At what instantaneous rate are the sales changing when t 2
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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. Show 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 ExampleProblem 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 Stay tuned with BYJU’S for more such interesting articles. Also, register to “BYJU’S – The Learning App” for loads of interactive, engaging Physics-related videos and an unlimited academic assist.
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 questionsasked Feb 18, 2015 in CALCULUS by anonymous asked Jan 22, 2015 in CALCULUS by anonymous asked Jan 22, 2015 in CALCULUS by anonymous asked Jan 8, 2015 in CALCULUS by anonymous asked Jan 22, 2015 in CALCULUS by anonymous asked Jan 20, 2015 in CALCULUS by anonymous asked Nov 1, 2014 in PHYSICS by anonymous asked May 2, 2015 in CALCULUS by anonymous asked May 2, 2015 in CALCULUS by anonymous asked Feb 4, 2015 in CALCULUS by anonymous asked Jan 22, 2015 in CALCULUS by anonymous asked Jan 22, 2015 in CALCULUS by anonymous asked Jan 22, 2015 in CALCULUS by anonymous asked Jul 26, 2014 in CALCULUS by anonymous How do you find the instantaneous rate of change?The instantaneous rate of change at some point x0 = a involves first the average rate of change from a to some other value x. So if we set h = a − x, then h = 0 and the average rate of change from x = a + h to x = a is ∆y ∆x = f(x) − f(a) x − a = f(a + h) − f(a) h . f(a + h) − f(a) h .
What is the instantaneous rate of change at a point?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.
How do you find the instantaneous rate of change from a graph at a point?To find the instantaneous rate of change using a graph, draw a line that only touches the graph at one point, known as a tangent line. Then find the slope of the tangent line to calculate the instantaneous rate of change.
What is an example of instantaneous rate of change?One example of an instantaneous rate of change is the speedometer on a car. It gives the instantaneous rate of change in position with time at each point.
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