Find the average rate of change of the function over the giv

Solution

We have given the problem function as f(x)= (x^3) - (1/4)e^x we have given the points as [0, 5] so, put x = 0 f(0) = (0)^3 - (1/4e^0) = 0 - 1/4 = -1/4 (as e^0 = 1) put x = 5 f(5) = (5)^3 - 1/4(e^5) = 125 - e^5/4 = 125 - 37.1 = 87.9 (as e = 2.7) the average rate of change of the function over the given interval will be f(5) - f(0)/(5 - 0) = (87.9 + 0.25)/5 = 17.63 Now, for instantaneous rate of change we have f(x)= (x^3) - (1/4)e^x put x = 5 f(5) = (5)^3 - 1/4(e^5) = 125 - e^5/4 = 125 - 37.1 = 87.9 Now, putting x = x1 then f(x1) = (x1^3) - (1/4)e^x1 The instantaneous rate of change at the point x = 5 is limit x1 tends to 5 [f(x1) - f(5)/(x1 - 5)] limit x1 tends to 5 [(x1^3) - (1/4)e^x1 - 87.9/(x1 - 5)] = 25 Hope this will help you!