Question 34 3.5 Linear Approximation 239described the population density (in cells per liter)as a function of t (in hours).34. In Example 4 of Section 1.5, we modeled the tidesfor Toms Cove in Assateague Beach, Virginia, onAugust 19, 2004 with the functiona. Find a function r (t) such thatITН(() — 1.8 сos– 11) +2.2P'(t) = r (t) P(t)b. Under the assumption that the growth rate ofthe population is proportional to the intensity oflight, what is the period of the light fluctuations inthe chamber?where His the height of the tide (in feet) and t is thetime (in hours after midnight). Find and interpretdHdt lt=635. The temperature in a swimming pool is given byc. Determine at what times (if any) the populationis decreasing in abundance.33. In a more ambitious experiment with algae, aresearch scientist manipulated the light in algaltanks so that the population density P(t) (in cellsper liter) as a function of t (in hours) is given byT(t) = 25 – 4 sin(nt/12) degrees Celsiuswhere t is measured in hours since midnight.a. Find T'(12) and interpret.b. During what hours is the temperature of the poolincreasing?36. The human heart goes through cycles of contrac-tion and relaxation (called systoles). During thesecycles, blood pressure goes up and down repeatedly;as the heart contracts, pressure rises, and as the heartrelaxes (for a split second), the pressure drops. Con-sider the following approximate function for theblood pressure of a patient:P(t) = 5,000 ecos t+ta. Find a function r (t) such thatP'(t) = r(t) P(t)b. Under the assumption that the growth rate ofthe population is proportional to the intensity oflight, what is the period of the light fluctuations in) mmHgP(t) = 100 + 20 costhe tank?c. Determine at what times (if any) the populationis decreasing in abundance.where t is measured in minutes. Find and interpretP'(t).3.5Linear ApproximationWe have seen that the tangent line is the line that just touches a curve at a singlepoint. In this section, we will discover that the tangent line can be used to provide areasonable apnroximation to a curve Using these linear approximations we will be

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34. In Example 4 of Section 1.5, we modeled the tides for Toms Cove in Assateague Beach, Virginia, on August 19, 2004 with the function H(t) = 1.8 cos [( – 11] + 2.2 where His the height of the tide (in feet) and t is the time (in hours after midnight). Find and interpret dH dt l=6