PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
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This 19 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Staff in Fall. Since its upload, it has received 5 views. For similar materials see /class/207051/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.
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Date Created: 09/19/15
Lecture 21 Section 32 Logarithmic Functions f Recall the graph of exponential function f a a gt 1 01 Since f is one to one it has an inverse function Def The logarithmic function with base a Where a gt 0 and a y 1 is written f loga 1 and is de ned by the relationship 3 loga 1 if and only if ex Write in exponential form 1 rogg 2 2 loge3x 1 2 Write in logarithmic form 41 5 8 Evaluate 1 10g2 64 2 loggl 3 log 3 1 410810 TOGO 6 10g2 1 7 logg 0 Properties of Logarithms 1 Recall If y loga x then 1 That is 1 gt 0 2 logal 3 loga a 4 Inverse Properties loga am for all real number 1 am for 1 gt O 5 OnetoOne Properties If loga 1 loga y then Evaluate 1 2log237r l 2 log 5 Solve log2I 3 log 9 Graphs 0f Logarithmic Functions Sketch y 2x and y 10g2 x Properties of the graph of logax Compare f logax and f 1x am fw10gaw was am 1 Domain 2 Range 3 Intercept 4 Asymptote 5 increasing if decreasing if 6 points on the graph Graph fI 10g351 1 The Natural Logarithmic Function 3 loge 1 ln 1 if and only if Note the following lnl lne Inverse Properties OnetoOne Property If lna lny then Evaluate 1 eln2m3 2 ln Solve ln12 51 ln6 Graph and nd the domain and vertical asymp tote of f 1 1nd 2 1H L 2 1 Common Logarithm Function 3 loglo 1 10g 1 if and only if Evaluate logl 10g 10 1 log 10000 log m Applications The loudness level of a sound D in decibels is given by I Where I is the intensity of a sound in watts per square meter 1 Determine the decibel level of a sound with an intensity of 10 2 watt per square meter 2 Determine the decibel level of a sound With an intensity of 1 watts per square meter 3 The intensity of a sound in part 2 is 100 times as great as the intensity in part By how much is the decible level increased Practice Determine the domain of each function 1 f 1 10g2 21f 2 gm log 12 3 Mm 1n iL 2 005 Z OO 8 S S Z OO Z 0 007 I 39Jamsuv Lecture 9 Section 15 Analyzing Graphs of Functions The graph of a function y f is the set of points x y in the cry plane that satisfy the equation Which of the following graphs represent 3 as a function of 51 Find the domain and range of each function 1y2 Vertical Line Test for Functions A set of points in the Jig plane is the graph of y as a function of 1 if and only if no vertical line intersects the graph at more than one point NOTE A function can have at most one y intercept Zeros of a Function Def The zeros of a function are the 1 Values for which Find the zeros of each function 1 fxIJ3 x2 4CIJ4 2 9x V4 512 Increasing and Decreasing Functions Def A function f is increasing on an open inter val I if for any x1 and 2 in I if 511 lt 562 then Def A function f is decreasing on an open in terval I if for any x1 and 2 in I if x1 lt x2 then Def A function f is constant on an open interval I if for any x1 and 2 in I then Relative Minimum and Maximum Def A function has a relative local mini mum at a at if there is an open interval I con taining a so that for all 1 y a in I Def A function has a relative local maxi mum at a f a if there is an open interval I con taining a so that for all 1 y a in I Consider the following graph 3 f and nd the following 1 lntervals on which f is increasing 2 lntervals on which f is decreasing 3 lntervals on which f is constant 4 Relative minimum values 5 Relative maximum values Average Rate of Change Def The average rate of change of f from 1 to 2 is mam 2 yfx NOTE The average rate of Change of a function from 1 to 2 is the Slope of the secant line through the points x1f511 and x2 f512 Given f x2 3 Find the average rate of Change of f from 1 t0 4 The distance 305 in feet traveled by a ball rolling down a ramp is given by the function 3t 5752 Where t is the time in seconds after the ball is released Find the balls average velocity from 1 t1 2 seconds to t2 21 seconds 2 t1 2 seconds to t2 201 seconds 3 t1 2 seconds to t2 2001 seconds NOTE In calculus this procedure leads to the concept of instantaneous velocity Even and Odd Functions Def A function f is even if for each a in the domain of f f x NOTE f is even if and only if Whenever the point x y is on the graph of f the point 1331 is also on the graph That is f is even if and only if the graph of f is symmetric with respect to Def A function is odd if for each 8 in the domain of f f x NOTE f is odd if and only if Whenever the point x y is on the graph of f the point 51 y is also on the graph That is f is odd if and only if the graph of f is symmetric with respect to NOTE What about symmetry With respect to the x axis Determine if the function is even odd or neither Graph the function 1 ms x2 3hxx2
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