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which then reduces to the function 1+x f(x, y) = 1. + y (a) Suppose that the changes in power x and drag y are small. Use the linear approximation of f to describe the effect of a small fractional increase in power versus a small fractional decrease in drag. (b) Based on the level curves of f(r, y) for the values c= -0.1,0,0.1,0.2, 0.3, what can you say about the speed if we increase the power P by 50% while the drag C remains unchanged? What if we decrease the drag C by 50% while the power P remains unchanged? What can you conclude from these changes?

which then reduces to the function 1+x f(x, y) = 1. + y (a) Suppose that the changes in power x and drag y are small. Use the linear approximation of f to describe the effect of a small fractional increase in power versus a small fractional decrease in drag. (b) Based on the level curves of f(r, y) for the values c= -0.1,0,0.1,0.2, 0.3, what can you say about the speed if we increase the power P by 50% while the drag C remains unchanged? What if we decrease the drag C by 50% while the power P remains unchanged? What can you conclude from these changes?

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2. The speed v of an object being propelled through water is given by 3(2P kC. v(P, C) = where P is the power being used to propel the object, C is the drag coefficient, and k is a positive constant. Swimmers can therefore increase their swimming speeds by increasing their power or reducing their drag coefficients. To compare the effect of increasing power versus reducing drag, we need to somehow compare the two in common units. A frequently used approach is to determine the percentage change 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 x corresponding to 100x percent), P changes from P to P+xP. Likewise, if the drag coefficient is changed by a fraction y, then C changes from C to C + yC. Then, the corresponding fractional change in speed is v(P+zР,С +УС) — v(Р,С) v(P,C)
Transcribed Image Text:2. The speed v of an object being propelled through water is given by 3(2P kC. v(P, C) = where P is the power being used to propel the object, C is the drag coefficient, and k is a positive constant. Swimmers can therefore increase their swimming speeds by increasing their power or reducing their drag coefficients. To compare the effect of increasing power versus reducing drag, we need to somehow compare the two in common units. A frequently used approach is to determine the percentage change 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 x corresponding to 100x percent), P changes from P to P+xP. Likewise, if the drag coefficient is changed by a fraction y, then C changes from C to C + yC. Then, the corresponding fractional change in speed is v(P+zР,С +УС) — v(Р,С) v(P,C)
which then reduces to the function 1+x f(x, y) 1. + y (a) Suppose that the changes in power x and drag y are small. Use the linear approximation of f to describe the effect of a small fractional increase in power versus a small fractional decrease in drag. (b) Based on the level curves of f(x, y) for the values c = -0.1,0, 0.1, 0.2, 0.3, what can you say about the speed if we increase the power P by 50% while the drag C remains unchanged? What if we decrease the drag C by 50% while the power P remains unchanged? What can you conclude from these changes?
Transcribed Image Text:which then reduces to the function 1+x f(x, y) 1. + y (a) Suppose that the changes in power x and drag y are small. Use the linear approximation of f to describe the effect of a small fractional increase in power versus a small fractional decrease in drag. (b) Based on the level curves of f(x, y) for the values c = -0.1,0, 0.1, 0.2, 0.3, what can you say about the speed if we increase the power P by 50% while the drag C remains unchanged? What if we decrease the drag C by 50% while the power P remains unchanged? What can you conclude from these changes?
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