Calculus BC Exam Review

AP Exam Review Flashcards

sin2 𝜃 + cos2 𝜃

Trigonometric Identity: Equals 1

sin 2𝐴

Trigonometric Identity: Equals 2 sin𝐴 cos 𝐴

Definition of the Derivative (Form 1)

𝑓′(𝑥) = lim ℎ→0 𝑓(𝑥 + ℎ) − 𝑓(𝑥) / ℎ

Definition of the Derivative (Form 2)

𝑓′(𝑐) = lim𝑥→𝑐 𝑓(𝑥) − 𝑓(𝑐) / 𝑥 − 𝑐

𝑑/𝑑𝑥 sin 𝑥

Derivative: Equals cos 𝑥

𝑑/𝑑𝑥 cos 𝑥

Derivative: Equals − sin 𝑥

𝑑/𝑑𝑥 tan 𝑥

Derivative: Equals sec2 𝑥

𝑑/𝑑𝑥 csc 𝑥

Derivative: Equals − csc 𝑥 cot 𝑥

𝑑/𝑑𝑥 sec 𝑥

Derivative: Equals sec 𝑥 tan 𝑥

𝑑/𝑑𝑥 cot 𝑥

Derivative: Equals − csc2 𝑥

𝑑/𝑑𝑥 sin−1 𝑥

Derivative: Equals 1 / √1 − 𝑥2

𝑑/𝑑𝑥 tan−1 𝑥

Derivative: Equals 1 / 1 + 𝑥2

𝑑/𝑑𝑥 sec−1 𝑥

Derivative: Equals 1 / |𝑥|√𝑥2 − 1

𝑑/𝑑𝑥 ln 𝑥

Derivative: Equals 1 / 𝑥

𝑑/𝑑𝑥 𝑒𝑥

Derivative: Equals 𝑒𝑥

𝑑/𝑑𝑥 𝑏𝑥

Derivative: Equals 𝑏𝑥 ln 𝑏

𝑔′(𝑥) for 𝑔(𝑥) = 𝑓−1(𝑥)

Equals 1 / 𝑓′(𝑔(𝑥))

Intermediate Value Theorem (IVT)

If 𝑓(𝑥) is continuous on [𝑎, 𝑏] and 𝑓(𝑎) ≠ 𝑓(𝑏), then for every value, 𝑀, between 𝑓(𝑎) and 𝑓(𝑏), there must exist some 𝑐 ∈ (𝑎, 𝑏) such that 𝑓(𝑐) = 𝑀.

Mean Value Theorem (MVT)

If 𝑓(𝑥) is continuous on [𝑎, 𝑏] and differentiable on (𝑎, 𝑏) then there exists at least one value 𝑥 = 𝑐 on (𝑎, 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏)−𝑓(𝑎)) / (𝑏−𝑎).

Rolle’s Theorem

If 𝑓(𝑥) is continuous on [𝑎, 𝑏] and differentiable on (𝑎, 𝑏) and 𝑓(𝑎) = 𝑓(𝑏), then there exists at least one value 𝑥 = 𝑐 on (𝑎, 𝑏) such that 𝑓′(𝑐) = 0.

∫ 1/𝑥 𝑑𝑢

Integral: Equals ln |𝑥| + 𝐶

∫ 𝑏𝑥 𝑑𝑥

Integral: Equals (𝑏𝑥 / ln 𝑏) + 𝐶

∫ 𝑒𝑥𝑑𝑥

Integral: Equals 𝑒𝑥 + 𝐶

∫ 𝑥𝑛𝑑𝑥

Integral: Equals (𝑥𝑛+1 / 𝑛 + 1) + 𝐶 (for 𝑛 ≠ −1)

Fundamental Theorem of Calculus (Part 1)

𝐹(𝑏) = 𝐹(𝑎) + ∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥 where 𝐹 is the antiderivative of 𝑓.

Fundamental Theorem of Calculus (Part 2)

𝑑/𝑑𝑥 ∫ 𝑓(𝑡) 𝑔(𝑥) 𝑎 𝑑𝑡 = 𝑓(𝑔(𝑥))𝑔′(𝑥)

Integration by Parts Formula

∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢

Extreme Value Theorem (EVT)

If 𝑓(𝑥) is continuous on [𝑎, 𝑏], then 𝑓 must attain a maximum and a minimum on [𝑎, 𝑏].

Riemann Sums

Approximations under 𝑓(𝑥) over [𝑎, 𝑏] using 𝑛 partitions

Right Riemann Sums (𝑅𝑛)

∆𝑥[𝑦2 + 𝑦3 + 𝑦4 + ⋯ + 𝑦𝑛]

Left Riemann Sums (𝐿𝑛)

∆𝑥[𝑦1 + 𝑦2 + 𝑦3 + ⋯ +𝑦𝑛−1]

Midpoint Riemann Sums (𝑀𝑛)

∆𝑥 [ (𝑦1 + 𝑦2)/2 + (𝑦2 + 𝑦3)/2 + ⋯ + (𝑦𝑛−1 + 𝑦𝑛)/2 ]

Trapezoidal Rule (𝑇𝑛)

1/2 ∆𝑥[𝑦1 + 2𝑦2 + 2𝑦3 + ⋯ +2𝑦𝑛−1 + 𝑦𝑛]

Area between 𝑓 and 𝑔 (where 𝑓(𝑥) > 𝑔(𝑥) on [𝑎, 𝑏])

∫ 𝑓(𝑥) − 𝑔(𝑥) 𝑏 𝑎 𝑑𝑥

Volume when rotating area between 𝑓 and 𝑔 about 𝑦 = 𝑘

𝜋 ∫ [𝑓(𝑥) − 𝑘]2 − [𝑔(𝑥) − 𝑘]2 𝑏 𝑎 𝑑𝑥

Average Rate of Change of 𝑓 on [𝑎, 𝑏]

(𝑓(𝑏) − 𝑓(𝑎)) / (𝑏 − 𝑎)

Average Value of 𝑓 on [𝑎, 𝑏]

(1 / (𝑏 − 𝑎)) ∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥

Improper Integrals (Type 1)

∫ 𝑓(𝑥) ∞ 𝑎 𝑑𝑥 = lim 𝑅→∞ ∫ 𝑓(𝑥) 𝑅 𝑎 𝑑𝑥

Improper Integrals (Type 2)

∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥 = lim 𝑅→𝑏− ∫ 𝑓(𝑥) 𝑅 𝑎 𝑑𝑥 (𝑖𝑓 𝑓(𝑏) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡)

Logistic Growth Equation (𝑑𝑃/𝑑𝑡)

𝑘𝑃 (1 − 𝑃/𝑀)

Logistic Growth Equation (𝑃(𝑡))

𝑀 / (1 + 𝐶𝑒−𝑘𝑡)

Euler’s Method Formula

𝑦𝑛 = 𝑦𝑛−1 + ∆𝑥𝑓′(𝑥𝑛−1)

Polar Coordinates: 𝑥

𝑟 cos 𝜃

Polar Coordinates: 𝑦

𝑟 sin 𝜃

Polar Coordinates: 𝑟

√(𝑥2 + 𝑦2)

Polar Coordinates: tan 𝜃

𝑦 / 𝑥

Polar Coordinates: Area (𝐴)

∫ 1/2 𝑟2𝑑𝜃 𝛽 𝛼

Parametric Equations: 𝑑𝑦/𝑑𝑥

𝑦′(𝑡) / 𝑥′(𝑡)

Parametric Equations: 𝑑2𝑦/𝑑𝑥2

(𝑑/𝑑𝑡 (𝑦′(𝑡) / 𝑥′(𝑡))) / 𝑥′(𝑡)

Vectors: Position

〈𝑥(𝑡), 𝑦(𝑡)〉

Vectors: Velocity

〈𝑥′(𝑡), 𝑦′(𝑡)〉

Vectors: Acceleration

〈𝑥′′(𝑡), 𝑦′′(𝑡)〉

Arc Length (Rectangular)

∫ √(1 + 𝑓′(𝑥)2) 𝑏 𝑎 𝑑𝑥

Arc Length (Parametric)

∫ √(𝑥′(𝑡)2 + 𝑦′(𝑡)2) 𝑏 𝑎 𝑑𝑡

Arc Length (Polar)

∫ √(𝑟(𝜃)2 + 𝑟′(𝜃)2) 𝛽 𝛼 𝑑𝜃

Speed (Parametric)

√(𝑥′(𝑡)2 + 𝑦′(𝑡)2)

Nth Term Test

Series diverges if lim𝑛→∞ 𝑎𝑛 ≠ 0

Geometric Series

converges if 0 < |𝑟| < 1, and diverges otherwise SUM=𝑐𝑟𝑀 / (1−𝑟)

p-Series

converges if p>1, diverges otherwise

Alternating Series Test

For ∑(−1)𝑛𝑎𝑛 if lim𝑛→∞ 𝑎𝑛 = 0 and 𝑎𝑛+1 < 𝑎𝑛, then the series converges.

Alternating Series Remainder

|𝑆 − 𝑆𝑛| ≤ 𝑎𝑛+1

Ratio Test

lim𝑛→∞ |𝑎𝑛+1 / 𝑎𝑛 | = 𝐿

Root Test

lim𝑛→∞ √(𝑎𝑛) = 𝐿

Taylor Series (centered at 𝑥 = 𝑐)

𝑃𝑛(𝑥) = 𝑓(𝑐) + 𝑓′(𝑐)(𝑥 − 𝑐) + 𝑓′′(𝑐) (𝑥 − 𝑐)2 / 2! + 𝑓′′′(𝑐) (𝑥 − 𝑐)3 / 3! + ⋯ + 𝑓(𝑛)(𝑐) (𝑥 − 𝑐)𝑛 / 𝑛! + ⋯

Lagrange Error Bound

|𝑓(𝑥) − 𝑃𝑛(𝑥)| ≤ |𝑓(𝑛+1)(𝑘) (𝑥−𝑐)𝑛+1 / (𝑛+1)!|, where |𝑓(𝑛+1)(𝑘)| is the maximum value of 𝑓(𝑛+1) on (𝑥, 𝑐)

Power Series Convergence (Centered at c)

a) converges at c; b) converges absolutely for |𝑥 − 𝑐| < 𝑅; c) converges absolutely for all x

Maclaurin Series

Taylor Series centered at x=0

1 / (1 − 𝑥)

Series: 1 + 𝑥 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + ⋯ + 𝑥𝑛 + ⋯ , Interval of Convergence: |𝑥| < 1

sin 𝑥

Series: 𝑥 − 𝑥3 / 3! + 𝑥5 / 5! − 𝑥7 / 7! + 𝑥9 / 9! − ⋯ + (−1)𝑛𝑥2𝑛+1 / (2𝑛 + 1)! + ⋯, Interval of Convergence: (−∞, ∞)

cos 𝑥

Series: 1 − 𝑥2 / 2! + 𝑥4 / 4! − 𝑥6 / 6! + 𝑥8 / 8! − ⋯ + (−1)𝑛𝑥2𝑛 / (2𝑛)! + ⋯, Interval of Convergence: (−∞, ∞)

𝑒𝑥

Series: 1 + 𝑥 + 𝑥2 / 2! + 𝑥3 / 3! + 𝑥4 / 4! + 𝑥5 / 5! + ⋯ + 𝑥𝑛 / 𝑛! + ⋯, Interval of Convergence: (−∞, ∞)