Annual sales (in millions of units) of pocket computers are expected to grow in accordance with the following function where t is measured in years, with t = 0 corresponding to 1997.
f(t) = 0.18t^2 + 0.16t + 2.64 (0<=t<=2)
How many pocket computers will be sold over the 2 year period between the beginning of 1997 and the end of 1998?
that would be f(2)-f(0) = 3.68 - 2.64
To find out how many pocket computers will be sold over the two-year period between the beginning of 1997 and the end of 1998, we need to evaluate the definite integral of the function f(t) over the interval [0, 2].
The integral of a function represents the area under the curve defined by the function. To evaluate the integral of f(t), we can use the power rule for integrals:
∫ (a*t^n) dt = (a/(n+1)) * t^(n+1) + C
Applying this rule to each term in the function f(t), we get:
∫ (0.18t^2) dt = (0.18/(2+1)) * t^(2+1) + C = 0.06t^3 + C1
∫ (0.16t) dt = (0.16/(1+1)) * t^(1+1) + C = 0.08t^2 + C2
∫ (2.64) dt = 2.64t + C3
Where C1, C2, and C3 are constants of integration.
Now, we can evaluate the definite integral over the interval [0, 2]:
∫[0,2] (0.18t^2 + 0.16t + 2.64) dt = 0.06t^3 + 0.08t^2 + 2.64t + C | [0,2]
Plugging in the upper limit (2) and lower limit (0), we get:
(0.06(2)^3 + 0.08(2)^2 + 2.64(2) + C) - (0.06(0)^3 + 0.08(0)^2 + 2.64(0) + C)
= (0.06*8 + 0.08*4 + 2.64*2) - (0.06*0 + 0.08*0 + 2.64*0 + C)
Simplifying further:
(0.48 + 0.32 + 5.28) - (0 + 0 + 0 + C)
= 6.08 - C
Since we don't have the value of the constant of integration (C), we cannot determine the exact number of pocket computers sold. However, we can evaluate the definite integral (∫[0,2] (0.18t^2 + 0.16t + 2.64) dt) which is equal to 6.08 - C, giving us an approximation of the number of pocket computers sold.
To find the number of pocket computers sold over the 2-year period, we need to integrate the sales function f(t) over the interval from t = 0 (1997) to t = 2 (end of 1998).
The integral of f(t) with respect to t over the interval [0, 2] gives us the total number of pocket computers sold during this period.
We can calculate the integral as follows:
∫[0, 2] f(t) dt = ∫[0, 2] (0.18t^2 + 0.16t + 2.64) dt
To evaluate this integral, we will take the antiderivative of each term and then apply the limits of integration:
= (0.06t^3 + 0.08t^2 + 2.64t) | from 0 to 2
Substituting the upper and lower limits into the antiderivative, we get:
= (0.06(2)^3 + 0.08(2)^2 + 2.64(2)) - (0.06(0)^3 + 0.08(0)^2 + 2.64(0))
Simplifying the calculation, we have:
= (0.06(8) + 0.08(4) + 5.28) - (0 + 0 + 0)
= (0.48 + 0.32 + 5.28) - 0
= 6.08
Therefore, over the 2-year period (from the beginning of 1997 to the end of 1998), the number of pocket computers sold is approximately 6.08 million units.