It is projected that t months from now, the average price per unit for gods in a certain sector of the economy will be P(t) = -t3 +7t2 +200t + 300 dollars.

c.Use calculus to estimate the change in the rate of price increase during the first half of the 6th month.
How can I solve this problem?

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P(t) = -t^3 + 7t^2 + 200t + 300 ?

P' (t) = -3t^2 + 14t + 200

beginning of 6th month ---> t=6
P'(6) = -108 + 84 + 200 = 176

middle of 6th month ---> t = 6.5
P'(6.5) = -126.75 + 91 + 200 = 164.25

Does that help?

To estimate the change in the rate of price increase during the first half of the 6th month, we need to calculate the derivative of the given price function, P(t), and then evaluate it at the midpoint of the first half of the 6th month.

Here's how you can solve this problem step-by-step:

Step 1: Find the derivative of the price function, P(t), with respect to time, t. The derivative will give you the rate at which the price is changing:
P'(t) = dP(t)/dt

Using the power rule of differentiation, where c is a constant:
P'(t) = -3t^2 + 14t + 200

Step 2: Determine the midpoint of the first half of the 6th month. Since a month consists of 30 days, the 6th month will have 180 days. The first half of the 6th month will be half of that, which is 90 days. To find the midpoint, divide the number of days by 2:
Midpoint = 90/2 = 45 days

Step 3: Substituting the midpoint value of t into the derivative function, P'(t), to calculate the change in the rate of price increase:
P'(45) = -3(45)^2 + 14(45) + 200

Using the order of operations (parentheses, exponents, multiplication/division, addition/subtraction), calculate the value of P'(45).

The final result will give you the estimated change in the rate of price increase during the first half of the 6th month.