Calculus Cheat Sheet

Calculus Cheat Sheet

Calculus can feel overwhelming, but with the right tools and key formulas, you can navigate through it with confidence. Here's a concise cheat sheet to guide you through the essentials of calculus.

1. Limits

Definition

The limit of a function describes its behavior as it approaches a specific point or infinity.

lim (x → c) f(x) = L

Common Rules:

  • Direct Substitution: If f(c) is defined, lim (x → c) f(x) = f(c).
  • Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) and lim (x → c) g(x) = lim (x → c) h(x) = L, then lim (x → c) f(x) = L.

Important Limits:

lim (x → 0) (sin(x)/x) = 1
lim (x → ∞) (1/x) = 0

2. Derivatives

Definition

The derivative measures the rate of change or slope of a function.

f'(x) = lim (h → 0) [(f(x+h) - f(x)) / h]

Basic Rules:

  • Power Rule: d/dx [x^n] = n * x^(n-1)
  • Product Rule: d/dx [u * v] = u'v + uv'
  • Quotient Rule: d/dx [u/v] = (u'v - uv') / v^2
  • Chain Rule: d/dx f(g(x)) = f'(g(x)) * g'(x)

Common Derivatives:

d/dx [sin(x)] = cos(x)
d/dx [cos(x)] = -sin(x)
d/dx [e^x] = e^x

3. Integrals

Definition

The integral represents the accumulation of quantities or the area under a curve.

  • Indefinite Integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x)
  • Definite Integral: ∫[a, b] f(x) dx gives the area from x = a to x = b.

Basic Rules:

  • ∫ x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)
  • ∫ e^x dx = e^x + C
  • ∫ sin(x) dx = -cos(x) + C
  • ∫ cos(x) dx = sin(x) + C

Fundamental Theorem of Calculus:

  1. F'(x) = f(x)
  2. ∫[a, b] f(x) dx = F(b) - F(a)

4. Common Techniques

Substitution Rule:

∫ f(g(x)) g'(x) dx = ∫ f(u) du

Integration by Parts:

∫ u * v' dx = uv - ∫ u' * v dx

Partial Fraction Decomposition:

Break rational functions into simpler fractions before integrating.

5. Key Formulas and Identities

Trigonometric Identities:

  • sin^2(x) + cos^2(x) = 1
  • 1 + tan^2(x) = sec^2(x)
  • 1 + cot^2(x) = csc^2(x)

Exponential and Logarithmic Rules:

  • d/dx [ln(x)] = 1/x
  • d/dx [a^x] = a^x * ln(a)
  • ∫ ln(x) dx = x * ln(x) - x + C

6. Applications of Calculus

Optimization:

  • Find f'(x).
  • Set f'(x) = 0 to find critical points.
  • Use the second derivative test to determine maxima or minima.
  • Differentiate both sides of an equation with respect to time t.

Area Between Curves:

∫[a, b] [f(x) - g(x)] dx

7. Practice Problems

Limits:

  1. lim (x → 2) [(x^2 - 4)/(x - 2)]
  2. lim (x → ∞) [(5x + 3)/(2x - 7)]

Derivatives:

  1. d/dx [3x^4 - 2x^2 + 7]
  2. d/dx [e^(2x) * sin(x)]

Integrals:

  1. ∫ (3x^2 - 4x + 5) dx
  2. ∫[0, π] sin(x) dx

With these core concepts and formulas, you'll be ready to tackle most calculus problems with ease. Keep practicing and remember: calculus is all about understanding change!