Calculus I (MATH 101) – Complete Exam Summary
1. Limits and Continuity
Key Concepts: - Limit definition: lim_{x->a} f(x) = L - One-sided limits: lim_{x->a^+} f(x), lim_{x->a^-} f(x) -
Limit laws: sum, difference, product, quotient - Special limits: infinity, indeterminate forms (0/0, ∞/∞) -
Continuity: f is continuous at x=a if lim_{x->a} f(x) = f(a)
Worked Examples: 1. lim_{x->2} (x^2 - 4)/(x - 2) = 4 2. lim_{x->0} (sin x)/x = 1
Mini Diagram: - Sketch showing a function approaching a limit from left and right
2. Derivatives
Key Formulas: - Power Rule: d/dx[x^n] = nx^{n-1} - Product Rule: (uv)' = u'v + uv' - Quotient Rule: (u/v)' = (u'v
- uv')/v^2 - Chain Rule: (f(g(x)))' = f'(g(x)) * g'(x) - Implicit Differentiation for F(x, y) = 0
Worked Examples: 1. Find dy/dx if x^2 + y^2 = 25 → dy/dx = -x/y 2. d/dx (x^2 * sin x) = 2xsin x + x^2cos x
3. Applications of Derivatives
Key Topics: - Critical points: solve f'(x) = 0 - Local maxima/minima - Optimization problems - Related rates:
dx/dt, dy/dt - Concavity & inflection points: f''(x) > 0 → concave up
Worked Examples: 1. Maximize area A = x(10-x) → x = 5 2. Related rates: If a balloon rises at 5 m/s, find rate
of shadow change at 10 m away
Mini Diagram: - Graph showing local max, min, and inflection points
4. Integration
Key Formulas: - Indefinite integral: ∫ x^n dx = x^{n+1}/(n+1) + C - Definite integral: ∫_a^b f(x) dx -
Fundamental Theorem: d/dx ∫_a^x f(t) dt = f(x) - Area under curve: A = ∫_a^b f(x) dx - Substitution: ∫ f(g(x))
g'(x) dx = ∫ f(u) du
Worked Examples: 1. ∫_0^2 3x^2 dx = 8 2. ∫ x*sin(x^2) dx → use u = x^2
Mini Diagram: - Area under y = x^2 from x=0 to x=2
5. Essential Formulas & Shortcuts
• Derivatives: trig, exponential, logarithmic functions
• Integrals: power, trig, exponential
1
1. Limits and Continuity
Key Concepts: - Limit definition: lim_{x->a} f(x) = L - One-sided limits: lim_{x->a^+} f(x), lim_{x->a^-} f(x) -
Limit laws: sum, difference, product, quotient - Special limits: infinity, indeterminate forms (0/0, ∞/∞) -
Continuity: f is continuous at x=a if lim_{x->a} f(x) = f(a)
Worked Examples: 1. lim_{x->2} (x^2 - 4)/(x - 2) = 4 2. lim_{x->0} (sin x)/x = 1
Mini Diagram: - Sketch showing a function approaching a limit from left and right
2. Derivatives
Key Formulas: - Power Rule: d/dx[x^n] = nx^{n-1} - Product Rule: (uv)' = u'v + uv' - Quotient Rule: (u/v)' = (u'v
- uv')/v^2 - Chain Rule: (f(g(x)))' = f'(g(x)) * g'(x) - Implicit Differentiation for F(x, y) = 0
Worked Examples: 1. Find dy/dx if x^2 + y^2 = 25 → dy/dx = -x/y 2. d/dx (x^2 * sin x) = 2xsin x + x^2cos x
3. Applications of Derivatives
Key Topics: - Critical points: solve f'(x) = 0 - Local maxima/minima - Optimization problems - Related rates:
dx/dt, dy/dt - Concavity & inflection points: f''(x) > 0 → concave up
Worked Examples: 1. Maximize area A = x(10-x) → x = 5 2. Related rates: If a balloon rises at 5 m/s, find rate
of shadow change at 10 m away
Mini Diagram: - Graph showing local max, min, and inflection points
4. Integration
Key Formulas: - Indefinite integral: ∫ x^n dx = x^{n+1}/(n+1) + C - Definite integral: ∫_a^b f(x) dx -
Fundamental Theorem: d/dx ∫_a^x f(t) dt = f(x) - Area under curve: A = ∫_a^b f(x) dx - Substitution: ∫ f(g(x))
g'(x) dx = ∫ f(u) du
Worked Examples: 1. ∫_0^2 3x^2 dx = 8 2. ∫ x*sin(x^2) dx → use u = x^2
Mini Diagram: - Area under y = x^2 from x=0 to x=2
5. Essential Formulas & Shortcuts
• Derivatives: trig, exponential, logarithmic functions
• Integrals: power, trig, exponential
1
