College of education mathematics department
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- Điều hướng trang này:
- 5.2. Application of holder inequality
| Exercise 5.4. A salmon swims upstream to cover a distance of . The speed of the water flow is . If the swimming speed of the fish in still water is then the energy consumed by the fish in t hours is given by the formula , where c is a constant number, E is in joules. Find the speed of the fish when the water is at rest so that the energy consumption is minimal.
Solution: With a self velocity of , and a water velocity of , then The speed of the salmon moving upstream is : . The time it takes for the salmon to cross upstream is : . Thus, the amount of energy consumed by the salmon is: (jun). Use Cauchy's inequality. The equal sign is reached when . So if the salmon's own speed is then its energy expenditure is the lowest. Exercise 5.5. From a square piece of corrugated iron with a side of 4dm, people cut out a fan with center O and radius OA=4dm (see picture) to roll it into a conical funnel (then OA coincides with OB). Calculate the height of the funnel. Solution: We have arc AB of length . Based on the problem, we can see that it is possible to form a vertex cone O, generating line OA. To roll into a conical funnel (when OA coincides with OB ), the circumference C of the base circle is equal to the arc length AB by 2π. Then the base radius is . Consider a right triangle OLA at I with OA=4dm,LA=R=1dm. h=OI where So . 5.2. Application of holder inequality5.2.1. Applications in calculus5.2.1.1. Integral inequality Exercise 5.6: Given two continuous functions f(x) and g(x) determined on [0;1] and get the same value on the interval [0;1]. Prove: Solution: Applying the consequence of the Hölder inequality, we get: (1) Since , so: Infer , it implies that : (2) From (1) and (2) it follows that: tải về 344.73 Kb. Chia sẻ với bạn bè của bạn:
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