Mathematics II Summary

• Polynomials
o 𝑦(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + ⋯ + 𝑎𝑛 𝑥 𝑛
o First degree polynomial
▪ Linear function
▪ 𝑦(𝑥) = 𝑎0 + 𝑎1 𝑥
o Second degree polynomial
▪ Parabolic function
▪ 𝑦(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2
▪ Root Formula (abc-formule)
−𝑏±√𝑏 2 −4𝑎𝑐
𝑥= 2𝑎
o Power Function
▪ 𝑦(𝑥) = 𝑐 ∙ 𝑥 𝑝
▪ 𝑥 = 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
▪ 𝑝 = 𝑡ℎ𝑒 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡
• Inverse Function
o Mirrored copy in the line y=x
o Asymptote’s x becomes y and vice versa
• Logarithm
o Logarithmic function is the inverse of the exponential function
o 𝑎log𝑎(𝑥) = 𝑥
o log 𝑎 (𝑥𝑦) = log 𝑎 (𝑥) + log 𝑎 (𝑦)
𝑥
o log 𝑎 (𝑦) = log 𝑎 (𝑥) − log 𝑎 (𝑦)
o log 𝑎 (𝑥 𝑝 ) = 𝑝 ∙ log 𝑎 (𝑥)
log(𝑥)
o log 𝑎 (𝑥) = log(𝑎)
• Derivative
𝑓(𝑥+∆𝑥)−𝑓(𝑥)
o 𝑓 ′ (𝑥) = lim
∆𝑥→0 ∆𝑥
o Tangent line
▪ 𝑦 = 𝑓(𝑎) + 𝑓′(𝑎) ∙ (𝑥 − 𝑎)
▪ 𝑎 = 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 ℎ𝑖𝑡𝑠 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ
o Standard derivatives
1
▪ 𝑓(𝑥) = ln(𝑥) ⟶ 𝑓 ′ (𝑥) = 𝑥
▪ 𝑓(𝑥) = 𝑎 𝑥 (𝑎 > 0) ⟶ 𝑓 ′ (𝑥) = 𝑎 𝑥 ln(𝑎)
1
▪ 𝑓(𝑥) = tan(𝑥) ⟶ 𝑓 ′ (𝑥) = cos(𝑥)2
▪ 𝑠𝑖𝑛2 (𝑥) + 𝑐𝑜𝑠 2 (𝑥) = 1
• Geometric sequence
𝑡𝑛
o 𝑟=𝑡 ⟶ 𝑡𝑛 = 𝑟 ∙ 𝑡𝑛−1
𝑛 −1
▪ r is the ratio aka multiplier
o 𝑡𝑛 = 𝑎 ∙ 𝑟 𝑛−1
▪ 𝑎 = 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
o Any geometric sequence with −1 < 𝑟 < 1 has a limit 0
o Any geometric sequence with 𝑟 < −1 𝑜𝑟 𝑟 > 1 does not have a limit