Final Notes for AB and BC - Trigonometric, Differentiation, Integration Formulas and Calculus Theorems

Oct 13, 2026

Upload by Muffet

FinalNotes整理AP Calculus AB与BC中常用的三角恒等式、求导法则与常见导数公式、积分基本公式与部分换元/反三角相关结果，并汇总极限与连续性、奇偶函数、周期性、介值定理、有理函数无穷处极限、渐近线、变化率、导数定义以及若干重要定理（如罗尔定理、平均值定理与极值定理）。适用于复习与快速查阅核心计算框架与常用结论。

LanguageEnglish

CategoryPresentations

FormatPDF

Pages16

Reading Time6 min

AccessFree to download

Trigonometric Formulas

Differentiation Formulas

Integration Formulas

Formulas and Theorems

Limits and Continuity

Even and Odd Functions

Periodicity

Intermediate-Value Theorem

Limits of Rational Functions as x→±∞

Horizontal and Vertical Asymptotes

Average and Instantaneous Rate of Change

Definition of Derivative

The Number e as a Limit

Rolle’s Theorem

Mean Value Theorem

Extreme-Value Theorem

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. AP CALCULUS AB and BC FinalNotes Trigonometric Formulas sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ sin(−θ ) = − sinθ cos(−θ ) = cos θ tan(−θ ) = − tan θ sin(A + B) = sin AcosB + sin B cos A sin(A − B) = sin AcosB − sin B cos Acos(A + B) = cos A cos B − sin Asin B cos(A − B) = cos A cos B + sin Asin B sin 2θ = 2 sinθcos θ 13. tan θ = scθθ = 1cot~~ ~~θ 14. cot θ = cossinθθ = ~~ ~~ta1n~~ ~~θ 15. sec θ = co1s~~ ~~θ [16.](16. csc)[ csc](16. csc) θ = si1n~~ ~~θ 17. cos2 θ = ~~ ~~12 ( 1+ cos 2 θ ) 18. sin 2 θ = ~~ ~~12 ( 1− cos 2 θ ) cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ −1 = 1− 2 sin 2 θ Differentiation Formulas ddx~~ ~~ (x n ) = nx n−1 ddx~~ ~~ ( fg) = fg ′ + gf ′ Product ruleddx~~ ~~ (~~ ~~fg~~ ~~) = gf~~ ~~′g−2~~ ~~fg~~ ~~′ Quotient ruleddx~~ ~~ f (g (x)) = f ′(g (x))g ′(x) Chain ruleddx~~ ~~ (sin x) = cos x ddx~~ ~~ (cos x) = − sin xddx~~ ~~ (tan x) = sec 2 xddx~~ ~~ (cot x) = − csc2 xddx~~ ~~ (sec x) = sec x tan xddx~~ ~~ (cscx) = − cscx cot x 11. ddx~~ ~~ (ex ) = ex 12. ddx~~ ~~ (a x ) = a x ln a 13. ddx~~ ~~ (ln x) = 1x 14. ddx~~ ~~ (Arc sin x) = 1~~ ~~x~~ ~~2~~ ~~ 15. ddx~~ ~~ (Arc tan x) = 1~~ ~~+1x~~ ~~2~~ ~~ 16. ddx~~ ~~ (Arc sec x) = |~~ ~~x~~ ~~|~~ ~~1x2~~ ~~−1 17. ddx~~ ~~ [ c ] = 0 18. ddx~~ ~~ cf ( x ) = cf ' ( x ) Integration Formulas 1. ∫a dx = ax + C 2. ∫x n dx = xn11 + C, n ≠ −1 3. ∫ ~~ ~~1x~~ ~~ dx = ln x + C 4. ∫ ex dx = ex + C 5. ∫a x dx = lanxa + C 6. ∫ ln x dx = x ln x − x + C 7. ∫ sin x dx = − cos x + C 8. ∫ cos x dx = sin x + C 9. ∫ tan x dx = ln sec x + C or − lncos x + C 10. ∫ cot x dx = ln sin x + C 11. ∫ sec x dx = ln sec x + tan x + C 12. ∫ csc x dx = − ln csc x + cot x + C 13. ∫ sec 2 x dx = tan x + C 14. ∫ sec x tan x dx = sec x + C 15. ∫ csc 2 x dx = − cot x + C 16. ∫ cscx cot x dx = − cscx + C 17. ∫ tan 2 x dx = tan x − x + C 18. ∫ ~~ ~~a~~ ~~2~~ ~~x~~ ~~2~~ ~~ = ~~ ~~1a~~ ~~ Arc tan􀂨􀂧􀂩 ~~ ~~xa~~ ~~ 􀂷􀂹􀂸 + C 19. ∫ ~~ ~~a~~ ~~d2~~ ~~x~~ ~~2~~ ~~ = Arc sin 􀂨􀂧􀂩 ~~ ~~xa~~ ~~ 􀂷􀂹􀂸 + C 20. ∫ ~~ ~~xxd2x−~~ ~~a~~ ~~2~~ ~~ = ~~ ~~1a~~ ~~ Arc sec ~~ ~~xa~~ ~~ + C = ~~ ~~1a~~ ~~ Arc cos ~~ ~~ax + C Formulas and Theorems 1. Limits and Continuity: A function y = f (x) is continuous at x = a if i) . f(a) exists ii) . f ( x ) exists iii) . f ( x ) = f (a) Otherwise, f is discontinuous at x = a. The limit a f ( x ) exists if and only if both corresponding one-sided limits exist and are equal – that is, a f ( x ) = L → a+ f ( x ) = L = a − f ( x ) 2. Even and Odd Functions 1. A function y = f (x) is even if f (−x) = f (x) for every x in the function’s domain. Every even function is symmetric about the y-axis. 2. A function y = f (x) is odd if f (−x) = − f (x) for every x in the function’s domain. Every odd function is symmetric about the origin. 3. Periodicity A function f (x) is periodic with period p (p > 0) if f (x + p) = f (x) for every value of x. Note: The period of the function y = Asin(Bx + C) or y = Acos(Bx + C) is 2πB~~ ~~ . The amplitude is A . The period of y = tan x is π . 4. Intermediate-Value Theorem A function y = f (x) that is continuous on a closed interval [a, b] takes on every value between f (a) and f (b) . Note: If f is continuous on [a, b] and f (a) and f (b) differ in sign, then the equation f (x) = 0 has at least one solution in the open interval (a, b) . 5. Limits of Rational Functions as x → ±∞ i) . xl ~~ ~~fg~~(~~(xx~~)~~) = 0 if the degree of f (x) < the degree of g (x) Example: lim ~~ ~~x23−~~ ~~2x = 0 x→∞ x + 3 ii) . lim f (x) is infinite if the degrees of f (x) > the degree of g (x) x→±∞ g (x) Example: lim ~~ ~~x32+~~ ~~2x = ∞ x→∞ x − 8 iii) . lim f (x) is finite if the degree of f (x) = the degree of g (x) x→±∞ g (x) Exam~~pl~~e: ~~ ~~2x120x−−3x5x2 = − ~~ ~~25 6. Horizontal and Vertical Asym

FAQs

What trigonometric identities are included in the notes?

The notes list key identities such as sin^2θ + cos^2θ = 1 and relationships involving tan, cot, sec, and csc, plus sum/difference formulas like sin(A+B), sin(A−B), and cos(A−B).

Which differentiation rules and derivative formulas are provided?

They include product rule, quotient rule, chain rule, plus standard derivatives for trig functions (sin, cos, tan, cot, sec, csc) and exponential/logarithmic forms like e^x, a^x, ln x, and selected arc functions.

What calculus theorems about limits and derivatives are summarized?

The notes cover limits and continuity, even/odd behavior and periodicity, intermediate value theorem, rational-function limits at infinity, asymptotes, rate of change, the derivative definition, and the main theorems: Rolle’s theorem, mean value theorem, and extreme-value theorem.

Oct 13, 2026

Upload by Muffet