Could someone answer this question so I understand it. Thanks
Given the supply function:
p = S(x) = 5(e^0.02x-1)
Find the average price (in dollars) over the interval [31,36].
Only the 0.02 is ^ in the e in the question.
Is this math, or Economics?
Normally, Supply is related to price, so your x would be price.
Normally, to find the average, you find the area under a curve, and divide by the range.
averageprice= 1/range *INT S(x) dx over range
= 1/5 * INT 5(e^.02x-1)
AT this point, I wonder about your statement that x is not in the exponent, I am figuring you are mistaken in that.
= INT (e.02x dx-INT dx
= 1/.02 e^.02x -x
now ovcr the range
50 e.72-50e.62-5
= 50(e1.16 -5
=154.7
double check all that
To find the average price over the interval [31, 36], we need to calculate the definite integral of the supply function within that interval and divide it by the length of the interval.
Step 1: Calculate the definite integral of the supply function within the given interval [31, 36].
The integral of the supply function is denoted as ∫S(x) dx. In this case, we integrate from 31 to 36:
∫(31 to 36) S(x) dx = ∫(31 to 36) 5(e^(0.02x) - 1) dx
Step 2: Evaluate the integral.
To integrate the function, we can use the property of the exponential function and the power rule of integration:
∫(31 to 36) 5(e^(0.02x) - 1) dx
= (5/0.02) ∫(31 to 36) e^(0.02x) dx - ∫(31 to 36) 5 dx
Step 3: Evaluate each integral separately.
Let's evaluate each integral step by step:
a) ∫(31 to 36) e^(0.02x) dx
To integrate e^(0.02x) with respect to x, we use the rule for integrating exponential functions:
= (5/0.02) [e^(0.02x) / 0.02] + C
Note: C represents the constant of integration.
b) ∫(31 to 36) 5 dx
When integrating a constant with respect to x, we multiply the constant by the interval length:
= 5 * (36 - 31)
Step 4: Evaluate the expression obtained from Step 3a and Step 3b and calculate the average price.
Substitute the limits of integration (31 and 36) into the expression obtained in Step 3a and evaluate it. Then calculate the average price:
Average price = (1 / (36-31)) * [(5/0.02) * (e^(0.02*36) / 0.02) - (5/0.02) * (e^(0.02*31) / 0.02) + 5 * (36 - 31)]
You can simplify the expression and calculate the numerical value of the average price using a calculator.
To find the average price over the interval [31,36], we need to compute the definite integral of the supply function from 31 to 36 and then divide it by the length of the interval.
First, let's clarify the notation. The expression "e^0.02x" means raising the number e (approximately 2.71828) to the power of 0.02x. The "^" symbol is the exponentiation operator, so "e^0.02x" is the same as "e^(0.02x)".
Now, let's calculate the average price. The supply function is given by:
p = S(x) = 5(e^(0.02x) - 1)
To find the average price over the interval [31,36], we need to evaluate the definite integral of the supply function from 31 to 36. The integral is denoted as ∫S(x)dx.
∫S(x)dx = ∫[31,36] 5(e^(0.02x) - 1)dx
To compute this integral, we can use the power rule of integration. The integral of e^(0.02x) is (1/0.02) * e^(0.02x), and the integral of 1 with respect to x is x. So, we have:
∫S(x)dx = 5 * ∫[31,36] (1/0.02) * e^(0.02x)dx - 5 * ∫[31,36]dx
Integrating the first term gives us:
5 * (1/0.02) * ∫[31,36] e^(0.02x)dx = 250 * [e^(0.02x)] [31,36]
Plugging in the limits of integration, we get:
250 * (e^(0.02*36) - e^(0.02*31))
To find the average price, we need to divide this value by the length of the interval, which is 36 - 31 = 5:
Average price = (250 * (e^(0.02*36) - e^(0.02*31))) / 5
Now, you can calculate this expression using a calculator or computer software to find the average price over the interval [31,36].