Final answer:
To find the value of c where the instantaneous rate of change equals the average rate of change for the function f(x) over [1, 3], we compute the derivative f'(x) and set it equal to the average rate of change between x=1 and x=3, then solve for c.
Step-by-step explanation:
The student is asking to find the value of c where the instantaneous rate of change of the function f(x)=2x³+3x²-40x+15 is the same as the average rate of change over the interval [1, 3]. The question involves understanding concepts of calculus, specifically the use of derivatives to find the instantaneous rate of change and the use of the slope formula (difference in function values over difference in x values) to find the average rate of change. To solve this, we will find the derivative of the function, evaluate it at c, and also compute the average rate of change over the interval [1, 3]. The value of c at which these two rates are equal will be our answer.
First, let's find the average rate of change from x=1 to x=3:
- Average rate of change = ∆f/∆x = (f(3) - f(1)) / (3 - 1) = ((2*3³ + 3*3² - 40*3 + 15) - (2*1³ + 3*1² - 40*1 + 15)) / 2
Then, let's find the derivative of the function, f'(x), and solve for c when f'(c) is equal to the average rate of change:
- f'(x) = 6x² + 6x - 40
- Set f'(c) equal to the average rate of change and solve for c.