect to time
  4. Substitute known values
  5. Solve for unknown rate

Linear Approximation

L(x) = f(a) + f'(a)(x - a)

Best linear approximation to f(x) near x = a
Differential: dy = f'(x)dx

Module 4: Exponential, Logarithmic & Inverse Functions

Exponential & Logarithmic Derivatives

Natural Functions

d/dx [e^x] = e^x
d/dx [ln x] = 1/x
d/dx [e^(f(x))] = e^(f(x)) · f'(x)
d/dx [ln(f(x))] = f'(x)/f(x)

General Exponential

d/dx [a^x] = a^x ln a
d/dx [log_a x] = 1/(x ln a)
d/dx [a^(f(x))] = a^(f(x)) ln a · f'(x)

Inverse Function Derivatives

If y = f^(-1)(x), then dy/dx = 1/(dx/dy) = 1/f'(y)

Inverse Trig

d/dx [arcsin x] = 1/√(1-x²)
d/dx [arccos x] = -1/√(1-x²)
d/dx [arctan x] = 1/(1+x²)

More Inverse Trig

d/dx [arccsc x] = -1/(|x|√(x²-1))
d/dx [arcsec x] = 1/(|x|√(x²-1))
d/dx [arccot x] = -1/(1+x²)

Module 5: Applications of Derivatives

L'Hôpital's Rule

If lim f(x)/g(x) gives form 0/0 or ∞/∞:
lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)

Also works for: 0·∞, ∞-∞, 1^∞, 0^0, ∞^0

Extreme Values

Critical Points: where f'(x) = 0 or f'(x) undefined

Absolute max/min on [a,b]:
Check critical points AND endpoints
First Derivative Test:
  1. Find critical points
  2. Test sign of f'(x) on each side
  3. f' changes + to -: local max
  4. f' changes - to +: local min

Mean Value Theorem

If f continuous on [a,b] and differentiable on (a,b):
∃ c ∈ (a,b) such that f'(c) = [f(b) - f(a)]/(b - a)

Shape of Graphs

First Derivative

f'(x) > 0: f increasing
f'(x) < 0: f decreasing
f'(x) = 0: critical point

Second Derivative

f''(x) > 0: f concave up
f''(x) < 0: f concave down
f''(x) = 0: possible inflection point

Module 6: Optimization & Antiderivatives

Optimization Strategy

  1. Define variables and draw diagram
  2. Write objective function to optimize
  3. Write constraint equation(s)
  4. Use constraints to eliminate variables
  5. Find critical points of objective function
  6. Test critical points and endpoints

Antiderivatives

If F'(x) = f(x), then F(x) is an antiderivative of f(x)
General antiderivative: F(x) + C

Power Rule

∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
∫ 1/x dx = ln|x| + C

Exponential

∫ e^x dx = e^x + C
∫ a^x dx = a^x/ln a + C

Trigonometric

∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C

Module 7: Integration

Riemann Sums & Definite Integrals

Area under curve ≈ Σ f(x_i) Δx
As n → ∞: ∫[a to b] f(x) dx = lim[n→∞] Σ f(x_i) Δx

Fundamental Theorem of Calculus

Part 1: If F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x)

Part 2: ∫[a to b] f(x) dx = F(b) - F(a)
where F'(x) = f(x)

Properties of Definite Integrals

Basic Properties

∫[a to a] f(x) dx = 0
∫[a to b] f(x) dx = -∫[b to a] f(x) dx
∫[a to b] c·f(x) dx = c∫[a to b] f(x) dx

Additivity

∫[a to b] [f(x) ± g(x)] dx = ∫[a to b] f(x) dx ± ∫[a to b] g(x) dx

∫[a to c] f(x) dx = ∫[a to b] f(x) dx + ∫[b to c] f(x) dx

Quick Reference Formulas

Common Derivatives

d/dx [sin x] = cos x
d/dx [cos x] = -sin x
d/dx [tan x] = sec²x
d/dx [sec x] = sec x tan x

Common Limits

lim[x→0] sin x/x = 1
lim[x→0] (1-cos x)/x = 0
lim[x→∞] (1 + 1/x)^x = e

Common Antiderivatives

∫ 1/(1+x²) dx = arctan x + C
∫ 1/√(1-x²) dx = arcsin x + C
∫ sec x tan x dx = sec x + C