Math 2400 Lecture Notes: Differentiation

1. Differentiability Versus Continuity 2 2. Differentiation Rules 3 3. Optimization 7 3.1. Intervals and interior points 7 3.2. Functions increasing or decreasing at a point 7 3.3. Extreme Values 8 3.4. Local Extrema and a Procedure for Optimization 10 3.5. Remarks on finding roots of f ′ 12 4. The Mean Value Theorem 12 4.1. Statement of the Mean Value Theorem 12 4.2. Proof of the Mean Value Theorem 13 4.3. The Cauchy Mean Value Theorem 14 5. Monotone Functions 15 5.1. The Monotone Function Theorems 15 5.2. The First Derivative Test 17 5.3. The Second Derivative Test 18 5.4. Sign analysis and graphing 19 5.5. A theorem of Spivak 21 6. Inverse Functions I: Theory 21 6.1. Review of inverse functions 21 6.2. The Interval Image Theorem 23 6.3. Monotone Functions and Invertibility 23 6.4. Inverses of Continuous Functions 24 6.5. Inverses of Differentiable Functions 25 7. Inverse Functions II: Examples and Applications 27 7.1. x 1 n 27 7.2. L(x) and E(x) 28 7.3. Some inverse trigonometric functions 31 References 33