[1] F. R. Adler, Modeling the Dynamics of Life: Calculus and Probability for Life
Scientists, 2nd ... [2] B. C. Berresford and A. M. Rockett, Applied Calculus, 5th ed.
Integration Applications Global Warming: The “greenhouse effect” is the rise in temperature that the Earth experiences because certain gases prevent heat from escaping the atmosphere. According to one study, the temperature is rising at the rate of 0.014t0.4 degrees Fahrenheit per year, where t is the number of years since 2000. Given that the average surface temperature of the earth was 57.8 degrees Fahrenheit in 2000, predict the temperature in 2200. (Adapted from [2].)
Drug Sensitivity: It is important for pharmaceutical companies to test the characteristics of the drugs they design. The strength of the drug is given by R(M ) where M measures the dosage, i.e. the amount of medicine absorbed in the blood. Suppose an experimental drug changes a patient’s temperature at the rate of R(M ) = 3M 2 /(M 3 + 1) degrees per milligram of the drug. Find the total change in temperature resulting from the first 4 milligrams of the drug.
Neurons: Neurons (nerve cells) provide most of the important functions of the nervous system, such as controlling muscle activity, sensing, thinking, remembering, and regulating glandular secretions. They consist of three basic parts: the dendrites, the cell body, and the axon. Neurons are responsible for transmitting information throughout the body. Electrical signals and chemical messengers are used in order for neurons to transmit and receive information. The dendrites receive information which is then passed down to the cell body and then travels down the axon. These pulse-like waves of voltage that travel along the axons are know as action potentials. Assume that the change in voltage v of a neuron with respect to time follows the differential equation dv 1 = 1.0 + − e0.01t , dt 1 + 0.2t over the course of 100 milliseconds, where t is measured in milliseconds and v in millivolts. Assume v(0) = −70. Integrate the equation to express voltage as a function of time. What is the voltage after 100 milliseconds? (Adapted from [1].)
Seed Dispersal: The following website applies integration by parts to a model for plant seed dispersal. http://www-rohan.sdsu.edu/ jmahaffy/courses/f00/math122/lectures/intparts/intparts.html
Poiseuille’s Law: Laminar flow within a tube, for example a blood vessel, has a velocity which increases with the distance from the wall of the tube. Poiseuille’s Law states that, assuming laminar flow, the velocity of blood r cm from the central axis of the artery is given by v(r) =
P (R2 − r2 ), 4ηl
where P is the pressure difference between the ends of a blood vessel of length l and radius R, and the viscosity of the blood in the vessel is given by η (viscosity measures the resistance of the fluid to flow). (a) The flow rate can be found by integrating the velocity function over the cross sectional area of the blood vessel. Imagine dividing the vessel into concentric circles. The flow rate is given by multiplying the circumference of each circle by the velocity of blood flowing through that circle, then adding together all of these circle from r = 0 to r = R. This can be written as the integral Z R P 2π (R2 − r2 )rdr. 4ηl 0 1
Find the expression for the flow rate. (b) Find the average velocity of blood in the artery (hint: the average velocity is found by taking the flow rate and dividing by the cross-sectional area).
References [1] F. R. Adler, Modeling the Dynamics of Life: Calculus and Probability for Life Scientists, 2nd ed. Brooks Cole, 2004. [2] B. C. Berresford and A. M. Rockett, Applied Calculus, 5th ed. Cengage Learning, 2008.
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