The marginal cost function for the manufacturing of headphones is given by C'(x)=20-x/200 where x is the number of headphones made. Use a right riemann sum with n=5 to estimate the cost of producing the first 5 headphones.
Since each rectangle has width=1, just add up the function values:
1(C(1)+C(2)+C(3)+C(4)+C(5))
But how do I get C to plug the values in? I take the antiderivative and the extra variable for the end is unknown. What point do I have to find that so I am able to plug values in?
My bad. The total cost is the area under the curve for C'. So the sums involve C'(x), not C(x).
To estimate the cost of producing the first 5 headphones using a right Riemann sum, we need to use the marginal cost function and divide the interval into subintervals.
The right Riemann sum formula is given by:
Σ [f(xi) * Δx]
Where:
Σ represents the sum
f(xi) represents the function evaluated at the right endpoint of each subinterval
Δx represents the width of each subinterval
In this case, the function f(x) represents the marginal cost function C'(x), which is given by C'(x) = 20 - x/200.
To calculate the right Riemann sum with n = 5, we need to divide the interval [0, 5] into 5 subintervals of equal width:
Δx = (b - a) / n
Δx = (5 - 0) / 5
Δx = 1
Now, let's calculate the right Riemann sum by evaluating the function at the right endpoint of each subinterval (xi) and multiplying it by the width of each subinterval (Δx):
Σ [f(xi) * Δx]
= [f(1) * 1] + [f(2) * 1] + [f(3) * 1] + [f(4) * 1] + [f(5) * 1]
Now let's substitute the function f(x) with C'(x) = 20 - x/200 and evaluate it at each xi:
= [(20 - 1/200) * 1] + [(20 - 2/200) * 1] + [(20 - 3/200) * 1] + [(20 - 4/200) * 1] + [(20 - 5/200) * 1]
= (20 - 1/200) + (20 - 2/200) + (20 - 3/200) + (20 - 4/200) + (20 - 5/200)
Now we can simplify the expression:
= (20 - 1/200) + (20 - 2/200) + (20 - 3/200) + (20 - 4/200) + (20 - 5/200)
= (20 - 0.005) + (20 - 0.01) + (20 - 0.015) + (20 - 0.02) + (20 - 0.025)
= 20 - 0.005 - 0.01 - 0.015 - 0.02 - 0.025 + 20
= 19.925
Therefore, the estimated cost of producing the first 5 headphones using a right Riemann sum is approximately 19.925.