If a company’s sales (in millions of dollars) for time t months is given by
f(t) = 0.003t^3 + 0.5t^2 + 2t + 3
a. Find f'(t).
b. Find f'(5).
c. What does this tell you about the company’s position in the month of May (i.e.,t = 5)?
f ' (t) = .009t^2 + t + 2
f ' (5) = .009(25) + 5 + 2 = 7.225
sales in May were rising at 7,225,000 units per month
To find the derivative of the function f(t), we can use the power rule, which states that if we have a function of the form f(t) = at^n, then its derivative is given by f'(t) = n*at^(n-1).
Now, let's find f'(t) using the power rule:
f(t) = 0.003t^3 + 0.5t^2 + 2t + 3
Taking the derivative of each term separately, we get:
f'(t) = 3 * 0.003t^(3-1) + 2 * 0.5t^(2-1) + 2 * 1t^(1-1) + 0
Simplifying this expression gives:
f'(t) = 0.009t^2 + t + 2
a. Therefore, f'(t) = 0.009t^2 + t + 2.
b. To find f'(5), we substitute t = 5 into the expression for f'(t):
f'(5) = 0.009(5)^2 + 5 + 2
= 0.009(25) + 5 + 2
= 0.225 + 5 + 2
= 7.225
c. The value f'(5) represents the instantaneous rate of change of sales with respect to time at t = 5, which tells us how fast the company's sales are changing at that point in time. In this case, f'(5) = 7.225, indicating that the company's sales are increasing at a rate of approximately 7.225 million dollars per month in May. Therefore, this suggests that the company's position in May is positive, as their sales are growing.