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Calculus 1 Complete Cheat Sheet
Calculus 1 Complete Cheat Sheet
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.…
Calculus 1 Complete Cheat Sheet
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<body>
<div class="navigation">
<a href="#module1" class="nav-link">Module 1: Limits</a>
<a href="#module2" class="nav-link">Module 2: Derivatives</a>
<a href="#module3" class="nav-link">Module 3: Advanced Derivatives</a>
<a href="#module4" class="nav-link">Module 4: Exp/Log/Inverse</a>
<a href="#module5" class="nav-link">Module 5: Applications</a>
<a href="#module6" class="nav-link">Module 6: Optimization</a>
<a href="#module7" class="nav-link">Module 7: Integration</a>
</div>
<div class="container">
<div class="header">
<h1>Calculus 1 Ultimate Cheat Sheet</h1>
<p>Functions • Limits • Derivatives • Applications • Integration</p>
</div>
<section id="module1" class="module">
<h2>Module 1: Functions, Limits & Continuity</h2>
<div class="topic">
<h3>Function Types</h3>
<div class="rule-grid">
<div class="rule-card">
<h4>Piecewise Functions</h4>
<div class="formula">
f(x) = { x² if x < 0
{ 2x if x ≥ 0
</div>
</div>
<div class="rule-card">
<h4>Composite Functions</h4>
<div class="formula">
(f ∘ g)(x) = f(g(x))
<br>Domain: x where g(x) is in domain of f
</div>
</div>
</div>
</div>
<div class="topic">
<h3>Limits</h3>
<div class="formula">
lim[x→a] f(x) = L means f(x) approaches L as x approaches a
</div>
<div class="limit-cases">
<div class="case-card">
<h5>Direct Substitution</h5>
<div class="formula">
If f is continuous at a:
<br>lim[x→a] f(x) = f(a)
</div>
</div>
<div class="case-card">
<h5>Factoring</h5>
<div class="formula">
lim[x→a] (x²-4)/(x-2)
<br>= lim[x→2] (x+2)(x-2)/(x-2)
<br>= lim[x→2] (x+2) = 4
</div>
</div>
<div class="case-card">
<h5>Rationalization</h5>
<div class="formula">
Multiply by conjugate:
<br>(√a + √b)(√a - √b) = a - b
</div>
</div>
<div class="case-card">
<h5>Squeeze Theorem</h5>
<div class="formula">
If g(x) ≤ f(x) ≤ h(x) and
<br>lim g(x) = lim h(x) = L
<br>then lim f(x) = L
</div>
</div>
</div>
</div>
<div class="topic">
<h3>Limits at Infinity</h3>
<div class="formula">
For rational functions f(x) = P(x)/Q(x):
<br>• If deg(P) < deg(Q): lim = 0
<br>• If deg(P) = deg(Q): lim = leading coefficient ratio
<br>• If deg(P) > deg(Q): lim = ±∞
</div>
</div>
<div class="topic">
<h3>Continuity</h3>
<div class="important">
f is continuous at x = a if:
<br>1. f(a) exists
<br>2. lim[x→a] f(x) exists
<br>3. lim[x→a] f(x) = f(a)
</div>
</div>
</section>
<section id="module2" class="module">
<h2>Module 2: Basic Derivatives</h2>
<div class="topic">
<h3>Definition of Derivative</h3>
<div class="formula">
f'(x) = lim[h→0] [f(x+h) - f(x)]/h
<br><br>
Geometric: slope of tangent line
<br>Physical: instantaneous rate of change
</div>
</div>
<div class="topic">
<h3>Basic Differentiation Rules</h3>
<div class="rule-grid">
<div class="rule-card">
<h4>Power Rule</h4>
<div class="formula">
d/dx [x^n] = nx^(n-1)
<br><br>
d/dx [c] = 0 (constant)
<br>d/dx [x] = 1
</div>
</div>
<div class="rule-card">
<h4>Sum/Difference</h4>
<div class="formula">
d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
<br><br>
d/dx [cf(x)] = cf'(x)
</div>
</div>
<div class="rule-card">
<h4>Product Rule</h4>
<div class="formula">
d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
<br><br>
"First times derivative of second
plus second times derivative of first"
</div>
</div>
<div class="rule-card">
<h4>Quotient Rule</h4>
<div class="formula">
d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²
<br><br>
"Low d-high minus high d-low,
all over low squared"
</div>
</div>
</div>
</div>
<div class="topic">
<h3>Chain Rule</h3>
<div class="formula">
d/dx [f(g(x))] = f'(g(x)) · g'(x)
<br><br>
"Derivative of outside times derivative of inside"
</div>
<div class="steps">
<strong>Chain Rule Steps:</strong>
<ol>
<li>Identify outer function f and inner function g</li>
<li>Find f'(x) and g'(x)</li>
<li>Evaluate f'(g(x))</li>
<li>Multiply: f'(g(x)) · g'(x)</li>
</ol>
</div>
</div>
</section>
<section id="module3" class="module">
<h2>Module 3: Advanced Derivatives</h2>
<div class="topic">
<h3>Implicit Differentiation</h3>
<div class="formula">
When y is defined implicitly by equation F(x,y) = 0:
<br>1. Differentiate both sides with respect to x
<br>2. Remember d/dx [y] = dy/dx = y'
<br>3. Solve for dy/dx
</div>
<div class="case-card">
<h5>Example: x² + y² = 25</h5>
<div class="formula">
d/dx [x² + y²] = d/dx [25]
<br>2x + 2y(dy/dx) = 0
<br>dy/dx = -x/y
</div>
</div>
</div>
<div class="topic">
<h3>Related Rates</h3>
<div class="steps">
<strong>Related Rates Strategy:</strong>
<ol>
<li>Draw diagram and label variables</li>
<li>Write equation relating variables</li>
<li>Differentiate implicitly with respect to time</li>
<li>Substitute known values</li>
<li>Solve for unknown rate</li>
</ol>
</div>
</div>
<div class="topic">
<h3>Linear Approximation</h3>
<div class="formula">
L(x) = f(a) + f'(a)(x - a)
<br><br>
Best linear approximation to f(x) near x = a
<br>Differential: dy = f'(x)dx
</div>
</div>
</section>
<section id="module4" class="module">
<h2>Module 4: Exponential, Logarithmic & Inverse Functions</h2>
<div class="topic">
<h3>Exponential & Logarithmic Derivatives</h3>
<div class="rule-grid">
<div class="rule-card">
<h4>Natural Functions</h4>
<div class="formula">
d/dx [e^x] = e^x
<br>d/dx [ln x] = 1/x
<br>d/dx [e^(f(x))] = e^(f(x)) · f'(x)
<br>d/dx [ln(f(x))] = f'(x)/f(x)
</div>
</div>
<div class="rule-card">
<h4>General Exponential</h4>
<div class="formula">
d/dx [a^x] = a^x ln a
<br>d/dx [log_a x] = 1/(x ln a)
<br>d/dx [a^(f(x))] = a^(f(x)) ln a · f'(x)
</div>
</div>
</div>
</div>
<div class="topic">
<h3>Inverse Function Derivatives</h3>
<div class="formula">
If y = f^(-1)(x), then dy/dx = 1/(dx/dy) = 1/f'(y)
</div>
<div class="rule-grid">
<div class="rule-card">
<h4>Inverse Trig</h4>
<div class="formula">
d/dx [arcsin x] = 1/√(1-x²)
<br>d/dx [arccos x] = -1/√(1-x²)
<br>d/dx [arctan x] = 1/(1+x²)
</div>
</div>
<div class="rule-card">
<h4>More Inverse Trig</h4>
<div class="formula">
d/dx [arccsc x] = -1/(|x|√(x²-1))
<br>d/dx [arcsec x] = 1/(|x|√(x²-1))
<br>d/dx [arccot x] = -1/(1+x²)
</div>
</div>
</div>
</div>
</section>
<section id="module5" class="module">
<h2>Module 5: Applications of Derivatives</h2>
<div class="topic">
<h3>L'Hôpital's Rule</h3>
<div class="important">
If lim f(x)/g(x) gives form 0/0 or ∞/∞:
<br>lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)
<br><br>
Also works for: 0·∞, ∞-∞, 1^∞, 0^0, ∞^0
</div>
</div>
<div class="topic">
<h3>Extreme Values</h3>
<div class="formula">
Critical Points: where f'(x) = 0 or f'(x) undefined
<br><br>
Absolute max/min on [a,b]:
<br>Check critical points AND endpoints
</div>
<div class="steps">
<strong>First Derivative Test:</strong>
<ol>
<li>Find critical points</li>
<li>Test sign of f'(x) on each side</li>
<li>f' changes + to -: local max</li>
<li>f' changes - to +: local min</li>
</ol>
</div>
</div>
<div class="topic">
<h3>Mean Value Theorem</h3>
<div class="formula">
If f continuous on [a,b] and differentiable on (a,b):
<br>∃ c ∈ (a,b) such that f'(c) = [f(b) - f(a)]/(b - a)
</div>
</div>
<div class="topic">
<h3>Shape of Graphs</h3>
<div class="rule-grid">
<div class="rule-card">
<h4>First Derivative</h4>
<div class="formula">
f'(x) > 0: f increasing
<br>f'(x) < 0: f decreasing
<br>f'(x) = 0: critical point
</div>
</div>
<div class="rule-card">
<h4>Second Derivative</h4>
<div class="formula">
f''(x) > 0: f concave up
<br>f''(x) < 0: f concave down
<br>f''(x) = 0: possible inflection point
</div>
</div>
</div>
</div>
</section>
<section id="module6" class="module">
<h2>Module 6: Optimization & Antiderivatives</h2>
<div class="topic">
<h3>Optimization Strategy</h3>
<div class="steps">
<ol>
<li>Define variables and draw diagram</li>
<li>Write objective function to optimize</li>
<li>Write constraint equation(s)</li>
<li>Use constraints to eliminate variables</li>
<li>Find critical points of objective function</li>
<li>Test critical points and endpoints</li>
</ol>
</div>
</div>
<div class="topic">
<h3>Antiderivatives</h3>
<div class="formula">
If F'(x) = f(x), then F(x) is an antiderivative of f(x)
<br>General antiderivative: F(x) + C
</div>
<div class="rule-grid">
<div class="rule-card">
<h4>Power Rule</h4>
<div class="formula">
∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
<br>∫ 1/x dx = ln|x| + C
</div>
</div>
<div class="rule-card">
<h4>Exponential</h4>
<div class="formula">
∫ e^x dx = e^x + C
<br>∫ a^x dx = a^x/ln a + C
</div>
</div>
<div class="rule-card">
<h4>Trigonometric</h4>
<div class="formula">
∫ sin x dx = -cos x + C
<br>∫ cos x dx = sin x + C
<br>∫ sec²x dx = tan x + C
</div>
</div>
</div>
</div>
</section>
<section id="module7" class="module">
<h2>Module 7: Integration</h2>
<div class="topic">
<h3>Riemann Sums & Definite Integrals</h3>
<div class="formula">
Area under curve ≈ Σ f(x_i) Δx
<br>As n → ∞: ∫[a to b] f(x) dx = lim[n→∞] Σ f(x_i) Δx
</div>
</div>
<div class="topic">
<h3>Fundamental Theorem of Calculus</h3>
<div class="important">
<strong>Part 1:</strong> If F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x)
<br><br>
<strong>Part 2:</strong> ∫[a to b] f(x) dx = F(b) - F(a)
<br>where F'(x) = f(x)
</div>
</div>
<div class="topic">
<h3>Properties of Definite Integrals</h3>
<div class="rule-grid">
<div class="rule-card">
<h4>Basic Properties</h4>
<div class="formula">
∫[a to a] f(x) dx = 0
<br>∫[a to b] f(x) dx = -∫[b to a] f(x) dx
<br>∫[a to b] c·f(x) dx = c∫[a to b] f(x) dx
</div>
</div>
<div class="rule-card">
<h4>Additivity</h4>
<div class="formula">
∫[a to b] [f(x) ± g(x)] dx = ∫[a to b] f(x) dx ± ∫[a to b] g(x) dx
<br><br>
∫[a to c] f(x) dx = ∫[a to b] f(x) dx + ∫[b to c] f(x) dx
</div>
</div>
</div>
</div>
</section>
<div class="module">
<h2 class="highlight">Quick Reference Formulas</h2>
<div class="rule-grid">
<div class="rule-card">
<h4>Common Derivatives</h4>
<div class="formula">
d/dx [sin x] = cos x
<br>d/dx [cos x] = -sin x
<br>d/dx [tan x] = sec²x
<br>d/dx [sec x] = sec x tan x
</div>
</div>
<div class="rule-card">
<h4>Common Limits</h4>
<div class="formula">
lim[x→0] sin x/x = 1
<br>lim[x→0] (1-cos x)/x = 0
<br>lim[x→∞] (1 + 1/x)^x = e
</div>
</div>
<div class="rule-card">
<h4>Common Antiderivatives</h4>
<div class="formula">
∫ 1/(1+x²) dx = arctan x + C
<br>∫ 1/√(1-x²) dx = arcsin x + C
<br>∫ sec x tan x dx = sec x + C
</div>
</div>
</div>
</div>
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