Math 142B Spring 2024
Terminology and Theorems

Updated 3/21/24

• Math 142A Terminology (for reference and review)

• Section 5.29
  • Rolle's theorem
  • mean value theorem
  • increasing, strictly increasing
  • decreasing, strictly decreasing
  • intermediate value theorem for derivatives
  • derivative of inverse function
• Section 5.30
  • generalized mean value theorem
  • L'Hôpital's rule
• Section 5.31
  • Taylor series for f about c
  • n^th remainder,   R_n(x)
  • Taylor's theorem
  • Taylor's theorem (Cauchy integral remainder)
  • Cauchy's form of R_n
  • Newton's method
  • secant method
• Section 6.32
  • partition of [a,b]
  • upper Darboux sum U(f,P) of f with respect to P
  • lower Darboux sum L(f,P) of f with respect to P
  • upper Darboux integral U(f) of f over [a,b]
  • lower Darboux integral L(f) of f over [a,b]
  • integrable
  • Riemann (Darboux) integral
  • mesh of a partition P,   mesh(P)
• Section 6.33
  • monotonic functions are integrable
  • continuous functions are integrable
  • linearity (constant multiple, additivity) (33.3)
  • absolute integrability (33.5)
  • piecwise integrability (33.6, 33.8)
  • intermediate value theorem for integrals (mean value theorem for integrals)
• Section 6.34
  • fundamental theorem of Calculus I
  • integration by parts
  • fundamental theorem of Calculus II
  • change of variable
• Section 4.24
  • pointwise convergence,   f_n converges pointwise
  • uniform convergence,   f_n converges uniformly
  • the uniform limit of continuous functions is continuous
• Section 4.25
  • uniformly Cauchy on a set S ⊆ ℝ
  • Weierstrass M-test
• Section 4.23
  • rules of convergence for power series (23.1)
• Section 4.26
  • differentiation and integration of power series
  • Abel's theorem
• Section 4.27
  • Bernstein polynomial for the function f
  • Weierstrass's approximation theorem