1. Find the average value have of the function h on the given interval.
h(x) = 2 cos4(x) sin(x), [0, π]
2. Consider the given function and the given interval.
f(x) = 6 sin(x) − 3 sin(2x), [0, π]
(a) Find the average value fave of f on the given interval.
(b) Find c such that fave = f(c). (Round your answers to three decimal places.)
3. Find the numbers b such that the average value of
f(x) = 3 + 10x − 9x2
on the interval [0, b] is equal to 4.
4. In a certain city the temperature (in °F) t hours after 9 AM was modeled by the function
T(t) = 48 + 19 sin
πt
12
.
Find the average temperature Tave during the period from 9 AM to 9 PM. (Round your answer to the nearest whole number.)
1. To find the average value of a function h(x) over an interval [a, b], you need to integrate the function over that interval and then divide the result by the length of the interval (b-a).
For the given function h(x) = 2cos^4(x)sin(x), on the interval [0, π], the average value is given by:
Average value = (1/(π-0)) * ∫(from 0 to π) [2cos^4(x)sin(x)] dx
To find the integral of this function, you can use integration techniques such as substitution or integration by parts.
2. (a) To find the average value of a function f(x) over an interval [a, b], you need to integrate the function over that interval and then divide the result by the length of the interval (b-a).
For the given function f(x) = 6sin(x) - 3sin(2x), on the interval [0, π], the average value is given by:
Average value = (1/(π-0)) * ∫(from 0 to π) [6sin(x) - 3sin(2x)] dx
To find the integral of this function, you can apply the properties of integrals and the trigonometric identities.
(b) To find the value of c such that fave = f(c), you need to set up an equation by equating the average value with f(c) and solve for c:
fave = f(c)
(1/(π-0)) * ∫(from 0 to π) [6sin(x) - 3sin(2x)] dx = f(c)
Evaluate the integral on the left-hand side and find the value of c that satisfies the equation.
3. To find the number b such that the average value of a function f(x) on the interval [0, b] is equal to a given value (4 in this case), you need to set up an equation and solve for b.
The average value of f on the interval [0, b] is given by:
Average value = (1/(b-0)) * ∫(from 0 to b) [3 + 10x - 9x^2] dx
Set the average value equal to the given value of 4:
(1/(b-0)) * ∫(from 0 to b) [3 + 10x - 9x^2] dx = 4
Solve this equation to find the value of b.
4. To find the average temperature Tave during the period from 9 AM to 9 PM, you need to integrate the temperature function T(t) over the 12-hour period (from t=0 to t=12) and then divide the result by the length of the period (12 hours).
The temperature function is given by:
T(t) = 48 + 19sin((πt)/12)
The average temperature is given by:
Tave = (1/(12-0)) * ∫(from 0 to 12) [48 + 19sin((πt)/12)] dt
Evaluate the integral on the right-hand side and round the result to the nearest whole number to find the average temperature.
Recall that the average value of f(x) on the interval [a,b] is
∫[a,b] f(x) dx
--------------------
b-a
So,
#1.
(∫[0,π] 2 cos4x sinx dx)/π = -4 / 15π
#2. f(x) = 6 sin(x) − 3 sin(2x), [0, π]
avg value is 12/π
So, where is f(x) = 12/π ?
#3. f(x) = 3 + 10x - 9x^2
You want
(∫[0,b] f(x) dx)/4 = 4
(3x + 5x^2 - 3x^3)[0,b] = 4*4
3b + 5b^2 - 3b^3 = 16
b = -1.4724
#4. Fix your formatting so we can read the actual function.
Then apply the rules followed above.