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Two really important ideas

Two really important ideas. Function Inverse & Exponential Function. Function Inverse. Going Driving. I start 10 miles away from my house and drive away from my house at 30 mph. If I know how long I’ve been driving, how far am I from my house?. Going Driving.

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Two really important ideas

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  1. Two really important ideas Function Inverse & Exponential Function

  2. Function Inverse

  3. Going Driving I start 10 miles away from my house and drive away from my house at 30 mph. If I know how long I’ve been driving, how far am I from my house?

  4. Going Driving I start 10 miles away from my house and drive away from my house at 30 mph. If I know how long I’ve been driving, how far am I from my house? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10

  5. Going Driving I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10

  6. Going Driving I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10 (d-10)/30=t

  7. Going driving

  8. I haven’t changed anything(except my point of view) (d-10)/30=t d=30t+10

  9. I haven’t changed anything(except my point of view) 40 miles 1 hours 40 miles 1 hours 10 miles 0 hours 0 hours 10 miles (d-10)/30=t d=30t+10

  10. Cubing I have an equation y=x3. I know x=2 and I want to figure out y. y=(2)3 y=8 I have an equation y=x3. I know y=8 and I want to figure out x. 8=(x)3 y=∛8=2

  11. Cubing to cube root

  12. Cubing to cube root y=x3 x=∛y

  13. Cubing to cube root The relationship between x and y stays the same Only my point of view changes y=x3 x=∛y

  14. Notation From x  y y=x3 ƒ(x)=x3

  15. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y

  16. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y

  17. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y ƒ-1(x)=∛x Because x and y don’t actually mean anything, I can change their names if I want.

  18. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y ƒ-1(x)=∛x Because x and y don’t actually mean anything, I can change their names if I want. This is not actually a good idea, but it’s popular in many math books

  19. How to find a function inverse • ƒ(x)=………….x…………. • Rewrite as y=……………x………… • Solve for y. x=~~~~y~~~~~~ • Rewrite as an inverse ƒ-1(y)=~~~~y~~~~~~ • OPTIONAL: change ys to xs. • ƒ-1(x)=~~~~x~~~~~~ • WARNING: Always check that your inverse is actually a function.

  20. Round trip I drive away from home for 1.25 hours at 30 miles per hour, then I turn around and drive back home at 30 miles per hour. y=number of miles I am from home x=number of hours since I started driving

  21. Round Trip

  22. Round Trip If I know x (time), I can figure out y (distance). y is a function of x. If I know y (distance), I can’t figure out y (time). x is NOT a function of y.

  23. Testing if the inverse is a function

  24. A shoe size that is size ‘x’ in the United States is size t(x) in Continental size, where t(x)=x+34.5 Find a function that will convert Continental shoe size to a US shoe size. • t-1(x) = 1/(x+34.5) • t-1(x) = 1/x + 34.5 • t-1(x) = 34.5 + x • t-1(x) = x – 1/34.5 • None of the above.

  25. A shoe size that is size ‘x’ in the United States is size t(x) in Continental size, where t(x)=x+34.5 Find a function that will convert Continental shoe size to a US shoe size. t(x)=x+34.5 y=x+34.5 y-34.5=x y-34.5=t-1(y) t-1(y)=y-34.5 t-1(x)=x-34.5 E

  26. Exponential Functions The “I’m going to lie to you a lot” version

  27. Exponential functions measure steady growth • If you really want to know what that means exactly, take differential equations (after Calculus) • Here’s the basic (lying) version • An exponential growth happens when something is making more of itself (in a “steady” way) • People, money, bacteria, etc…

  28. Example • One dollar makes one dollar every year. $1 $1 Year 0 Year 1

  29. Example • One dollar makes one dollar every year. $1 $1 $1 Year 0 Year 1

  30. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  31. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  32. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  33. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  34. Example • Every year I keep what I have and add what I have. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  35. Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  36. Example • Every year I double my money $1 y=1(2x) y=# of $ x=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  37. Exponential Growth

  38. Exponential Decay

  39. Which of the following functions represent that of exponential decay? • f(x)=(1/2)x • f(x)=(1/2)-x • f(x)=(1/3)-x • (b) and (c) • None of the above

  40. Which of the following functions represent that of exponential decay? • f(x)=(1/2)x • f(x)=(1/2)-x • f(x)=(1/3)-x • (b) and (c) • None of the above

  41. Anatomy • The standard form of the exponential is y=abx • a is called the initial value (y-intercept) • b is called the growth factor. • When 0<b<1, you have exponential decay • A non-standard form is y=ac-x • a is the initial value (y-intercept) • c-1=1/c is the growth factor. • When 0<1/c<1, you have exponential decay

  42. The compound interest formula • P dollars are invested at r% per year compounded n times per year. After t years, I have A dollars.

  43. The compound interest example • 100dollarsare invested at 7% per year compounded monthly. How many dollars do I have after 5 years? • P=100,r=7/100,n=12 (12 months a year),t=5 Find A.

  44. The compound interest example • I invested some money at 3% per year compounded quarterly. After 9 yearsI had $1000. How much did I start with? • R=3/100, n=4,t=9,A=1000,find P.

  45. Asymptotes • All exponentials y=abx have asymptote y=0

  46. Asymptotes • Adding c moves a graph up by c.

  47. Asymptotes • Any function y=abx+c has asymptote y=c

  48. Consider the function below: Which of the following statements matches with this function? As x approaches infinity, f(x) approaches 0. As x approaches negative infinity, f(x) approaches 0. As x approaches infinity, f(x) approaches -4. As x approaches negative infinity, f(x) approaches -4. None of the above

  49. D) As x approaches negative infinity, f(x) approaches -4.

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