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Calculus I (MAT 145) Dr. Day Fri day January 18 , 2013

Calculus I (MAT 145) Dr. Day Fri day January 18 , 2013. Finish : Characteristics of Functions (1.1 - 1.3 ) Exponential Functions (1.5) Inverse Functions and Logarithmic Functions (1.6) Assignments. Function Notation. What does f ( x ) = 3 x + 5 represent?

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Calculus I (MAT 145) Dr. Day Fri day January 18 , 2013

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  1. Calculus I (MAT 145)Dr. Day Friday January 18, 2013 • Finish: Characteristics of Functions (1.1 - 1.3) • Exponential Functions (1.5) • Inverse Functions and Logarithmic Functions (1.6) • Assignments MAT 145

  2. Function Notation • What does f(x) = 3x + 5 represent? • Using y = f(x) as just described, determine and simplify each of the following. • f(2) (b) f(−4) (c) f() (d) f(x-1) • f(2x+4) (f) f(t2+7t) (g) f(f(x)) (h) f(x + h) (i) f(x + h) – f(x) MAT 145

  3. Families of Functions Algebraic—functions formed by using algebraic operations (+, -, x, ÷, nth root of a , a to the mth power , such that m and n are rational) on coefficients and variables • Polynomial (Constant, Linear, Quadratic, Cubic, Quartic, and others) • Power • Root • Rational Transcendental—non-algebraic functions (i.e., those that “transcend” algebra) • Trigonometric • Exponential • Logarithmic • Others—Piecewise, Absolute Value, Greatest Integer, and others MAT 145

  4. Operating on Functions Algebra of functions: apply one or more of the four basic operations (+,−,,) to two or more functions. Composition of functions: putting together two ormore functions in a particular way. When two functions are composed, one is evaluated within the other. MAT 145

  5. Exponential Functions(PIP page 9; Text section 1.5) Exponential functions are of the form where The input variable, the exponent, can be any real number. The base must be a positive real number. Why make this restriction? MAT 145

  6. Use and Apply Exponential Functions (1) The graph of contains the points (1,6) and (4,162). Determine C and a. (2) Some versions of biological warfare involve releasing highly effective fast-growing toxins into the atmosphere. Suppose that a toxin enters my body at 12 noon with an initial count of 20 bacteria cells. The cells double every 15 minutes. (a) If growth occurs as described above, and goes unchecked, represent the number of bacteria cells P in my body t hours after entering my body. (b) At what time will there be 1 million bacteria cells present? (3) A certain radio-active element has a half-life of 60 days. Suppose there are 8.5 grams present today, January 23, 2012. (a) If decay occurs as described above, represent the amount A, in grams, of the element that is present t days from now. (b) On what date (day, month, year) will there first be less than 2 grams present? MAT 145

  7. Functions and Their Inverses(PIP p 10-11; Text section 1.6) • Representing the inverse of a relationship. • Graphs, tables, symbols: f(x) = 5x – 2; g(x) = (4x + 3)/(x-2) • Does a function have an inverse that also is a function? • One-to-one functions • Inverses of exponential functions: Logarithmic functions MAT 145

  8. Logarithmic Functions • Inverses of exponential functions: Logarithmic functions exponent: y base: a MAT 145

  9. Assignments • WebAssign • Section 1.3 due tonight • Section 1.5 due tomorrow night • Section 1.6 due Wednesday night • Test #1: Friday Jan 27; practice test available • WA: Techno problems with “grade student work” (1.1 #8) • Email, MAT 145 homepage, WA announcements/resources MAT 145

  10. Reminders/Assignments • Assignments • PIP Packet #54 • WebAssign : Sec 1.2 and 1.3 • Quiz #3: function characteristics (1.1 to 1.3) MAT 145

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