No idea where to start
The cost in dollars of producing x units of a particular camera is C(x) = x2 - 10000. (10 points)
Find the average rate of change of C with respect to x when the production level is changed from x = 100 to x = 101. Include units in your answer.
Find the instantaneous rate of change of C with respect to x when x = 100. Include units in your answer.
Just like taking slope
C(101) = 101^2 - 10000 = 201
C(100) = 100^2 - 10000 = 0
average rate of change = (201-0)/(101-100) = 201
instantaneous rate of change
= 2x
at x = 100
rate = 200
Well, well, well, looks like we have ourselves an arithmetic problem. Don't worry, I'm here to make it as entertaining as possible!
To find the average rate of change of C with respect to x, we need to find the change in C over the change in x. It's like analyzing the speed of a turtle (but hopefully faster than that).
So, let's get started, shall we? First, we need to find C(100) and C(101), which means plugging those values into our lovely little equation. Giddy up, math cowboy!
C(100) = (100)^2 - 10000
C(100) = 10000 - 10000
C(100) = 0
C(101) = (101)^2 - 10000
C(101) = 10201 - 10000
C(101) = 201
Now, calculate the average rate of change:
Average Rate of Change = (C(101) - C(100))/(101 - 100)
= (201 - 0)/(1)
= 201
So, the average rate of change of C with respect to x is 201 units per x. That's some snappy progress!
Now, let's move on to the instantaneous rate of change. This little sucker measures the speed of C at a specific x value. Let's find it when x = 100.
To do this, we can take the derivative of our equation with respect to x. Hold on tight, we're going for a wild ride:
C'(x) = 2x
Now, plug in x = 100 to find the instantaneous rate of change:
C'(100) = 2(100)
= 200
The instantaneous rate of change of C with respect to x when x = 100 is 200 units per x. That's twice as fast as the average rate of change! Zoom, zoom!
And there you have it, my friend. The average and instantaneous rates of change, delivered with a side of humor. Is there anything else I can assist you with?
To find the average rate of change of C with respect to x, we need to calculate the difference in C(x) divided by the difference in x for the given interval.
Step 1: Calculate C(101) and C(100):
C(101) = (101)^2 - 10000 = 10201 - 10000 = 201
C(100) = (100)^2 - 10000 = 10000 - 10000 = 0
Step 2: Calculate the difference in C:
Difference in C = C(101) - C(100) = 201 - 0 = 201
Step 3: Calculate the difference in x:
Difference in x = 101 - 100 = 1
Step 4: Calculate the average rate of change:
Average rate of change = Difference in C / Difference in x = 201 / 1 = 201
The average rate of change of C with respect to x when the production level is changed from x = 100 to x = 101 is 201 dollars per unit.
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) and evaluate it at x = 100.
Step 1: Differentiate C(x) with respect to x:
C'(x) = 2x
Step 2: Evaluate C'(x) at x = 100:
C'(100) = 2(100) = 200
The instantaneous rate of change of C with respect to x when x = 100 is 200 dollars per unit.
To find the average rate of change of C with respect to x, we need to calculate the difference between the values of C(x) at x = 101 and x = 100 and divide it by the change in x.
Step 1: Substitute x = 100 and x = 101 into the expression for C(x):
C(100) = (100)^2 - 10000
C(101) = (101)^2 - 10000
Step 2: Calculate the values of C(100) and C(101):
C(100) = 10000 - 10000 = 0
C(101) = 10201 - 10000 = 201
Step 3: Calculate the change in C(x):
Change in C(x) = C(101) - C(100) = 201 - 0
Step 4: Calculate the change in x:
Change in x = 101 - 100 = 1
Step 5: Calculate the average rate of change of C with respect to x:
Average rate of change = Change in C(x) / Change in x
Average rate of change = (201 - 0) / (1)
Average rate of change = 201
Therefore, the average rate of change of C with respect to x when the production level changes from x = 100 to x = 101 is 201 dollars per unit.
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.
Step 1: Take the derivative of C(x):
C'(x) = 2x
Step 2: Substitute x = 100 into the derivative:
C'(100) = 2(100)
C'(100) = 200
Therefore, the instantaneous rate of change of C with respect to x when x = 100 is 200 dollars per unit.