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## Math 142B Spring 2024 Terminology and Theorems

#### Updated 3/21/24

Note: The following list is a minimal collection of the important terms and theorems for this course.  You simply must be familiar with them and how they are used:  in the case of theorems, you should be familiar with how they are proved.  This basic knowledge is analogous to the vocabulary of a language:  it is impossible to speak the language without it.

• 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'HÃ´pital's rule
• Section 5.31
• Taylor series for f about c
• nth remainder,   Rn(x)
• Taylor's theorem
• Taylor's theorem (Cauchy integral remainder)
• Cauchy's form of Rn
• 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,   fn converges pointwise
• uniform convergence,   fn 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

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