The instantaneous rate of change is the change in the rate at a particular instant, and it is same as the change in the derivative value at a specific point. For a graph, the instantaneous rate of change at a specific point is the same as the tangent line slope. That is, it is a curve slope.
Another way to better grasp this definition is with the differential quotient and limits. The average rate of y shift with respect to x is the quotient of difference.
The Formula of Instantaneous Rate of Change represented with limit exists in,
With respect to x, when x=a and y = f[x]
Solved Example
Problem 1: Compute the Instantaneous rate of change of the function f[x] = 3x2 + 12 at x = 4 ?
Answer:
Known Function,
y = f[x] = 3x2 + 12
f'[x] = 3[2x] + 0
f'[x] =6x
Thus, the instantaneous rate of change at x = 4
f'[4] = 6[4]
f'[4] = 24
Problem 2: Compute the Instantaneous rate of change of the function f[x] = 5x3 – 4x2 + 2x + 1 at x = 2?
Answer:
Known Function,
y = f[x] = 5x3 – 4x2 + 2x + 1
f'[x] = 5[3x2] – 4[2x] + 2 + 0
f'[x] = 15x2 – 8x + 2
Thus, the instantaneous rate of change at x = 2
f'[2] = 15[2]2 – 8[2] + 2 = 60 – 16 + 2 = 46
f'[2] = 46
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3 Answers
[a].
The function is P[x] = 20x - 0.01x² - 100.
Substitute x = 20 in the above equation.
P[20] = 20[20] - 0.01[20]² - 100
P[20] = 400 - 4 - 100
P[20] = $296.
The profit for 20 trees is $296.
answered Oct 25, 2014 by Expert[b].
The function is P[x] = 20x - 0.01x² - 100.
Calculate the average change in the interval [a, b] by using formula: [f[b] - f[a]] / [b - a].
Substitute a = 11 and b = 16 in the above formula.
Average change = [P[16] - P[11]] / [16 - 11]
= {[20[16] - 0.01[16]² - 100] - [20[11] - 0.01[11]² - 100]} / 5
= {217.44 - 118.79} / 5
= 98.65 / 5
= 19.73
The Average change in profit sales from 11 to 16 tress is $19.73 per tree.
answered Oct 25, 2014 by casacop Expert[c].
The function is P[x] = 20x - 0.01x² - 100.
To find the instantaneous rate of change, differentiate with respect to x.
P'[x] = 20 - 0.02x
Substitute x = 20 in the above equation.
P'[20] = 20 - 0.02[20]
P'[20] = 19.6
The instantaneous rate of change of profit at sales level 20 tress is $19.6 per tree.
answered Oct 25, 2014 by casacop ExpertRelated questions
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