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PERIMETER, AREA & VOLUME Rectangle 𝑃𝑃 = 2𝑙𝑙 + 2𝑤𝑤 𝐴𝐴 = 𝑙𝑙𝑤𝑤 Square 𝑃𝑃 = 4𝑠𝑠 𝐴𝐴 = 𝑠𝑠2 Triangle P = add all sides A = 1 2 𝑏𝑏ℎ Parallelogram P= add all sides A = bh Trapezoid P = add all sides A= 1 2 (𝑏𝑏1 + 𝑏𝑏2)h Circle 𝐶𝐶 = 𝜋𝜋𝜋𝜋 = 2𝜋𝜋𝜋𝜋 𝐴𝐴 = 𝜋𝜋𝜋𝜋2 Arc Length S = 𝜃𝜃𝜋𝜋 in radians S= 𝜋𝜋 180 𝜃𝜃𝜋𝜋in degrees Circle Sector Area A = 𝜃𝜃 2 𝜋𝜋2 in radians A= 𝜃𝜃 360 𝜋𝜋𝜋𝜋2in degrees Rectangular solid S = 2𝑙𝑙𝑤𝑤 + 2𝑙𝑙ℎ + 2𝑤𝑤ℎ V = 𝑙𝑙𝑤𝑤ℎ Cube 𝑆𝑆𝐴𝐴 = 6𝑠𝑠2 V = s3 Cylinder 𝑆𝑆𝐴𝐴 = 2𝜋𝜋𝜋𝜋2 + 2𝜋𝜋𝜋𝜋ℎ 𝑉𝑉 = 𝜋𝜋𝜋𝜋2ℎ Cone 𝑆𝑆𝐴𝐴 = 𝜋𝜋𝜋𝜋𝑠𝑠 + 𝜋𝜋𝜋𝜋2 𝑉𝑉 = 1 3 𝜋𝜋𝜋𝜋2ℎ A = 𝜋𝜋𝜋𝜋√𝜋𝜋2 + ℎ2 Sphere SA = 4πr2 V = 4 3 πr3 A = 4𝜋𝜋𝜋𝜋2 Rectangle Pyramid SA = 𝑙𝑙𝑤𝑤 + 2𝑙𝑙𝑠𝑠 + 2𝑤𝑤𝑠𝑠 V = 1 3 𝑙𝑙𝑤𝑤ℎ EXPONENT LAWS 𝑥𝑥0 = 1 if x ≠ 0 𝑥𝑥1 = 𝑥𝑥 𝑥𝑥−𝑛𝑛 = 1 𝑥𝑥𝑛𝑛 if x ≠ 0 𝑥𝑥𝑚𝑚. 𝑥𝑥𝑛𝑛 = 𝑥𝑥𝑚𝑚+𝑛𝑛 (𝑥𝑥𝑚𝑚)𝑛𝑛 = 𝑥𝑥𝑚𝑚.𝑛𝑛 𝑥𝑥𝑚𝑚 ÷ 𝑥𝑥𝑛𝑛 = 𝑥𝑥𝑚𝑚 𝑥𝑥𝑛𝑛 = 𝑥𝑥𝑚𝑚−𝑛𝑛 if x ≠0 (𝑥𝑥𝑥𝑥)𝑚𝑚 = 𝑥𝑥𝑚𝑚𝑥𝑥𝑚𝑚 �𝑥𝑥 𝑦𝑦 � 𝑛𝑛 = 𝑥𝑥𝑛𝑛 𝑦𝑦𝑛𝑛 if y ≠0 𝑥𝑥 𝑚𝑚 𝑛𝑛 = √𝑥𝑥𝑚𝑚𝑛𝑛 if (a ≥ 0, m ≥0, n>0) PROPERTIES OF LOGARITHMS 𝑥𝑥 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝑥𝑥 ⇔ 𝑥𝑥 = 𝑎𝑎𝑦𝑦where a >0, a ≠ 0 𝑎𝑎𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝑀𝑀 = 𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂(𝑀𝑀𝑀𝑀) = 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝑀𝑀 + 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎N 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂 � 𝑀𝑀 𝑀𝑀 � = 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝑀𝑀 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝑀𝑀𝑥𝑥 = 𝑥𝑥𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂𝑀𝑀 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏𝒂𝒂 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂 = 𝑙𝑙𝑛𝑛𝑀𝑀 𝑙𝑙𝑛𝑛𝒂𝒂 SPECIAL PRODUCTS 𝑥𝑥2 − 𝑥𝑥2 = (𝑥𝑥 + 𝑥𝑥)(𝑥𝑥 − 𝑥𝑥) 𝑥𝑥3 ± 𝑥𝑥3 = (𝑥𝑥 ± 𝑥𝑥)(𝑥𝑥2 ∓ 𝑥𝑥𝑥𝑥 + 𝑥𝑥2) BINOMIAL THEOREM (𝑥𝑥 ± 𝑥𝑥)2 = 𝑥𝑥2 ± 2𝑥𝑥𝑥𝑥 + 𝑥𝑥2 (𝑥𝑥 ± 𝑥𝑥)3 = 𝑥𝑥3 ± 3𝑥𝑥2𝑥𝑥 + 3𝑥𝑥𝑥𝑥2 ± 𝑥𝑥3 (𝑥𝑥 + 𝑥𝑥)𝑛𝑛 = 𝑥𝑥𝑛𝑛 + 𝑛𝑛𝑥𝑥𝑛𝑛−1𝑥𝑥 + 𝑛𝑛(𝑛𝑛−1) 2 𝑥𝑥𝑛𝑛−2𝑥𝑥2 + … + �𝑛𝑛 𝑘𝑘� 𝑥𝑥𝑛𝑛−𝑘𝑘 + … + 𝑛𝑛𝑥𝑥𝑥𝑥𝑛𝑛−1 + 𝑥𝑥𝑛𝑛 where �𝑛𝑛 𝑘𝑘� = 𝑛𝑛(𝑛𝑛−1)…(𝑛𝑛−𝑘𝑘+1) 1•2•3•…•𝑘𝑘 PYTHAGOREAN THEOREM leg2+ leg2 = hypotenuse2 DISTANCE FORMULA 𝜋𝜋 = �(𝑥𝑥2 − 𝑥𝑥1)2 + (𝑥𝑥2 − 𝑥𝑥1)2 sin𝛉𝛉 = 𝑙𝑙𝑜𝑜𝑜𝑜 ℎ𝑥𝑥𝑜𝑜 𝑐𝑐𝑙𝑙𝑠𝑠𝜽𝜽 = 𝑎𝑎𝜋𝜋𝑎𝑎 ℎ𝑥𝑥𝑜𝑜 𝑡𝑡𝑎𝑎𝑛𝑛𝜽𝜽 = 𝑙𝑙𝑜𝑜𝑜𝑜 𝑎𝑎𝜋𝜋𝑎𝑎 𝑐𝑐𝑙𝑙𝑡𝑡𝜽𝜽 = 𝑎𝑎𝜋𝜋𝑎𝑎 𝑙𝑙𝑜𝑜𝑜𝑜 𝑐𝑐𝑠𝑠𝑐𝑐𝜽𝜽 = ℎ𝑥𝑥𝑜𝑜 𝑙𝑙𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑐𝑐𝜽𝜽 = ℎ𝑥𝑥𝑜𝑜 𝑎𝑎𝜋𝜋𝑎𝑎 𝑡𝑡𝑎𝑎𝑛𝑛𝜽𝜽 = 𝑠𝑠𝑠𝑠𝑛𝑛𝜽𝜽 𝑐𝑐𝑙𝑙𝑠𝑠𝜽𝜽 𝑐𝑐𝑙𝑙𝑡𝑡𝜽𝜽 = 𝑐𝑐𝑙𝑙𝑠𝑠𝜽𝜽 𝑠𝑠𝑠𝑠𝑛𝑛𝜽𝜽 𝑠𝑠𝑠𝑠𝑛𝑛𝜽𝜽 = 1 𝑐𝑐𝑠𝑠𝑐𝑐𝜽𝜽 𝑐𝑐𝑙𝑙𝑠𝑠𝜽𝜽 = 1 𝑠𝑠𝑠𝑠𝑐𝑐𝜽𝜽 𝑐𝑐𝑠𝑠𝑐𝑐𝜽𝜽 = 1 𝑠𝑠𝑠𝑠𝑛𝑛𝜽𝜽 𝑠𝑠𝑠𝑠𝑐𝑐𝜽𝜽 = 1 𝑐𝑐𝑙𝑙𝑠𝑠𝜽𝜽 𝑡𝑡𝑎𝑎𝑛𝑛𝜽𝜽 = 1 𝑐𝑐𝑙𝑙𝑡𝑡𝜽𝜽 𝑐𝑐𝑙𝑙𝑡𝑡𝜽𝜽 = 1 𝑡𝑡𝑎𝑎𝑛𝑛𝜽𝜽 𝑠𝑠𝑠𝑠𝑛𝑛2𝜽𝜽 + 𝑐𝑐𝑙𝑙𝑠𝑠2𝜽𝜽 = 1 𝑡𝑡𝑎𝑎𝑛𝑛2𝜽𝜽 + 1 = 𝑠𝑠𝑠𝑠𝑐𝑐2𝜽𝜽 𝑐𝑐𝑙𝑙𝑡𝑡2𝜽𝜽 + 1 = 𝑐𝑐𝑠𝑠𝑐𝑐2𝜽𝜽 𝑠𝑠𝑠𝑠𝑛𝑛2𝜽𝜽 = 1 − 𝑐𝑐𝑙𝑙𝑠𝑠𝟐𝟐𝜽𝜽 2 𝑐𝑐𝑙𝑙𝑠𝑠2𝜽𝜽 = 1 + 𝑐𝑐𝑙𝑙𝑠𝑠𝟐𝟐𝜽𝜽 2 𝑡𝑡𝑎𝑎𝑛𝑛2𝜽𝜽 = 1 − 𝑐𝑐𝑙𝑙𝑠𝑠𝟐𝟐𝜽𝜽 1 + 𝑐𝑐𝑙𝑙𝑠𝑠𝟐𝟐𝜽𝜽 𝑠𝑠𝑠𝑠𝑛𝑛(𝟐𝟐𝜽𝜽) = 2𝑠𝑠𝑠𝑠𝑛𝑛𝜽𝜽𝑐𝑐𝑙𝑙𝑠𝑠𝜽𝜽 𝑐𝑐𝑙𝑙𝑠𝑠(𝟐𝟐𝜽𝜽) = 𝑐𝑐𝑙𝑙𝑠𝑠2𝜽𝜽 − 𝑠𝑠𝑠𝑠𝑛𝑛2𝜽𝜽 = 2𝑐𝑐𝑙𝑙𝑠𝑠2𝜽𝜽 − 1 = 1 − 2𝑠𝑠𝑠𝑠𝑛𝑛2𝜽𝜽 𝑡𝑡𝑎𝑎𝑛𝑛(𝟐𝟐𝜽𝜽) = 2𝑡𝑡𝑎𝑎𝑛𝑛𝜽𝜽 1 − 𝑡𝑡𝑎𝑎𝑛𝑛2𝜽𝜽 Law of Sines 𝑠𝑠𝑠𝑠𝑛𝑛𝜶𝜶 𝑎𝑎 = 𝑠𝑠𝑠𝑠𝑛𝑛𝜷𝜷 𝑏𝑏 = 𝑠𝑠𝑠𝑠𝑛𝑛𝜸𝜸 𝑐𝑐 Law of Cosines 𝑎𝑎2 = 𝑏𝑏2 + 𝑐𝑐2 − 2𝑏𝑏𝑐𝑐𝒄𝒄𝒄𝒄𝒄𝒄𝜶𝜶 𝑏𝑏2 = 𝑎𝑎2 + 𝑐𝑐2 − 2𝑎𝑎𝑐𝑐𝒄𝒄𝒄𝒄𝒄𝒄𝜷𝜷 𝑐𝑐2 = 𝑎𝑎2 + 𝑏𝑏2 − 2𝑎𝑎𝑏𝑏𝒄𝒄𝒄𝒄𝒄𝒄𝜸𝜸 Sum and Difference 𝑠𝑠𝑠𝑠𝑛𝑛(𝜶𝜶 ± 𝜷𝜷) = 𝑠𝑠𝑠𝑠𝑛𝑛𝜶𝜶𝑐𝑐𝑙𝑙𝑠𝑠𝜷𝜷 ± 𝑐𝑐𝑙𝑙𝑠𝑠𝜶𝜶𝑠𝑠𝑠𝑠𝑛𝑛𝜷𝜷 𝑐𝑐𝑙𝑙𝑠𝑠(𝜶𝜶 ± 𝜷𝜷) = 𝑐𝑐𝑙𝑙𝑠𝑠𝜶𝜶𝑐𝑐𝑙𝑙𝑠𝑠𝜷𝜷 ∓ 𝑠𝑠𝑠𝑠𝑛𝑛𝜶𝜶𝑠𝑠𝑠𝑠𝑛𝑛𝜷𝜷 𝑡𝑡𝑎𝑎𝑛𝑛(𝜶𝜶 ± 𝜷𝜷) = 𝑡𝑡𝑎𝑎𝑛𝑛𝜶𝜶 ± 𝑡𝑡𝑎𝑎𝑛𝑛𝜷𝜷 1 ∓ 𝑡𝑡𝑎𝑎𝑛𝑛𝜶𝜶𝑡𝑡𝑎𝑎𝑛𝑛𝜷𝜷 Sum to Product 𝑠𝑠𝑠𝑠𝑛𝑛𝜶𝜶 + 𝑠𝑠𝑠𝑠𝑛𝑛𝜷𝜷 = 2𝑠𝑠𝑠𝑠𝑛𝑛 � 𝜶𝜶 + 𝜷𝜷 𝟐𝟐 � 𝑐𝑐𝑙𝑙𝑠𝑠 � 𝜶𝜶 − 𝜷𝜷 𝟐𝟐 � 𝑠𝑠𝑠𝑠𝑛𝑛𝜶𝜶 − 𝑠𝑠𝑠𝑠𝑛𝑛𝜷𝜷 = 2𝑐𝑐𝑙𝑙𝑠𝑠 � 𝜶𝜶 + 𝜷𝜷 𝟐𝟐 � 𝑠𝑠𝑠𝑠𝑛𝑛 � 𝜶𝜶 − 𝜷𝜷 𝟐𝟐 � 𝑐𝑐𝑙𝑙𝑠𝑠𝜶𝜶 + 𝑐𝑐𝑙𝑙𝑠𝑠𝜷𝜷 = 2𝑐𝑐𝑙𝑙𝑠𝑠 � 𝜶𝜶 + 𝜷𝜷 𝟐𝟐 � 𝑐𝑐𝑙𝑙𝑠𝑠 � 𝜶𝜶 − 𝜷𝜷 𝟐𝟐 � 𝑐𝑐𝑙𝑙𝑠𝑠𝜶𝜶 − 𝑐𝑐𝑙𝑙𝑠𝑠𝜷𝜷 = −2𝑠𝑠𝑠𝑠𝑛𝑛 � 𝜶𝜶 + 𝜷𝜷 𝟐𝟐 � 𝑠𝑠𝑠𝑠𝑛𝑛 � 𝜶𝜶 − 𝜷𝜷 𝟐𝟐 � Product to Sum 𝑠𝑠𝑠𝑠𝑛𝑛𝜶𝜶𝑠𝑠𝑠𝑠𝑛𝑛𝜷𝜷 = 𝑐𝑐𝑙𝑙𝑠𝑠(𝜶𝜶 − 𝜷𝜷) − 𝑐𝑐𝑙𝑙𝑠𝑠(𝜶𝜶 + 𝜷𝜷) 2 𝑐𝑐𝑙𝑙𝑠𝑠𝜶𝜶𝑐𝑐𝑙𝑙𝑠𝑠𝜷𝜷 = 𝑐𝑐𝑙𝑙𝑠𝑠(𝜶𝜶 − 𝜷𝜷) + 𝑐𝑐𝑙𝑙𝑠𝑠(𝜶𝜶 + 𝜷𝜷) 2 𝑠𝑠𝑠𝑠𝑛𝑛𝜶𝜶𝑐𝑐𝑙𝑙𝑠𝑠𝜷𝜷 = 𝑠𝑠𝑠𝑠𝑛𝑛(𝜶𝜶 + 𝜷𝜷) + 𝑠𝑠𝑠𝑠𝑛𝑛(𝜶𝜶 − 𝜷𝜷) 2 𝑐𝑐𝑙𝑙𝑠𝑠𝜶𝜶𝑠𝑠𝑠𝑠𝑛𝑛𝜷𝜷 = 𝑠𝑠𝑠𝑠𝑛𝑛(𝜶𝜶 + 𝜷𝜷) − 𝑠𝑠𝑠𝑠𝑛𝑛(𝜶𝜶 − 𝜷𝜷) 2 /ep2016 SANTA ANA COLLEGE 1530 West 17th Street, Santa Ana CA 92704 THE MATH CENTER www.sac.edu/MathCenter Room L-204 Phone: (714) 564-6678 OPERATIONAL HOURS Monday thru Thursday 9:00AM – 7:50PM Friday 10:00AM – 12:50PM Saturday 12:00PM – 4:00PM "Who has not been amazed to learn that the function𝑥𝑥 = 𝑠𝑠𝑥𝑥, like a phoenix rising from its own ashes, is its own derivative?" Francois le Lionnais DERIVATIVES Definition: Derivative: 𝑓𝑓′(𝑥𝑥) = ℎ→0 𝑙𝑙𝑙𝑙𝑚𝑚 𝑓𝑓(𝑥𝑥+ℎ)−𝑓𝑓(𝑥𝑥) ℎ if this limit exists. Applications: If 𝑥𝑥 = 𝑓𝑓(𝑥𝑥) then, • 𝑚𝑚 = 𝑓𝑓 ′(𝑎𝑎) is the slope of the tangent line to y=f(x) at x=a and the equation of the tangent line at 𝑥𝑥 = 𝑎𝑎 is given by 𝑥𝑥 = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓 ′(𝑎𝑎)(𝑥𝑥 − 𝑎𝑎). • 𝑓𝑓′(𝑎𝑎)is the instantaneous rate of change of 𝑓𝑓(𝑥𝑥)at 𝑥𝑥 = 𝑎𝑎. • If 𝑓𝑓(𝑥𝑥)is the position of an object at time 𝑥𝑥, then 𝑓𝑓 ′(𝑎𝑎) is the velocity of the object at 𝑥𝑥 = 𝑎𝑎 Critical points: 𝑥𝑥 = 𝑐𝑐 is the critical point of 𝑓𝑓(𝑥𝑥) = 𝑐𝑐 provided either 1. 𝑓𝑓 ′(𝑐𝑐) = 0 or 2. 𝑓𝑓 ′(𝑐𝑐) does not exist. Increasing/Decreasing • If 𝑓𝑓 ′(𝑥𝑥) > 0 for all x in an interval I, then f(x) is increasing on the interval I. • If 𝑓𝑓 ′(𝑥𝑥) < 0 for all x in an interval I, then f(x) is decreasing on the interval I. • If 𝑓𝑓 ′ (𝑥𝑥) = 0 for all x in an interval I, then f(x) is constant on the interval I. Concavity • If 𝑓𝑓 ′′(𝑥𝑥) > 0 for all x in an interval I, then f(x) is concave up on the interval I. • If 𝑓𝑓 ′′(𝑥𝑥) < 0 for all x in an interval I, then f(x) is concave down on the interval I. Inflection Points 𝑥𝑥 = 𝑐𝑐 is an inflection point of f(x)if the concavity changes at 𝑥𝑥 = 𝑐𝑐. COMMON DERIVATIVES 1) 𝑐𝑐’ = 0 2) [𝑓𝑓(𝑥𝑥) + 𝑙𝑙(𝑥𝑥)]’ = 𝑓𝑓’(𝑥𝑥) + 𝑙𝑙’(𝑥𝑥) 3) [𝑓𝑓(𝑥𝑥)𝑙𝑙(𝑥𝑥)]’ = 𝑓𝑓(𝑥𝑥)𝑙𝑙’(𝑥𝑥) + 𝑓𝑓’(𝑥𝑥)𝑙𝑙(𝑥𝑥) 4) [𝑓𝑓(𝑙𝑙(𝑥𝑥))]’ = 𝑓𝑓’(𝑙𝑙(𝑥𝑥))𝑙𝑙’(𝑥𝑥) 5) [𝑐𝑐𝑓𝑓(𝑥𝑥)]’ = 𝑐𝑐𝑓𝑓’(𝑥𝑥) 6) [𝑓𝑓(𝑥𝑥) – 𝑙𝑙(𝑥𝑥)]’ = 𝑓𝑓’(𝑥𝑥) – 𝑙𝑙’(𝑥𝑥) 7) �𝑓𝑓(𝑥𝑥) 𝑙𝑙(𝑥𝑥) � ′ = 𝑙𝑙(𝑥𝑥)𝑓𝑓′(𝑥𝑥)− 𝑓𝑓(𝑥𝑥)𝑙𝑙′(𝑥𝑥) [𝑙𝑙(𝑥𝑥)]2 8) (𝑥𝑥𝑛𝑛)′ = 𝑛𝑛𝑥𝑥𝑛𝑛−1 9) [𝑠𝑠𝑥𝑥]′ = 𝑠𝑠𝑥𝑥 10) [𝑎𝑎𝑥𝑥]′ = 𝑎𝑎𝑥𝑥𝑙𝑙𝑛𝑛𝒂𝒂 11) [𝑙𝑙𝑛𝑛|𝒙𝒙|]′ = 1 𝑥𝑥 12) [𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎𝒙𝒙]′ = 1 𝑥𝑥𝑙𝑙𝑛𝑛𝒂𝒂 13) (𝑠𝑠𝑠𝑠𝑛𝑛𝒙𝒙)′ = cos𝒙𝒙 14) (𝑐𝑐𝑙𝑙𝑠𝑠𝒙𝒙)′ = −𝑠𝑠𝑠𝑠𝑛𝑛𝒙𝒙 15) (𝑡𝑡𝑎𝑎𝑛𝑛𝒙𝒙) ′ = 𝑠𝑠𝑠𝑠𝑐𝑐2𝒙𝒙 16) (𝑐𝑐𝑙𝑙𝑡𝑡𝒙𝒙)′ = −𝑐𝑐𝑠𝑠𝑐𝑐2𝒙𝒙 17) (𝑠𝑠𝑠𝑠𝑐𝑐𝒙𝒙)′ = 𝑠𝑠𝑠𝑠𝑐𝑐𝒙𝒙𝑡𝑡𝑎𝑎𝑛𝑛𝒙𝒙 18) (𝑐𝑐𝑠𝑠𝑐𝑐𝒙𝒙)′ = −𝑐𝑐𝑠𝑠𝑐𝑐𝒙𝒙𝑐𝑐𝑙𝑙𝑡𝑡𝒙𝒙 19) (𝑠𝑠𝑠𝑠𝑛𝑛−1 𝒙𝒙)′ = 1 √1−𝑥𝑥2 20) (𝑐𝑐𝑙𝑙𝑠𝑠−1 𝒙𝒙) ′ = − 1 √1−𝑥𝑥2 21) (𝑡𝑡𝑎𝑎𝑛𝑛−1 𝒙𝒙)′ = 1 1+𝑥𝑥2 22) (𝑐𝑐𝑙𝑙𝑡𝑡−1 𝒙𝒙) ′ = − 1 1+𝑥𝑥2 23) (𝑠𝑠𝑠𝑠𝑐𝑐−1 𝒙𝒙)′ = 1 |𝑥𝑥|√𝑥𝑥2−1 24) (𝑐𝑐𝑠𝑠𝑐𝑐−1 𝒙𝒙)′ = − 1 |𝑥𝑥|�𝑥𝑥2−1 25) (𝑠𝑠𝑠𝑠𝑛𝑛ℎ𝒙𝒙)′ = cosh𝒙𝒙 26) (𝑐𝑐𝑙𝑙𝑠𝑠ℎ𝒙𝒙)′ = 𝑠𝑠𝑠𝑠𝑛𝑛ℎ𝒙𝒙 27) (𝑡𝑡𝑎𝑎𝑛𝑛ℎ𝒙𝒙) ′ = 𝑠𝑠𝑠𝑠𝑐𝑐ℎ2𝒙𝒙 28) (𝑐𝑐𝑙𝑙𝑡𝑡ℎ𝒙𝒙)′ = −𝑐𝑐𝑠𝑠𝑐𝑐ℎ2𝒙𝒙 29) (𝑠𝑠𝑠𝑠𝑐𝑐ℎ𝒙𝒙)′ = −𝑠𝑠𝑠𝑠𝑐𝑐ℎ𝒙𝒙𝑡𝑡𝑎𝑎𝑛𝑛ℎ𝒙𝒙 30) (𝑐𝑐𝑠𝑠𝑐𝑐𝒙𝒙)′ = −𝑐𝑐𝑠𝑠𝑐𝑐ℎ𝒙𝒙𝑐𝑐𝑙𝑙𝑡𝑡ℎ𝒙𝒙 31) (𝑠𝑠𝑠𝑠𝑛𝑛ℎ−1 𝒙𝒙)′ = 1 √1+𝑥𝑥2 32) (𝑐𝑐𝑙𝑙𝑠𝑠ℎ−1 𝒙𝒙)′ = 1 √𝑥𝑥2−1 33) (𝑡𝑡𝑎𝑎𝑛𝑛ℎ−1 𝒙𝒙)′ = 1 1−𝑥𝑥2 34) (𝑐𝑐𝑙𝑙𝑡𝑡ℎ−1 𝒙𝒙)′ = 1 1−𝑥𝑥2 35) (𝑠𝑠𝑠𝑠𝑐𝑐ℎ−1 𝒙𝒙)′ = − 1 |𝑥𝑥|√1−𝑥𝑥2 36) (𝑐𝑐𝑠𝑠𝑐𝑐ℎ−1 𝒙𝒙)′ = − 1 |𝑥𝑥|�𝑥𝑥2+1 INTEGRATION Definition: Suppose 𝑓𝑓(𝑥𝑥) is continuous on [𝑎𝑎, 𝑏𝑏]. Divide [𝑎𝑎, 𝑏𝑏] into n subintervals of width ∆𝑥𝑥 and choose 𝑥𝑥𝑙𝑙 ∗ from each interval. Then � 𝑓𝑓(𝑥𝑥)𝜋𝜋𝑥𝑥 𝑏𝑏 𝑎𝑎 = lim 𝑛𝑛→∞ � 𝑓𝑓(𝑥𝑥𝑙𝑙 ∗ ∞ 𝑙𝑙=1 )∆𝑥𝑥 where ∆𝑥𝑥 = (𝑏𝑏−𝑎𝑎) 𝑛𝑛 Fundamental Theorem of Calculus: Suppose 𝑓𝑓(𝑥𝑥) is continuous on [𝑎𝑎, 𝑏𝑏], then Part I: 𝑙𝑙(𝑥𝑥) =∫ 𝑓𝑓(𝑡𝑡)𝜋𝜋𝑡𝑡𝑥𝑥 𝑎𝑎 is also continuous on [𝑎𝑎, 𝑏𝑏] and 𝑙𝑙′(𝑥𝑥) = 𝑑𝑑 𝑑𝑑𝑥𝑥 ∫ 𝑓𝑓(𝑡𝑡)𝜋𝜋𝑡𝑡 = 𝑓𝑓(𝑥𝑥)𝑥𝑥 𝑎𝑎 where 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑏𝑏. Part II: 𝑑𝑑 𝑑𝑑𝑥𝑥 ∫ 𝑓𝑓(𝑥𝑥)𝜋𝜋𝑥𝑥 = 𝐹𝐹(𝑏𝑏)𝑏𝑏 𝑎𝑎 − 𝐹𝐹(𝑎𝑎) where 𝐹𝐹(𝑥𝑥) is any anti-derivative of 𝑓𝑓(𝑥𝑥), i.e, a function such that 𝐹𝐹’ = 𝑓𝑓. Applications: INTEGRALS 1) ∫ 𝑢𝑢𝑛𝑛𝜋𝜋𝑢𝑢 = 𝑢𝑢𝑛𝑛+1 𝑛𝑛+1 + 𝑐𝑐, 𝑛𝑛 ≠ −1 2) ∫ 𝑑𝑑𝑢𝑢 𝑢𝑢 = 𝑙𝑙𝑛𝑛|𝒖𝒖| + 𝑐𝑐 3) ∫ 𝑠𝑠𝑢𝑢𝜋𝜋𝑢𝑢 = 𝑠𝑠𝑢𝑢 + 𝑐𝑐 4) ∫ 𝑎𝑎𝑢𝑢𝜋𝜋𝑢𝑢 = 𝑎𝑎𝑢𝑢 𝑙𝑙𝑛𝑛𝒂𝒂 + c 5) ∫ 𝑙𝑙𝑛𝑛𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑢𝑢𝑙𝑙𝑛𝑛𝒖𝒖 − 𝑢𝑢 + 𝑐𝑐 6) ∫ 1 𝑢𝑢𝑙𝑙𝑛𝑛𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛|𝑙𝑙𝑛𝑛𝒖𝒖| + 𝑐𝑐 7) ∫ 𝑠𝑠𝑠𝑠𝑛𝑛𝒖𝒖 𝜋𝜋𝑢𝑢 = −𝑐𝑐𝑙𝑙𝑠𝑠𝒖𝒖 + 𝑐𝑐 8) ∫ 𝑐𝑐𝑙𝑙𝑠𝑠𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑠𝑠𝑠𝑠𝑛𝑛𝒖𝒖 + 𝑐𝑐 9) ∫ 𝑡𝑡𝑎𝑎𝑛𝑛𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛|𝑠𝑠𝑠𝑠𝑐𝑐𝒖𝒖| + 𝑐𝑐 10) ∫ 𝑐𝑐𝑙𝑙𝑡𝑡𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛|𝑠𝑠𝑠𝑠𝑛𝑛𝒖𝒖| + 𝑐𝑐 11) ∫ 𝑠𝑠𝑠𝑠𝑐𝑐𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛|𝑠𝑠𝑠𝑠𝑐𝑐𝒖𝒖 + 𝑡𝑡𝑎𝑎𝑛𝑛𝒖𝒖| + 𝑐𝑐 12) ∫ 𝑐𝑐𝑠𝑠𝑐𝑐𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛|𝑐𝑐𝑠𝑠𝑐𝑐𝒖𝒖 − 𝑐𝑐𝑙𝑙𝑡𝑡𝒖𝒖| + 𝑐𝑐 13) ∫ 𝑠𝑠𝑠𝑠𝑐𝑐2𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑡𝑡𝑎𝑎𝑛𝑛 𝒖𝒖 + 𝑐𝑐 14) ∫ 𝑐𝑐𝑠𝑠𝑐𝑐2𝒖𝒖 𝜋𝜋𝑢𝑢 = −𝑐𝑐𝑙𝑙𝑡𝑡𝒖𝒖 + 𝑐𝑐 15) ∫ 𝑠𝑠𝑠𝑠𝑐𝑐𝒖𝒖 𝑡𝑡𝑎𝑎𝑛𝑛𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑠𝑠𝑠𝑠𝑐𝑐𝒖𝒖 + 𝑐𝑐 16) ∫ 𝑐𝑐𝑠𝑠𝑐𝑐𝒖𝒖 𝑐𝑐𝑙𝑙𝑡𝑡𝒖𝒖 𝜋𝜋𝑢𝑢 = −𝑐𝑐𝑠𝑠𝑐𝑐𝒖𝒖 + 𝑐𝑐 17) ∫ 𝑑𝑑𝑢𝑢 √𝑎𝑎2−𝑢𝑢2 = 𝑠𝑠𝑠𝑠𝑛𝑛−1 𝒖𝒖 𝒂𝒂 + 𝑐𝑐, 𝑎𝑎 > 0 18) ∫ 𝑑𝑑𝑢𝑢 𝑎𝑎2+𝑢𝑢2 = 1 𝑎𝑎 𝑡𝑡𝑎𝑎𝑛𝑛−1 𝒖𝒖 𝒂𝒂 + 𝑐𝑐 19) ∫ 𝑑𝑑𝑢𝑢 𝑢𝑢√𝑢𝑢2−𝑎𝑎2 = 1 𝑎𝑎 𝑠𝑠𝑠𝑠𝑐𝑐−1 𝒖𝒖 𝒂𝒂 + 𝑐𝑐 20) ∫ 𝑑𝑑𝑢𝑢 𝒂𝒂𝟐𝟐−𝒖𝒖𝟐𝟐 = 1 2𝑎𝑎 𝑙𝑙𝑛𝑛 �𝒖𝒖+𝒂𝒂 𝒖𝒖−𝒂𝒂 �+c 21) ∫ 𝑑𝑑𝑢𝑢 𝒖𝒖𝟐𝟐−𝒂𝒂𝟐𝟐 = 1 2𝑎𝑎 𝑙𝑙𝑛𝑛 �𝒖𝒖−𝒂𝒂 𝒖𝒖+𝒂𝒂 �+c 22) ∫ 𝑠𝑠𝑠𝑠𝑛𝑛−1 𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑢𝑢 𝑠𝑠𝑠𝑠𝑛𝑛−1 𝒖𝒖 + √1 − 𝑢𝑢2 + 𝑐𝑐 23) ∫ 𝑐𝑐𝑙𝑙𝑠𝑠−1 𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑢𝑢 𝑐𝑐𝑙𝑙𝑠𝑠−1 𝒖𝒖 + √1 − 𝑢𝑢2 + 𝑐𝑐 24) ∫ 𝑡𝑡𝑎𝑎𝑛𝑛−1 𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑢𝑢 tan−1 𝒖𝒖 − 1 2 ln(1 + 𝑢𝑢2) + 𝑐𝑐 25) ∫ 𝑠𝑠𝑠𝑠𝑛𝑛ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑐𝑐𝑙𝑙𝑠𝑠ℎ𝒖𝒖 + 𝑐𝑐 26) ∫ 𝑐𝑐𝑙𝑙𝑠𝑠ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑠𝑠𝑠𝑠𝑛𝑛ℎ𝒖𝒖 + 𝑐𝑐 27) ∫ 𝑡𝑡𝑎𝑎𝑛𝑛ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛 (𝑐𝑐𝑙𝑙𝑠𝑠ℎ𝒖𝒖) + 𝑐𝑐 28) ∫ 𝑐𝑐𝑙𝑙𝑡𝑡ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛|𝑠𝑠𝑠𝑠𝑛𝑛ℎ𝒖𝒖| + 𝑐𝑐 29) ∫ 𝑠𝑠𝑠𝑠𝑐𝑐ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑡𝑡𝑎𝑎𝑛𝑛−1|𝑠𝑠𝑠𝑠𝑛𝑛ℎ𝒖𝒖| + 𝑐𝑐 30) ∫ 𝑐𝑐𝑠𝑠𝑐𝑐ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑙𝑙𝑛𝑛 �𝑡𝑡𝑎𝑎𝑛𝑛ℎ 1 2 𝒖𝒖� + 𝑐𝑐 31) ∫ 𝑠𝑠𝑠𝑠𝑐𝑐ℎ2𝒖𝒖 𝜋𝜋𝑢𝑢 = 𝑡𝑡𝑎𝑎𝑛𝑛ℎ 𝒖𝒖 + 𝑐𝑐 32) ∫ 𝑐𝑐𝑠𝑠𝑐𝑐ℎ2𝒖𝒖 𝜋𝜋𝑢𝑢 = −𝑐𝑐𝑙𝑙𝑡𝑡ℎ𝒖𝒖 + 𝑐𝑐 33) ∫ 𝑠𝑠𝑠𝑠𝑐𝑐ℎ𝒖𝒖 𝑡𝑡𝑎𝑎𝑛𝑛ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = −𝑠𝑠𝑠𝑠𝑐𝑐ℎ𝒖𝒖 + 𝑐𝑐 34) ∫ 𝑐𝑐𝑠𝑠𝑐𝑐ℎ𝒖𝒖 𝑐𝑐𝑙𝑙𝑡𝑡ℎ𝒖𝒖 𝜋𝜋𝑢𝑢 = −𝑐𝑐𝑠𝑠𝑐𝑐ℎ𝒖𝒖 + 𝑐𝑐 35) ∫ 𝑢𝑢𝜋𝜋𝑢𝑢 = 𝑢𝑢𝑢𝑢 − ∫ 𝑢𝑢𝜋𝜋𝑢𝑢 Area: 𝐴𝐴 = ∫ 𝑓𝑓(𝑥𝑥)𝜋𝜋𝑥𝑥𝑏𝑏 𝑎𝑎 Area between Curves: • 𝑥𝑥 = 𝑓𝑓(𝑥𝑥); 𝐴𝐴 = ∫ (𝑢𝑢𝑜𝑜𝑜𝑜𝑠𝑠𝜋𝜋 − 𝑙𝑙𝑙𝑙𝑤𝑤𝑠𝑠𝜋𝜋 𝑓𝑓𝑢𝑢𝑛𝑛𝑡𝑡𝑠𝑠𝑙𝑙𝑛𝑛)𝜋𝜋𝑥𝑥𝑏𝑏 𝑎𝑎 • 𝑥𝑥 = 𝑓𝑓(𝑥𝑥); 𝐴𝐴 = ∫ (𝜋𝜋𝑠𝑠𝑙𝑙ℎ𝑡𝑡 − 𝑙𝑙𝑠𝑠𝑓𝑓𝑡𝑡 𝑓𝑓𝑢𝑢𝑛𝑛𝑡𝑡𝑠𝑠𝑙𝑙𝑛𝑛) 𝜋𝜋𝑥𝑥𝑏𝑏 𝑎𝑎 Volumes: V= ∫ 𝐴𝐴𝜋𝜋𝑠𝑠𝑎𝑎(𝑥𝑥) 𝜋𝜋𝑥𝑥𝑏𝑏 𝑎𝑎 Volume of Revolution Rings V= ∫ 2𝜋𝜋(𝑙𝑙𝑢𝑢𝑡𝑡𝑠𝑠𝜋𝜋 𝜋𝜋2 − 𝑠𝑠𝑛𝑛𝑛𝑛𝑠𝑠𝜋𝜋 𝜋𝜋2)𝑏𝑏 𝑎𝑎 Cylinders V= ∫ 𝑐𝑐𝑠𝑠𝜋𝜋𝑐𝑐𝑢𝑢𝑚𝑚𝑓𝑓𝑠𝑠𝜋𝜋𝑠𝑠𝑛𝑛𝑐𝑐𝑠𝑠 ∙ ℎ𝑠𝑠𝑠𝑠𝑙𝑙ℎ𝑡𝑡 ∙ 𝑡𝑡ℎ𝑠𝑠𝑐𝑐𝑘𝑘𝑛𝑛𝑠𝑠𝑠𝑠𝑠𝑠𝑏𝑏 𝑎𝑎 Work: If a force of 𝐹𝐹(𝑥𝑥) moves an object in 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑏𝑏, then the work done is W=∫ 𝐹𝐹(𝑥𝑥)𝜋𝜋𝑥𝑥𝑏𝑏 𝑎𝑎 Average Function Value: The average value of 𝑓𝑓(𝑥𝑥) on 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑏𝑏 is𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑙𝑙𝑎𝑎 = 1 𝑏𝑏−𝑎𝑎 ∫ 𝑓𝑓(𝑥𝑥)𝜋𝜋𝑥𝑥𝑏𝑏 𝑎𝑎 “Success is the sum of small efforts, repeated day in and day out.” Robert Collier “Go down deep enough into anything and you will find mathematics.” Dean Schlicter