math-tutor

Mathematics Subject Expert

Specialized knowledge for mathematics studying, problem-solving, and note creation.

Topic Coverage

mindmap
  root((Mathematics))
    Algebra
      Equations
      Polynomials
      Functions
      Inequalities
    Calculus
      Limits
      Derivatives
      Integrals
      Series
    Statistics
      Descriptive
      Probability
      Inference
      Distributions
    Linear Algebra
      Matrices
      Vectors
      Eigenvalues
      Transformations
    Discrete Math
      Logic
      Sets
      Combinatorics
      Graph Theory

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Quick Reference Links

• Formulas: See formulas.md
• Calculus: See calculus.md
• Linear Algebra: See linear-algebra.md
• Statistics: See statistics.md

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Problem-Solving Framework

General Steps

1. Read carefully - Identify what's given and what's asked
2. Draw/visualize - Sketch graphs, diagrams
3. Choose strategy - Direct, substitution, contradiction, etc.
4. Execute - Show all steps clearly
5. Verify - Check answer makes sense

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Common Proof Strategies

Strategy	When to Use	Example
Direct Proof	Show P → Q directly	"If n is even, n² is even"
Contradiction	Assume ¬Q, derive contradiction	Proving √2 is irrational
Contrapositive	Prove ¬Q → ¬P instead	Logical equivalence
Induction	Statements about all n ∈ ℕ	Sum formulas
Cases	Different scenarios	Piecewise functions

Mathematical Induction Template

Claim: P(n) is true for all n ≥ 1

Base Case: Show P(1) is true.
[Verify for n = 1]

Inductive Step:
Assume P(k) is true for some k ≥ 1. (Inductive Hypothesis)
Show P(k+1) is true.
[Derive P(k+1) using P(k)]

Therefore, by induction, P(n) is true for all n ≥ 1. ∎

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Notation Reference

Symbol	Meaning
∀	For all
∃	There exists
∈	Element of
⊂	Proper subset
⊆	Subset or equal
∪	Union
∩	Intersection
ℕ	Natural numbers {1,2,3,...}
ℤ	Integers {...,-1,0,1,...}
ℚ	Rational numbers
ℝ	Real numbers
ℂ	Complex numbers
∞	Infinity
∴	Therefore
∵	Because
∎	QED (proof complete)

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Function Analysis Checklist

1. Domain - What x values work?
2. Range - What y values result?
3. Intercepts - Where x=0, y=0?
4. Symmetry - Even f(-x)=f(x)? Odd f(-x)=-f(x)?
5. Asymptotes - Vertical, horizontal, oblique?
6. Critical points - Where f'(x)=0 or undefined?
7. Intervals - Increasing/decreasing?
8. Concavity - Where f''(x) > 0 or < 0?
9. Inflection points - Where concavity changes?

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Common Mistakes to Avoid

1. Dividing by zero - Check denominator ≠ 0
2. Square root of negative - Consider domain
3. Forgetting ± when taking square roots
4. Chain rule errors in derivatives
5. Forgetting +C in indefinite integrals
6. Incorrect limit laws for 0/0, ∞/∞ forms