We start with straightforward global extrema problems, then we look at some optimization and at the end we do some related rates problems.

If you want to refer to sections of Methods Survey while working the exercises, you can click here and it will appear in a separate full-size window. Similarly, here we offer Theory.

In the following problems, find global extrema of the given function over the given set.

- 1.

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- 2.

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- 3.

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- 4.

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- 5.

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- 6.

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- 7.

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- 8.

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- 9.

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Solve the following optimization problems.

- 10. Find a real number
*x*that minimizes2 *x*+*x*^{2}.

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- 11. Find the
*x*that minimizes over the set of positive numbers.*x*+ 100/*x*

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- 12. Among rectangles with perimeter 40, which one has the
largest area?

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- 13. What point on the line
is nearest to the origin?*y*= 2*x*+ 2

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- 14. What point on the hyperbola
is nearest to the point*x*^{2}/2 −*y*^{2}= 1(3,0) ?

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- 15. Among rectangles inscribed into a circle of radius
*R*, which one has the largest area?

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- 16. Among rectangles inscribed into a half-circle of radius
*R*, which one has the largest area?

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- 17. Find which cylinder inscribed into a ball of radius
*R*has the largest volume.

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- 18. A corridor 3 m wide turns into a corridor 4 m wide, the two
corridors are perpendicular. What is the longest beam that you can drag on
the floor through this turn? The beam is so thin that you can ignore its
thickness and
just work with a straight segment.

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Remark: We drag the beam in order to keep the problem two-dimensional. If we allowed other positions, then obviously the best way to carry it is so that one ends trails the ground and the other touches the ceiling. Then the length of the longest beam is given by the length of the beam's projection onto the floor - and we are back to the 2D problem we ask about here. Thus the problem that we ask about also pretty much solves the general situation.

If there is no ceiling, then we can take beam of arbitrary length - we simply carry it in vertical position.

If we had to consider the width of the object - e.g. if the question was what is the longest sofa 1 m wide one can move through - then the problem becomes much more difficult, not in idea, but in calculations. If you are curious, check out this note.

Solve the following "related rates" problems.

Note: If you feel uneasy about units, just replace meters with feet and km
with miles and everything will work just the same. Can't help you with
radians, getting around them is too much work to be worth it.

- 19. A fisherman is sitting on a bank of a straight river. A fish
takes a bite when it is 2 meters from the fisherman exactly opposite him,
and starts swimming at
5 m/sec along the river. How fast is the line reeling off?

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- 20. We have a magical rectangle whose one side can be
shortened/lengthened and the other adjusts automatically so that the area of
the rectangle stays constant at
*A*. At what rate does one side shorten if we lengthen the other side at the rate*k*?

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- 21. You stand at a lamppost that has light 6 m above the ground.
You are 2 meters tall. You start walking away at
1 m/sec. How fast is the length of your shadow increasing when you are 4 meters from the lamp?

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- 22. A lamp lies on the ground 10 meters from a vertical wall. A
man 2 meters tall walks at
1/2 m/sec from the lamp directly towards this wall. How fast is the height of his shadow changing when he is halfway to the wall? Is the shade growing or getting shorter?

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- 23. An airplane flies over a radar site at elevation
5 km, it flies straight at780 km/hr. What relative speed will the radar measure when the plane is 13 km away?

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- 24. A camera positioned on the ground tracks a rocket that took
off from a launching pad that is 1 km from the camera. The rocket flies
straight up with acceleration
10 m/sec At what rate is the camera swiveling up 20 seconds after the take-off?^{2}.

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- 25. Consider a triangle with a certain angle
*a*and adjacent sides of length 4. How fast is its area changing when we start changing*a*?

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- 26. A ladder 6 meters long is lying on the ground perpendicularly
to a vertical wall, with one end right against this wall.
We tie a rope to this adjacent end and start pulling it up along the wall
at
1/2 m/sec. How fast does the other end move along the ground when the end with the rope is half the way up?

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- 27. A ladder 6 meters long is lying on the ground perpendicularly
to a vertical wall, with one end right against this wall.
We tie a rope to the opposite end and start pulling it towards the wall
(that is also
6 m tall) at1/2 m/sec, forcing the ladder to start turning about the adjacent end. How fast is the angle between the ladder and the ground changing?

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- 28. A conveyor belt pours sand on a heap at the rate
0.1 m The falling sand forms a regular cone whose height is always equal to the radius of its base. How fast is the height of the heap increasing when it is 2 meters tall?^{3}/sec.

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- 29. You are on a biking trip with your friend and at a crossing
decide to split. The two roads take you in straight and perpendicular
directions. You bike at
20 km/hr, your friend bikes at15 km/hr. How fast is your distance growing after 2 hours?

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- 30. Two roads run parallel and straight
10 km apart. On each road there is a car, one goes at50 km/hr, the other at56 km/hr in the same direction. At time 0 they are level. How fast is their distance increasing 4 hours later?

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- 31. A cylinder is rolling down a slope, picking up snow along the
way, thus increasing its radius. How fast is its volume growing depending
on the rate of change of radius?

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- 32. Picture a wheel powered by a steam engine directly via piston
and rod. The piston moves to and fro in a straight line, at its end there is
a rod 4 meters long whose other end is fixed to the rim of the wheel that has
diameter 1 meter. The center of this wheel is exactly on the axis of the
piston. The wheel turns at a constant speed so that the whole rig moves at
10 m/sec. Find the relative velocity of the piston with respect to the engine.

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