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Final

MAT133Y1 Final: MAT133 FINAL FORMULA SHEET

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Department
Mathematics
Course
MAT133Y1
Professor
D Jill Tate
Semester
Fall

Description
MAT133 FINAL FORMULA SHEET Effective π‘Ÿ β–― Continuously 𝑆 = 𝑃𝑒 β–― Rate π‘Ÿβ–―= (1 + )π‘“βˆ’ 1 Compounding 𝑃 = 𝑆𝑒 β–―β–―β–― PV Annuities β–― β–―β–― FV Annuities (1 + π‘Ÿ) βˆ’ 1 (t=0) 1 βˆ’ (1 + π‘Ÿ) (t=n) 𝑆 = 𝑅 βˆ— π‘Ÿ r = effective 𝐴 = 𝑅 βˆ— π‘Ÿ period rate FV Annuity 1 + π‘Ÿ β–―β–―β–―βˆ’ 1 PV Annuity 1 βˆ’ 1 + π‘Ÿ β–―β–―β–― 𝑆 = 𝑅 βˆ— ( βˆ’ 1) 𝐴 = 𝑅 βˆ— ( + 1) Due π‘Ÿ Due π‘Ÿ Monthly 𝐴 Perpetuity 𝑅 𝑅 = β–―β–― (No FV) 𝐴 = π‘Ÿ Mortgage 1 βˆ’ (1 + 𝑖) Payment 𝑖 Effective β–―β–― 1 βˆ’ (1 + 𝑖)β–―β–― π‘Ÿ β–― Bond Price 𝑃 = 𝑉(1 + 𝑖) + π‘Ÿπ‘‰ βˆ— Monthly 𝑖 = (1 + 2 𝑖 Interest Rate β–― Indefinite Definite Integrals 𝑓 π‘₯ 𝑑π‘₯ = 𝐹 π‘₯ + 𝐢 Integrals 𝑓 π‘₯ 𝑑π‘₯ = [𝐹 𝑏 βˆ’ 𝐹 π‘Ž ] β–―β–― π‘˜π‘‘π‘₯ = π‘˜π‘₯ + 𝐢 ▯𝑓 π‘₯ Β± 𝑔 π‘₯ 𝑑π‘₯ β–―β–―β–― β–― π‘₯ 𝑑π‘₯ = π‘₯ + 𝐢 = 𝑓 π‘₯ 𝑑π‘₯ π‘Ž + 1 β–―β–― 1 𝑑π‘₯ = ln π‘₯ + 𝐢 More Common Β± ▯𝑔 π‘₯ 𝑑π‘₯ Common π‘₯ Integrals (including β–― β–― Integrals β–― β–― 𝑓 π‘₯ 𝑑π‘₯ = βˆ’ 𝑓 π‘₯ 𝑑π‘₯ 𝑒 𝑑π‘₯ = 𝑒 + 𝐢 definite) β–― β–― β–― β–― β–― 𝑏 𝑑π‘₯ = 𝑏 + 𝐢 𝑓 π‘₯ 𝑑π‘₯ = 𝑓 π‘₯ 𝑑π‘₯ 𝑙𝑛𝑏 β–― β–― β–― π‘˜π‘“ π‘₯ 𝑑π‘₯ = π‘˜ 𝑓 π‘₯ 𝑑π‘₯ + 𝑓 π‘₯ 𝑑π‘₯ β–― β–― Substitution 𝑓 𝑔 π‘₯ 𝑔 π‘₯ 𝑑π‘₯ = 𝑓 𝑒 𝑑𝑒 By Parts 𝑒𝑑𝑣 = 𝑒𝑣 βˆ’ 𝑣𝑑𝑒 π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑒 = 𝑔 π‘₯ ,𝑑𝑒 = 𝑔′(π‘₯) Improper β–― β–― 𝑓 π‘₯ 𝑑π‘₯ = lim 𝑓 π‘₯ 𝑑π‘₯ Integrals β–― β–―β†’β–― β–― ±∞ β–― Average 1 β–― β†’ π‘‘π‘–π‘£π‘’π‘Ÿπ‘”π‘’π‘›π‘‘ 𝑓 π‘₯ 𝑑π‘₯ = lim 𝑓 π‘₯ 𝑑π‘₯ 𝑓 = 𝑓 π‘₯ 𝑑π‘₯ π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ β–―β–― β–―β†’β–―β–― Value 𝑏 βˆ’ π‘Ž β–― β†’ π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘”π‘’π‘›π‘‘ β–― β–― β–― 𝑓 π‘₯ = 𝑓 π‘₯ 𝑑π‘₯ + 𝑓 π‘₯ 𝑑π‘₯ β–―β–― β–―β–― β–― Consumer β–―β–― β–―β–― & Area β–―β–― Calculation 𝐴 = 𝑦1 βˆ’ 𝑦2 𝑑π‘₯ + 𝑦2 βˆ’ 𝑦1 + β‹― Producer 𝐢𝑆 = 𝑓 π‘ž βˆ’ 𝑝0 π‘‘π‘ž β–―β–―β–― β–―β–― Surplus β–―β–―β–― (y=…x) A = π‘₯1 βˆ’ π‘₯2 𝑑𝑦 + π‘₯2 βˆ’ π‘₯1 𝑑𝑦 + β‹― (p=…q) (x=…y) β–―β–― β–―β–― 𝑃𝑆 = 𝑝0 βˆ’ 𝑔 π‘ž π‘‘π‘ž (q0, p0) = β–― equilibrium PV Continuous β–― FV Continuous β–― Annuity 𝑓 𝑑 𝑒▯▯▯𝑑𝑑 𝑓 𝑑 𝑒▯(β–―β–―β–―)𝑑𝑑 T = years β–― Annuity β–― πœ•π‘§ 𝑓 π‘₯ + β„Ž,𝑦 βˆ’ 𝑓(π‘₯,𝑦) πœ•π‘§ 𝑓 π‘₯,𝑦 + β„Ž βˆ’ 𝑓(π‘₯,𝑦) = 𝑓 π‘₯,𝑦 = lim = 𝑓▯π‘₯,𝑦 = lim πœ•π‘₯ β–― β–―β†’β–― β„Ž πœ•π‘¦ β–―β†’β–― β„Ž Competitive Products Complementary Products β–―β–―β–― β–―β–―β–― β–―β–―β–― β–―β–―β–― > 0 π‘Žπ‘›π‘‘ > 0 < 0 π‘Žπ‘›π‘‘ < 0 β–―β–―β–― β–―β–―β–― β–―β–―β–― β–―β–―β–― Chain Rule Critical Point 𝑙𝑒𝑑 𝑧 = 𝑓 π‘₯,𝑦 ,π‘₯ = π‘₯ π‘Ÿ,𝑠 ,𝑦 = π‘Ÿ,𝑠 𝑓 π‘₯,𝑦 = 0 β–― πœ•π‘§ = πœ•π‘§ Γ— πœ•π‘₯ + πœ•π‘§ Γ— πœ•π‘¦
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