Find the average rate of change of g(x) = -2x2 + 36x - 142 on the interval from x = -5 to x = 10.
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To find the average rate of change, we need to calculate the slope of the secant line that connects two points on the graph of the function. The slope of a line is given by the difference in y-coordinates divided by the difference in x-coordinates.
In this case, we need to find the difference quotient of the function g(x) = -2x^2 + 36x - 142 over the interval from x = -5 to x = 10.
Step 1: Evaluate g(x) at x = -5
g(-5) = -2(-5)^2 + 36(-5) - 142
= -2(25) - 180 - 142
= -50 - 180 - 142
= -372
Step 2: Evaluate g(x) at x = 10
g(10) = -2(10)^2 + 36(10) - 142
= -2(100) + 360 - 142
= -200 + 360 - 142
= 18
Step 3: Find the difference in y-coordinates
Δy = g(10) - g(-5)
= 18 - (-372)
= 18 + 372
= 390
Step 4: Find the difference in x-coordinates
Δx = 10 - (-5)
= 10 + 5
= 15
Step 5: Calculate the average rate of change
Average rate of change = Δy / Δx
= 390 / 15
= 26
Therefore, the average rate of change of g(x) = -2x^2 + 36x - 142 on the interval from x = -5 to x = 10 is 26.