Precalc0110to0111

Precalc0110to0111 - (Section 1.10: Difference Quotients)...

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(Section 1.10: Difference Quotients) 1.10.1 SECTION 1.10: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES • Define average rate of change (and average velocity) algebraically and graphically. • Be able to identify, construct, and evaluate (and simplify) various forms of difference quotients. PART A: DISCUSSION • This section will revisit Section 0.14 on slopes of lines, Section 1.1 on function evaluations, and Section 1.2 on graphs. • A difference quotient is used to find the slope of a secant line to a graph or to find an average rate of change (perhaps an average velocity ). Difference quotients will form the basis for derivatives, tangent lines, and instantaneous rates of change in Section 1.11. PART B: SECANT LINES and AVERAGE RATE OF CHANGE The secant line to the graph of a function f on the interval a , b ± ² ³ ´ , where a < b , is the line that passes through the points a , fa () and b , fb . The average rate of change of f on a , b ± ² ³ ´ is equal to the slope of this secant line, which is given by: rise run = ± b ± a . We call this a difference quotient , because it has the form: difference of outputs difference of inputs . • (See Footnote 1 on assumptions about f .)
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(Section 1.10: Difference Quotients) 1.10.2 PART C: AVERAGE VELOCITY The following development of average velocity will help explain the association between slope and average rate of change. Example 1 (Average Velocity) A car is driven due north 100 miles during a two-hour trip. What is the average velocity of the car? • Let t = the time (in hours) elapsed since the beginning of the trip. • Let y = st () , where s is the position function for the car (in miles). s gives the signed distance of the car from the starting position. •• The position ( s ) values would be negative if the car were south of the starting position. • Let s 0 = 0 , meaning that y = 0 corresponds to the starting position. Therefore, s 2 = 100 (miles). The average velocity on the time-interval a , b ± ² ³ ´ is the average rate of change of position with respect to time . That is, change in position change in time = ± s ± t where ± (uppercase delta) denotes “change in” = sb ± sa b ± a , a difference quotient Here, the average velocity on 0, 2 ± ² ³ ´ is: s 2 ± s 0 2 ± 0 = 100 ± 0 2 = 50 miles hour or mi hr or mph ² ³ ´ µ · TIP 1 : The unit of velocity is the unit of slope given by: unit of s unit of t .
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(Section 1.10: Difference Quotients) 1.10.3 The average velocity is 50 mph on 0, 2 ± ² ³ ´ in the three scenarios below. It is the slope of the orange secant line. We will define instantaneous velocity (or simply velocity ) in Section 1.11. • Here, the velocity is constant (50 mph). • Here, the velocity is increasing; the car is accelerating. • Here, the car overshoots the destination and then backtracks. WARNING 1 : The car’s velocity is negative in value when it is backtracking; this happens when the graph falls.
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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Precalc0110to0111 - (Section 1.10: Difference Quotients)...

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