Derivatives¶
Part 4 of the lecture notes. Derivative, rules of differentiation, theorems of differential calculus, Taylor, curve sketching, Newton.
-
1. The derivative function
Line through two points · Tangent line · Derivative of a function at a point and derivative function
-
2. Derivatives of elementary functions
Derivatives of elementary functions
-
3. Corner points, cusps, points with vertical/horizontal tangent
Right derivative and left derivative · Corner points · Points with vertical/horizontal tangent · Cusps
-
4. Rules for computing derivatives
Rules for computing derivatives
-
5. Mean value theorem, maxima and minima
Local and global maxima/minima · Fermat's theorem and stationary points · Mean value theorem (Lagrange's theorem) · Differential monotonicity test theorem · …
-
6. L'Hôpital's rule and differentiability
L'Hôpital's rule · Limit of the derivative and differentiability
-
7. Second derivative
Second derivative and second derivative function · Second derivative, concavity and convexity · Convexity and derivatives · Convexity and tangent lines · …
-
8. Differential calculus and approximations
First-order asymptotic expansions · Derivatives of polynomials · Taylor formula/expansion with Peano remainder · Taylor formula/expansion with Lagrange remainder
-
9. Curve sketching
Graphs of real functions of a real variable · Examples
-
10. Newton's method
Newton's method · Error estimates for Newton's method