Use the Fundamental Theorem of Calculus to find the area of the region bounded by the x-axis and the graph of y = 4 x3 − 4 x.
Answer:
(1) Use the Fundamental Theorem of Calculus to find the average value of f(x) = e0.9 x between x = 0 and x = 2.
Answer:
(2) Draw the graph of f(x) on your calculator. Find the average value on the graph. Finally give the exact x value for which f(x) = the average value.
Answer for the x value:
To use the Fundamental Theorem of Calculus to find the area of the region bounded by the x-axis and the graph of a function, you need to follow these steps:
1. Integrate the function from the lower x-value to the upper x-value.
In this case, we need to integrate the function y = 4x^3 - 4x from x = a to x = b.
∫[a to b] (4x^3 - 4x) dx
2. Evaluate the antiderivative of the function.
The antiderivative of 4x^3 is (4/4)x^4 = x^4, and the antiderivative of -4x is (-4/2)x^2 = -2x^2.
∫[a to b] (4x^3 - 4x) dx = [x^4 - 2x^2] from a to b
3. Evaluate the definite integral by substituting the upper x-value and the lower x-value into the antiderivative and subtracting the result.
Area = [b^4 - 2b^2] - [a^4 - 2a^2]
For the given function y = 4x^3 - 4x, and the region bounded by the x-axis and the graph of the function, we need to find the area by following the above steps.
Now, for the second question:
To find the average value of a function using the Fundamental Theorem of Calculus, follow these steps:
1. Evaluate the definite integral of the function from the lower x-value to the upper x-value.
In this case, we need to integrate the function f(x) = e^(0.9x) from x = a to x = b.
∫[a to b] e^(0.9x) dx
2. Evaluate the antiderivative of the function.
The antiderivative of e^(0.9x) is (1/0.9)e^(0.9x) = (10/9)e^(0.9x).
∫[a to b] e^(0.9x) dx = [(10/9)e^(0.9x)] from a to b
3. Evaluate the definite integral by substituting the upper x-value and the lower x-value into the antiderivative and subtracting the result.
Average Value = [(10/9)e^(0.9b)] - [(10/9)e^(0.9a)] / (b - a)
For the given function f(x) = e^(0.9x), and the interval [0, 2], we need to find the average value by following the above steps.
Lastly, to find the exact x-value for which f(x) equals the average value, you need to solve the equation f(x) = avg, where f(x) is the function and avg is the average value.