Math Tutor

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

Common Proof Strategies

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. ∎

Notation Reference

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?

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