Calc
Storyboard Text
- CHAPTER 1...
- Hmmm.... Okay, I'll have to do some analysis back in my lab
- Yes! For now, we can only produce between 10 and 90 units at a time.
- This should be doable, is there anything else I should know?
- This means that 'x' is bounded on the interval [10,90]. I have determined the first derivative to be Using the first derivative, I can find the intervals of increasing and decreasing for the production cost. This should give me a better idea of where I should be looking.
- Our production cost is modeled byWhat number of production units 'x' would give me the minimum cost?
- In the lab...
- My work indicates a minimum, but just to be certain, I can confirm this using two methods. The first is the 2nd Derivative Test, which will determine whether this value is a local maximum or minimum, and the second is the Closed Interval Method, which will determine if this value is an absolute maximum or minimum on the interval:
- dwdw
- Second derivative test:Closed interval method:
- C'(x)=0 at ≈ 42.45134 (crit. pt.)First derivative test:C'(40)=-0.61, C'(45)≈0.653This means C(x) is decreasing on [10,42.45134), and increasing on (42.45134,90].
- The cost function is concave up throughout the current closed interval. It’s also important to note that this means there are no points of inflection on this graph since the 2nd derivative never changes signs on the interval.
- The 2nd Derivative Test has confirmed that the critical point I found is a local minimum, and the closed interval method has confirmed that the aforementioned critical point is an absolute minimum on the interval [10,90] of C(x). For my own knowledge in the future, I will also check the concavity of the cost function.
- C''(x)=0 when x ≈ 136.84 and C''(90)=10.2, C''(10)≈0.214.
- The production cost is minimized when producing ~42.4513 units, which rounds to 42 units. This will reduce the production cost to ~$342.58 per unit.
- This is great! Initially, we assumed that producing the lowest number of units would have the lowest cost, thankfully you showed us the real best way.
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