MHF 4U 04 Rates of Change Sampler
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1 Name: Class: Date: MHF 4U 04 Rates of Change Sampler 1. The tables below shows the concentration of CO 2 in the air in a room over time. The best estimate of the instantaneous rate of change of the CO 2 level at t=0 is... time(s) CO time(s) CO time(s) CO a b c d. 0 e The slope of the tangent to the graph of y= 2x 3 +x at x=2 is a. 18 b. 25 c. 31 d. 14 1
2 Name: 3. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown. The instantaneous rate of change of y with respect to x at x=4 is... a. 0 b. 0.5 c. 1 d
3 Name: 4. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown. Which of the following is most likely to be true? a. The instantaneous rate of change of f at x=-2 is 0 b. The instantaneous rate of change of f at x=-1 is 2 3 c. The instantaneous rate of change of f at x=2 is 0 d. The instantaneous rate of change of f at x=6 is 3 3
4 Name: 5. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown. Which of the following is most likely to be true? a. The instantaneous rate of change at x=-1 does not exist b. The average rate of change between x= -1 and x= 2 is -1 c. The instantaneous rate of change at x=2 does not exist d. The average rate of change between x= 2 and x=4 is 1 6. A man kept track of his weight and daily food consumption every day for several months. He drew a graph with weight on the vertical axis and time on the horizontal axis. What units could be used for instantaneous rate of change on this graph? a. kg/day c. cal/day b. years/kg d. kg/cal 7. The graph illustrates the motion of a person walking toward or away from a CBR. Distance from the CBR is recorded on the vertical axis in metres. Elapsed time since the CBR was started up is recorded on the horizontal axis in seconds. Which of the following best describes the motion? The person is... a. walking away from the CBR and speeding up. b. walking toward the CBR and speeding up. c. walking away from the CBR and slowing down d. walking toward the CBR and slowing down. 4
5 MHF 4U 04 Rates of Change Sampler Answer Section MULTIPLE CHOICE 1. A For instantaneous rate of change, we want the limit of slopes of line segments with one endpoint at 0. The best estimate of the limit is B Type the equation into Y1 on the calculator, GRAPH (and make sure the window is appropriate), then push 2nd PRGM to select DRAW. Push 5 to select TANGENT(, then push 2 and the calculator will display X=2 on the bottom of the graph. Press ENTER, and it will draw the tangent line at x=2 and will display the equation (approximate) on the bottom of the screen. In this case the equation listed is y=25x+-32. Its slope (25) is the instantaneous rate of change of the function at x=2. 3. C The instantaneous rate of change is the slope of the tangent at (4,0). This slope is 1 (comparing y=x-4 to y=mx+b.) 4. A From the graph, the instantaneous rate of change of f at x=-1 does not exist because there is a corner. On one side, the slope is 0, but on the other side it is 2 3 Also the instantaneous rate of change of f at x=2 does not exist because there is a corner. On the left side, the slope is 2 but on the other side it is approaching 0. 3 At x=-2, the the instantaneous rate of change of f is 0 because the slope is 0. At any point between (-1,1) and (2,-1), (but not including those points), the slope is 2, so the instantaneous rate 3 of change is 2 3. At (4,0), the instantaneous rate of change is 1 because the slope of the tangent line is 1 (equation is y=1x-4) At (6,3), the instantaneous rate of change is 2 because the slope of the tangent line is 2 (equation is y=2x-9) 1
6 5. A 6. A To the left of x=-1, the instantaneous rate of change is the same as the average rate of change anywhere along it because it is a straight line. This rate is 2 (from (-2,0) to (-1,2). The instantaneous rate of change of f at x=-1 does not exist because the graph is discontinuous there. The actual function value is -1 when x is -1. Anywhere between x=-1 and x=2, the slope is 0, so both instantaneous and average rate of change is 0. If we approach x=2 along the curve from the right, the slope approaches 0, so there is no corner there. The instantaneous rate of change at x=2 is 0. At (4,0), the instantaneous rate of change is 1 because the slope of the tangent line is 1 (equation is y=1x-4) At (6,3), the instantaneous rate of change is 2 because the slope of the tangent line is 2 (equation is y=2x-9) The average rate of change between... (-2,0) and (-1,-1) is -1 (rise of -1, run of 1) between (-1,1) and (2,-1) is 0; between (2,-1) and (4.0) is 0.5 (rise of 1, run of 2); between (4,0) and (6,3) it is 1.5. slope would be change in weight, so the units could be kg/day time difference 7. B With a negative slope, we know the distance from the CBR is decreasing, so the person is walking towards the CBR. The slope is becoming more and more negative, so the speed is also increasing. 2
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