FDWKCalcSM_ch3

# FDWKCalcSM_ch3 - 86 Section 3.1 54(a f(2 = x 2 a 2 x =(2 a...

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86 Section 3.1 54. (a) fx a x () 2 22 =−− =− 42 2 a a (b) 242 2 =−= 2 48 4 2 (c) For x 2, f is continuous. For x = 2, we have lim ( ) ( ) ( ) xx f →→ −+ == = 24 as long as a 2. 55. (a) gx x x x 3 2 (b) x x = + = 32 2 21 3 1 = + 3 3 x x 3 3 1 = Chapter 3 Derivatives Section 3.1 Derivative of a Function (pp. 99–108) Exploration 1 Reading the Graphs 1. The graph in Figure 3.3b represents the rate of change of the depth of the water in the ditch with respect to time. Since y is measured in inches and x is measured in days, the derivative dy dx would be measured in inches per day. Those are the units that should be used along the y -axis in Figure 3.3b. 2. The water in the ditch is 1 inch deep at the start of the first day and rising rapidly. It continues to rise, at a gradually decreasing rate, until the end of the second day, when it achieves a maximum depth of 5 inches. During days 3, 4, 5, and 6, the water level goes down, until it reaches a depth of 1 inch at the end of day 6. During the seventh day it rises again, almost to a depth of 2 inches. 3. The weather appears to have been wettest at the beginning of day 1 (when the water level was rising fastest) and driest at the end of day 4 (when the water level was declining the fastest). 4. The highest point on the graph of the derivative shows where the water is rising the fastest, while the lowest point (most negative) on the graph of the derivative shows where the water is declining the fastest. 5. The y -coordinate of point C gives the maximum depth of the water level in the ditch over the 7-day period, while the x -coordinate of C gives the time during the 7-day period that the maximum depth occurred. The derivative of the function changes sign from positive to negative at C , indicating that this is when the water level stops rising and begins falling. 6. Water continues to run down sides of hills and through underground streams long after the rain has stopped falling. Depending on how much high ground is located near the ditch, water from the first day’s rain could still be flowing into the ditch several days later. Engineers responsible for flood control of major rivers must take this into consideration when they predict when floodwaters will “crest,” and at what levels. Quick Review 3.1 1. hh h h +− = ++ − 0 2 0 2 4 44 4 ( ) =+ =+= h h 0 4 404 2. x x + + = + = 2 3 2 23 2 5 2 3. Since y y 1 for y y y y <= 01 0 , lim . 4. x x x = 28 2 2 2 = + = lim ( ) ( ) h x 4 2 4 2 8 5. The vertex of the parabola is at (0, 1). The slope of the line through (0, 1) and another point ( h , h 2 + 1) on the parabola is . h h h 2 11 0 = Since lim , h h = 0 0 the slope of the line tangent to the parabola at its vertex is 0. 6. Use the graph of f in the window [ 6, 6] by [ 4, 4] to find that (0, 2) is the coordinate of the high point and (2, 2) is the coordinate of the low point. Therefore, f is increasing on ( −∞ , 0] and [2, ).

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## This note was uploaded on 09/26/2010 for the course MATH 135 taught by Professor Noone during the Summer '08 term at Rutgers.

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FDWKCalcSM_ch3 - 86 Section 3.1 54(a f(2 = x 2 a 2 x =(2 a...

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