Understanding Calculus Cheat Sheet

Key Definitions

• Limit: The value that a function approaches as the input approaches a point. ✅

• Derivative: The rate of change of a function with respect to a variable. ✅

• Integral: The accumulation of quantities, representing area under a curve. ✅

• Function: A relation where each input has exactly one output. ✅

Important Concepts

Limits

• Evaluating limits at points and approaching infinity.

• One-sided limits (left-hand and right-hand limits).

Derivatives

• Notation: f'(x), dy/dx, D[f(x)].

• Rules: Power, Product, Quotient, and Chain Rule.

Integrals

• Definite Integral: ∫[a to b] f(x) dx gives the area under the curve from a to b. ✅

• Indefinite Integral: ∫ f(x) dx = F(x) + C, where F' = f. ✅

Key Formulas

• Derivative:

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

• Power Rule:

d/dx [x^n] = n*x^(n-1) ✅

• Integral of Power:

∫ x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1) ✅

• Fundamental Theorem of Calculus:

∫[a to b] f'(x) dx = f(b) - f(a) ✅

Common Mistakes to Avoid

Common Mistake: Forgetting to apply the constant multiple rule in differentiation. ⚠️

Common Mistake: Confusing definite and indefinite integrals. ⚠️

Memory Tricks

Tip: "LIMES" — Limits, Integrals, Mean Value Theorem, Evaluating Derivatives, Sequences.

Key Diagrams

Limit: 
      |
      |       *
      |      *
      |     *
      |    *
      |___*____________
           x

Derivative:
      |
      |           *
      |          *
      |         *
      |        *
      |_______*________
           x

Integral:
      |     *
      |    *
      |   
      | *
      |____________
           x