Problems 1-5 of this applied project. ovad100-13Eshic die13 1505 224910in RuleThe speed v of an object being propelled through water is given by1/31SECTION 14.5 The Chain Rule2PkCv(P, C) =where P is the power being used to propel the object, C is the drag coefficient, and k is a positivereducing their drag coefficients. But how effective is each of these?constant. Athletes can therefore increase their swimming speeds by increasing their power ofTo compare the effect of increasing power versus reducing drag, we need to somehow com-pare the two in common units. The most common approach is to determine the percentagechange in speed that results from a given percentage change in power and in drag.If we work with percentages as fractions, then when power is changed by a fraction x (with xcorresponding to 100x percent), P changes from P to P + xP. Likewise, if the drag coefficient ischange in speed resulting from both effects isungchanged by a fraction y, this means that it has changed from C to C + yC. Finally, the fractionalv(P + xP, C + yC) − v(P, C)v(P, C)f(x, y)1. Expression 1 gives the fractional change in speed that results from a change x in power and achange y in drag. Show that this reduces to the function-9771/3x-(1)y- 1Given the context, what is the domain of f?2. Suppose that the possible changes in power x and drag y are small. Find the linear approxima-tion to the function f(x, y). What does this approximation tell you about the effect of a smallincrease in power versus a small decrease in drag?3. Calculate fxx (x, y) and fyy(x, y). Based on the signs of these derivatives, does the linearapproximation in Problem 2 result in an overestimate or an underestimate for an increase inpower? What about for a decrease in drag? Use your answer to explain why, for changes inpower or drag that are not very small, a decrease in drag is more effective.4. Graph the level curves of f(x, y). Explain how the shapes of these curves relate to youranswers to Problems 2 and 3.Recall that the Chain Rule for functions of a single variable gives the rule for differentiat-ing a composite function: If y = f(x) and x = g(t), where f and g are differentiable func-tions, then y is indirectly a differentiable function of t anddy dx Compare with the actual new BMI.42. Suppose you need to know an equation of the tangent plane toa surface S at the point P(2, 1, 3). You don't have an equationAPPLIED PROJECTTotal number ofrecords brokeny1814...............20002002FIGURE 1 Number of world records set in long-course men's and women's freestyle swimming event 1990–2011It might be surprising that a simple reduction in drag could have such a big effect onperformance. We can gain some insight into this using a simple mathematical model.²102--1990THE SPEEDO LZR RACERperformance. One of the best known is the introduction, in 2008, of the Speedo LZR racer. It wasMany technological advances have occurred in sports that have contributed to increased athleticclaimed that this full-body swimsuit reduced a swimmer's drag in the water. Figure 1 shows thenumber of world records broken in men's and women's long-course freestyle swimming eventsto claim that such suits are a form of technological doping. As a result all full-body suits werefrom 1990 to 2011.' The dramatic increase in 2008 when the suit was introduced led peoplebanned from competition starting in 2010.WomenMen19921994the result of(b) Explain why fx and fy are not continuous at (0, 0).1996021998200420062008200620101. L. Foster et al., "Influence of Full Body Swimsuits on Competitive Performance, Procedia Engineering34 (2012): 712-17.2. Adapted from http://plus.maths.org/content/swimming.bris olde02.014.

Question

Problems 1-5 of this applied project. ovad100-13Eshic die13 1505 224910in RuleThe speed v of an object being propelled through water is given by1/31SECTION 14.5 The Chain Rule2PkCv(P, C) =where P is the power being used to propel the object, C is the drag coefficient, and k is a positivereducing their drag coefficients. But how effective is each of these?constant. Athletes can therefore increase their swimming speeds by increasing their power ofTo compare the effect of increasing power versus reducing drag, we need to somehow com-pare the two in common units. The most common approach is to determine the percentagechange in speed that results from a given percentage change in power and in drag.If we work with percentages as fractions, then when power is changed by a fraction x (with xcorresponding to 100x percent), P changes from P to P + xP. Likewise, if the drag coefficient ischange in speed resulting from both effects isungchanged by a fraction y, this means that it has changed from C to C + yC. Finally, the fractionalv(P + xP, C + yC) − v(P, C)v(P, C)f(x, y)1. Expression 1 gives the fractional change in speed that results from a change x in power and achange y in drag. Show that this reduces to the function-9771/3x-(1)y- 1Given the context, what is the domain of f?2. Suppose that the possible changes in power x and drag y are small. Find the linear approxima-tion to the function f(x, y). What does this approximation tell you about the effect of a smallincrease in power versus a small decrease in drag?3. Calculate fxx (x, y) and fyy(x, y). Based on the signs of these derivatives, does the linearapproximation in Problem 2 result in an overestimate or an underestimate for an increase inpower? What about for a decrease in drag? Use your answer to explain why, for changes inpower or drag that are not very small, a decrease in drag is more effective.4. Graph the level curves of f(x, y). Explain how the shapes of these curves relate to youranswers to Problems 2 and 3.Recall that the Chain Rule for functions of a single variable gives the rule for differentiat-ing a composite function: If y = f(x) and x = g(t), where f and g are differentiable func-tions, then y is indirectly a differentiable function of t anddy dx Compare with the actual new BMI.42. Suppose you need to know an equation of the tangent plane toa surface S at the point P(2, 1, 3). You don't have an equationAPPLIED PROJECTTotal number ofrecords brokeny1814...............20002002FIGURE 1 Number of world records set in long-course men's and women's freestyle swimming event 1990–2011It might be surprising that a simple reduction in drag could have such a big effect onperformance. We can gain some insight into this using a simple mathematical model.²102--1990THE SPEEDO LZR RACERperformance. One of the best known is the introduction, in 2008, of the Speedo LZR racer. It wasMany technological advances have occurred in sports that have contributed to increased athleticclaimed that this full-body swimsuit reduced a swimmer's drag in the water. Figure 1 shows thenumber of world records broken in men's and women's long-course freestyle swimming eventsto claim that such suits are a form of technological doping. As a result all full-body suits werefrom 1990 to 2011.' The dramatic increase in 2008 when the suit was introduced led peoplebanned from competition starting in 2010.WomenMen19921994the result of(b) Explain why fx and fy are not continuous at (0, 0).1996021998200420062008200620101. L. Foster et al., &#34;Influence of Full Body Swimsuits on Competitive Performance, Procedia Engineering34 (2012): 712-17.2. Adapted from http://plus.maths.org/content/swimming.bris olde02.014.

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