SME
Calculus I · Problem type
The steps of an optimization problem
Use this for “A closed can must hold 1000 cm³. What radius and height use the least metal?”
Put in order
Put the steps for the can of 1000 cm³ with the least metal in order.
1
?
2
?
3
?
4
?
5
?
6
?
Assumptions
•
Optimization finds where a quantity is largest or smallest, which is where its derivative is 0 or at an end.
•
A constraint ties the variables together, so the quantity can be written with one variable.
•
Check the answer is a maximum or a minimum: compare the ends, or use the sign of the second derivative.
Topic check
See what you know: 5 questions on this topic
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Related lessons
Related rates and optimization
Main lesson
The largest field along a river
Use this for “A gardener has 16 m of fence for a plot along a straight river, with no fence on the river side. What sides give the largest area?”
Highs and lows of a cubic
Use this for “Find the local maximum and minimum of f(x) = x³ − 6x² + 9x + 1, and where the graph changes how it bends.”
A sliding ladder
Use this for “A 10 m ladder leans on a wall and its foot slides away at 1 m/s. How fast is the top sliding down when the foot is 6 m from the wall?”
Two cars moving apart
Use this for “Two cars leave a crossroads, one east at 15 km/h and one north at 20 km/h. How fast is the gap between them growing when they are 12 km and 5 km out?”
Filling a cone tank
Use this for “Water pours at 0.03 m³/s into a cone tank, point down, 1.5 m in rim radius and 3 m deep. How fast does the level rise when the water is 1 m deep?”