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# Week 4 - Trigonometry

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El keynote de la semana 4 de mi curso de trigonometría.

El keynote de la semana 4 de mi curso de trigonometría.

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• ### Transcript

• 1. Day 161. OpenerSketch the graph of f(x) = (1/2)x.
• 2. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x.
• 3. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x.
• 4. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x.
• 5. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x.
• 6. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x
• 7. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y
• 8. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2
• 9. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 -1
• 10. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 -1 0
• 11. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 -1 0 1
• 12. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 -1 0 1 2
• 13. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 4 -1 0 1 2
• 14. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 4 -1 2 0 1 2
• 15. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 4 -1 2 0 1 1 2
• 16. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 4 -1 2 0 1 1 1/2 2
• 17. Day 161. Opener 0<a<1Sketch the graph of f(x) = (1/2)x. x y -2 4 -1 2 0 1 1 1/2 2 1/4
• 18. Day 171. OpenerFind the exact value ofa) log 2 8 1b) log 3 3c) log 5 25
• 19. Day 171. OpenerFind the exact value ofa) log 2 8 =3 1b) log 3 3c) log 5 25
• 20. Day 171. OpenerFind the exact value ofa) log 2 8 =3 1b) log 3 = -1 3c) log 5 25
• 21. Day 171. OpenerFind the exact value ofa) log 2 8 =3 1b) log 3 = -1 3c) log 5 25 =2
• 22. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5b) log e b = −3c) log 3 5 = c
• 23. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5 a5 = 4b) log e b = −3c) log 3 5 = c
• 24. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5 a5 = 4b) log e b = −3 e-3 = bc) log 3 5 = c
• 25. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5 a5 = 4b) log e b = −3 e-3 = bc) log 3 5 = c 3c = 5
• 26. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5b) log e b = −3c) log 3 5 = c
• 27. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5 a5 = 4b) log e b = −3c) log 3 5 = c
• 28. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5 a5 = 4b) log e b = −3 e-3 = bc) log 3 5 = c
• 29. Change each logarithmic expression to anequivalent expression containing an exponent.a) log a 4 = 5 a5 = 4b) log e b = −3 e-3 = bc) log 3 5 = c 3c = 5
• 30. 2. Graphs of Logarithmic functions. y = loga x
• 31. 2. Graphs of Logarithmic functions.Sketch the graph of the function belowFind the domain and range. y = log 2 ( x + 2 )
• 32. 2. Graphs of Logarithmic functions.Sketch the graph of the function belowFind the domain and range. y = log 1 ( x ) − 2 3
• 33. 2. Graphs of Logarithmic functions.Sketch the graph of the function belowFind the domain and range. y = log 3 ( x + 1) − 2 2
• 34. 3. Exercises.Sketch the graph of the functions belowFind the domain and range.1. f (x) = ln ( x + 4 )2. f (x) = − log x3. g(x) = log 2 ( x − 3) + 2 34. h(x) = log 3 ( x − 4 ) + 35. y = log 4 ( x + 1) − 2 3
• 35. Day 181. OpenerSolve the equationa) 7 x+6 = 7 3x−4 2 x+3 x2b) 3 =3 −100 x x−4c) 2 = ( 0.5 )d) 4 x−3 = 8 4−x 3−2 x ⎛ 1 ⎞ x 2 xe) 4 ⋅ ⎜ ⎟ = 8 ⋅(2 ) ⎝ 2 ⎠
• 36. Day 181. OpenerSolve the equationa) 7 x+6 = 7 3x−4 =5 2 x+3 x2b) 3 =3 −100 x x−4c) 2 = ( 0.5 )d) 4 x−3 = 8 4−x 3−2 x ⎛ 1 ⎞ x 2 xe) 4 ⋅ ⎜ ⎟ = 8 ⋅(2 ) ⎝ 2 ⎠
• 37. Day 181. OpenerSolve the equationa) 7 x+6 = 7 3x−4 =5 2 x+3 x2 = -1, 3b) 3 =3 −100 x x−4c) 2 = ( 0.5 )d) 4 x−3 = 8 4−x 3−2 x ⎛ 1 ⎞ x 2 xe) 4 ⋅ ⎜ ⎟ = 8 ⋅(2 ) ⎝ 2 ⎠
• 38. Day 181. OpenerSolve the equationa) 7 x+6 = 7 3x−4 =5 2 x+3 x2 = -1, 3b) 3 =3 x−4c) 2 −100 x = ( 0.5 ) = -4/99d) 4 x−3 = 8 4−x 3−2 x ⎛ 1 ⎞ x 2 xe) 4 ⋅ ⎜ ⎟ = 8 ⋅(2 ) ⎝ 2 ⎠
• 39. Day 181. OpenerSolve the equationa) 7 x+6 = 7 3x−4 =5 2 x+3 x2 = -1, 3b) 3 =3 x−4c) 2 −100 x = ( 0.5 ) = -4/99d) 4 x−3 = 8 4−x = 18/5 3−2 x ⎛ 1 ⎞ x 2 xe) 4 ⋅ ⎜ ⎟ = 8 ⋅(2 ) ⎝ 2 ⎠
• 40. Day 181. OpenerSolve the equationa) 7 x+6 = 7 3x−4 =5 2 x+3 x2 = -1, 3b) 3 =3 x−4c) 2 −100 x = ( 0.5 ) = -4/99d) 4 x−3 = 8 4−x = 18/5 3−2 x ⎛ 1 ⎞ x 2 xe) 4 ⋅ ⎜ ⎟ = 8 ⋅(2 ) =3 ⎝ 2 ⎠
• 41. 2. Solving a logarithmic equationSolve the equation log 4 ( 5 + x ) = 3
• 42. 2. Solving a logarithmic equationSolve the equation log 6 ( 4x − 5 ) = log 6 ( 2x + 1)
• 43. 2. Solving a logarithmic equationThe population N(t) (in millions) of the United States tyears after 1980 may be approximated by the formulaN(t) = 227e0.007t. When will the population be twice whatit was in 1980?
• 44. 3. Compound Interest Formula nt ⎛ r ⎞ A = P ⎜ 1+ ⎟ ⎝ n ⎠whereP = Principalr = annual interest rate expressed as a decimaln = number of interest periods per yeart = number of years P is investedA = amount after t years
• 45. 3. Compound Interest FormulaSuppose that \$1000 is invested at an interest rate of 9%compounded monthly. Find the new amount of principalafter 5 years, after 10 years, and after 15 years. Illustrategraphically the growth of the investment.