Calculate the average rate of change of f(x) over the interval [-4, -1] using the formula [ BLANK ] The value of f(-1) is [BLANK
] The value of f(-4) is .[BLANK] The average rate of change of f(x) over the interval [-4, -1] is .[BLANK] First set of answers: A. [f(-4)-f(-1)]/[-5]
B.[f(-1)-f(-4)]/[-5]
C.[f(-1)-f(-4)]/[3]
D.[f(-4)-f(-1)]/[3]
Second set of answers: A.3 Third set: A.-4 Final set A.-2
B.2 B.-3 B.-1.5
C.-1 C.4 C.2
D.4
Answers: The formula is [f(-1)-f(-4)]/[3] The value of f(-1) is 3 The value of f(-4) is -3 The average rate of change is 2
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Explanation:
For the first blank, we use the formula [ f(b) - f(a) ]/[ b - a ] where 'a' and 'b' are the endpoints for the x interval
In this case, a = -4 and b = -1. When you plug those values into the formula above, you get... [ f(b) - f(a) ]/[ b - a] [ f(-1) - f(-4)]/[ -1 - (-4) ] [ f(-1) - f(-4)]/[ -1+4 ] [ f(-1) - f(-4)]/[ 3 ] which is why the answer is choice C for the first blank
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To compute the value of f(-1), we draw a vertical line through -1 on the x axis. This vertical line crosses the diagonal function graph at the point (-1,3). The y value of this point is what we want. Plugging in x = -1 leads to y = 3. This is why f(-1) = 3 If you want, you can draw a horizontal line through (-1,3) and you'll see it touching 3 on the y axis.
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Follow similar steps as above to compute f(-4). Draw a vertical line through x = -4 on the x axis. Mark the point where the vertical line crosses the diagonal line. This point is (-4,-3). Optionally draw a horizontal line over til you hit the y axis and you'll find that y = -3 corresponds to x = -4
This is why f(-4) = -3
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We'll use the last three sections to compute the average rate of change. Everything combines together building up to this moment.
From the first part, we had the formula [ f(b) - f(a) ]/[ b - a ] [ f(-1) - f(-4)]/[ 3 ]
We can replace the "f(-1)" with 3 since we found that f(-1) = 3 Similarly, f(-4) = -3 so we can replace the "f(-4)" with -3
Doing those replacements and simplifying leads to...
Note: because the entire graph is a straight line, the average rate of change for any interval a < x < b is going to be equal to the slope m. In this case, the slope of the line is m = 2/1 = 2. We move up 2 units each time we move to the right 1 unit along the diagonal line.
