The Cost (in dollars) of producing x units of a certain commodity is C(x) = 500 + 10x + 0.005x
2
i. Find the average rate of change of C with respect x when the production level is from x = 100 to
x = 105 units. [2]
ii. Find the instantaneous rate of change of C with respect to x when x = 100.
Where the average change=∆y/∆x
marginal change in production=(y2-y1)/(x2-x1)
At x=100
C(x) = 500 + 10x + 0.005x²?
If yes then this is all you need to do
C(100)=500+10(100)+0.005(100)²
=500+1000+50=1550
at x=105
C(105)=500+10(105)+0.005(105)²
=500+1050+55.125
=1,605.125
Y2=1605.125 ,y1=1550
X1=100 X2=105
average change=∆y/∆x=(1605.125 -1550)/(101-100)=55.125
Instant change of x would respect to c just say
C(x)=500 + 10x + 0.005x²?
C'(x)=10+2x(0.005)
C'(x)=10+0.01x
Where x=100
C'(100)=10+0.01(100)=10+1
=11
Correction
average change=∆y/∆x=(1605.125 -1550)/(105-100)=55.125/5=11.025
i. Let's find the average rate of change of C with respect to x using the given formula.
Average Rate of Change (ARC) = (C(105) - C(100)) / (105 - 100)
To find C(105), substitute x = 105 in the given function:
C(105) = 500 + 10(105) + 0.005(105)^2
C(105) = 500 + 1050 + 55.125
C(105) = 1605.125
To find C(100), substitute x = 100 in the given function:
C(100) = 500 + 10(100) + 0.005(100)^2
C(100) = 500 + 1000 + 50
C(100) = 1550
Now, let's calculate the average rate of change:
ARC = (C(105) - C(100)) / (105 - 100)
ARC = (1605.125 - 1550) / (105 - 100)
ARC = 55.125 / 5
ARC = 11.025
ii. The instantaneous rate of change can be found by taking the derivative of the cost function with respect to x:
C'(x) = 10 + 0.01x
To find the instantaneous rate of change when x = 100, substitute x = 100 in the derivative function:
C'(100) = 10 + 0.01(100)
C'(100) = 10 + 1
C'(100) = 11
Therefore, the instantaneous rate of change of C with respect to x when x = 100 is 11.
To find the average rate of change of C with respect to x when the production level is from x = 100 to x = 105 units, we need to calculate the difference in C(x) divided by the difference in x.
i. Average rate of change = (C(105) - C(100)) / (105 - 100)
First, let's find C(105) using the given equation:
C(105) = 500 + 10(105) + 0.005(105)^2
Simplifying this:
C(105) = 500 + 1050 + 0.005(11025)
C(105) = 1550 + 55.125
C(105) = 1605.125
Next, let's find C(100):
C(100) = 500 + 10(100) + 0.005(100)^2
Simplifying this:
C(100) = 500 + 1000 + 0.005(10000)
C(100) = 1500 + 50
C(100) = 1550
Now we can find the average rate of change:
Average rate of change = (1605.125 - 1550) / (105 - 100)
Average rate of change = 55.125 / 5
Average rate of change = 11.025
Therefore, the average rate of change of C with respect to x when the production level is from x = 100 to x = 105 units is 11.025 dollars per unit.
ii. To find the instantaneous rate of change of C with respect to x when x = 100, we need to find the derivative of C(x) with respect to x and evaluate it at x = 100.
C(x) = 500 + 10x + 0.005x^2
Taking the derivative of C(x) with respect to x:
C'(x) = 10 + 0.01x
Now we can evaluate C'(x) at x = 100:
C'(100) = 10 + 0.01(100)
C'(100) = 10 + 1
C'(100) = 11
Therefore, the instantaneous rate of change of C with respect to x when x = 100 is 11 dollars per unit.
To find the average rate of change of C with respect to x, we need to calculate the difference in the values of C(x) at the two different production levels and divide it by the difference in the values of x.
i. Average rate of change of C with respect to x from x = 100 to x = 105:
Step 1: Substitute the values of x into the equation C(x) = 500 + 10x + 0.005x^2:
C(100) = 500 + 10(100) + 0.005(100)^2
= 500 + 1000 + 0.005(10000)
= 500 + 1000 + 50
= 1550
C(105) = 500 + 10(105) + 0.005(105)^2
= 500 + 1050 + 0.005(11025)
= 500 + 1050 + 55.125
= 1605.125
Step 2: Calculate the average rate of change:
Average Rate of Change = (C(105) - C(100)) / (105 - 100)
= (1605.125 - 1550) / (5)
= 55.125 / 5
= 11.025
Therefore, the average rate of change of C with respect to x when the production level is from x = 100 to x = 105 units is 11.025 dollars.
ii. To find the instantaneous rate of change of C with respect to x when x = 100, we can take the derivative of the function C(x) with respect to x.
Step 1: Differentiate the equation C(x) = 500 + 10x + 0.005x^2:
C'(x) = 10 + (2)(0.005)(x)
= 10 + 0.01x
Step 2: Substitute x = 100 into the derivative:
C'(100) = 10 + 0.01(100)
= 10 + 1
= 11
Therefore, the instantaneous rate of change of C with respect to x when x = 100 is 11 dollars.