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1.5 Logarithms

1.5 Logarithms. In other words, a function is one-to-one on domain D if: . whenever. A relation is a function if: for each x there is one and only one y . A relation is a one-to-one if also: for each y there is one and only one x . .

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1.5 Logarithms

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  1. 1.5 Logarithms

  2. In other words, a function is one-to-one on domain D if: whenever A relation is a function if: for each x there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one x.

  3. To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. one-to-one not one-to-one not a function (also not one-to-one)

  4. Inverse functions: Given an x value, we can find a y value. Solve for x: Inverse functions are reflections about y = x. Switch x and y: (eff inverse of x)

  5. WINDOW GRAPH a parametrically: Y= example 3: Graph: for

  6. b Find the inverse function: WINDOW Y= > GRAPH example 3: Graph: for Switch x & y: Change the graphing mode to function.

  7. is called the natural log function. is called the common log function. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Two raised to what power is 16? Example: The most commonly used bases for logs are 10: and e:

  8. In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work. is called the natural log function. is called the common log function.

  9. 7 Here is a useful keyboard shortcut for the newer TI-89 Titanium calculators. (Unfortunately the shortcut does not work on the older TI-89s.) returns: If you enter: you get: If you enter: you get: Even though we will be using natural logs in calculus, you may still need to find logs with other bases occasionally. (base 10) (base 2)

  10. 9 returns: If you enter: you get: If you enter: you get: And while we are on the topic of TI-89 Titanium keyboard shortcuts: (square root) (fifth root)

  11. Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

  12. Example 6: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent. 17.9 years In real life you would have to wait 18 years. p*

  13. Indonesian Oil Production (million barrels per year): Example 7: Use the natural logarithm regression equation to estimate oil production in 1982 and 2000. How do we know that a logarithmic equation is appropriate? In real life, we would need more points or past experience.

  14. ENTER ENTER STO 6 3 5 L 1 L 2 2nd 2nd MATH 2nd Indonesian Oil Production: 20.56 million 42.10 70.10 60 70 90 2nd 60,70,90 { 2nd } L 1 2nd (on a Ti-89) , LnReg The calculator should return: Statistics Regressions Done

  15. STAT CALC 9 L 1 L 2 2nd 2nd VARS , , ENTER (on a Ti-84)… LnReg Y-VARS FUNCTION… Y1 The calculator gives you an equation and constants:

  16. WINDOW GRAPH 2nd Plot1…On Type: Scatter ENTER Y= Xlist: L1 Ylist: L2 We can use the calculator to plot the new curve along with the original points:

  17. WINDOW GRAPH

  18. ENTER CALC ENTER 82 2ND 100 CALC 2ND What does this equation predict for oil production in 1982 and 2000? TRACE This lets us see values for the distinct points. Enters an x-value of 82. In 1982, production was 59 million barrels. Moves to the line. Enters an x-value of 100. p In 2000, production was 84 million barrels.

  19. 1.6 Trig Functions

  20. When you use trig functions in calculus, you must use radian measure for the angles. The best plan is to set the calculator mode to radians and use when you need to use degrees. 2nd o Trigonometric functions are used extensively in calculus.

  21. Cosine is an even function because: Even and Odd Trig Functions: “Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change. Secant is also an even function, because it is the reciprocal of cosine. Even functions are symmetric about the y - axis.

  22. Sine is an odd function because: Even and Odd Trig Functions: “Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the sign of the y value also changes. Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function. Odd functions have origin symmetry.

  23. is a stretch. is a shrink. The rules for shifting, stretching, shrinking, and reflecting the graph of a function apply to trigonometric functions. Vertical stretch or shrink; reflection about x-axis Vertical shift Positive d moves up. Horizontal shift Horizontal stretch or shrink; reflection about y-axis Positive c moves left. The horizontal changes happen in the opposite direction to what you might expect.

  24. is the amplitude. is the period. B A C D When we apply these rules to sine and cosine, we use some different terms. Vertical shift Horizontal shift

  25. Trig functions are not one-to-one. However, the domain can be restricted for trig functions to make them one-to-one. These restricted trig functions have inverses. p*

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