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Unformatted text preview: Chapter 7 The Chain Rule, Related Rates, and Implicit Differentiation 7.1 For each of the following, find the derivative of y with respect to x . (a) y 6 + 3 y 2 x 7 x 3 = 0 (b) e y + 2 xy = √ 3 (c) y = x cos x 7.2 Consider the growth of a cell, assumed spherical in shape. Suppose that the radius of the cell increases at a constant rate per unit time. (Call the constant k , and assume that k > 0.) (a) At what rate would the volume, V , increase ? (b) At what rate would the surface area, S , increase ? (c) At what rate would the ratio of surface area to volume S/V change? Would this ratio increase or decrease as the cell grows? [Remark: note that the answers you give will be expressed in terms of the radius of the cell.] 7.3 Growth of a circular fungal colony A fungal colony grows on a flat surface starting with a single spore. The shape of the colony edge is circular (with the initial site of the spore at the center of the circle.) Suppose the radius of the colony increases at a constant rate per unit time. (Call this constant C .) (a) At what rate does the area covered by the colony change ? (b) The biomass of the colony is proportional to the area it occupies (factor of proportionality α ). At what rate does the biomass increase? v.2005.1  September 4, 2009 1 Math 102 Problems Chapter 7 7.4 Limb development During early development, the limb of a fetus increases in size, but has a constant proportion. Suppose that the limb is roughly a circular cylinder with radius r and length l in proportion l/r = C where C is a positive constant. It is noted that during the initial phase of growth, the radius increases at an approximately constant rate, i.e. that dr/dt = a. At what rate does the mass of the limb change during this time? [Note: assume that the density of the limb is 1 gm/cm 3 and recall that the volume of a cylinder is V = Al where A is the base area (in this case of a circle) and l is length.] 7.5 A rectangular trough is 2 meter long, 0 . 5 meter across the top and 1 meter deep. At what rate must water be poured into the trough such that the depth of the water is increasing at 1 m / min when the depth of the water is 0 . 7 m? 7.6 Gas is being pumped into a spherical balloon at the rate of 3 cm 3 /s . (a) How fast is the radius increasing when the radius is 15 cm ? (b) Without using the result from (a), find the rate at which the surface area of the balloon is increasing when the radius is 15 cm . 7.7 A point moves along the parabola y = 1 4 x 2 in such a way that at x = 2 the xcoordinate is increasing at the rate of 5 cm/s. Find the rate of change of y at this instant. 7.8 Boyle’s Law In chemistry, Boyle’s law describes the behaviour of an ideal gas: This law relates the volume occupied by the gas to the temperature and the pressure as follows: PV = nRT where n, R are positive constants....
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 Winter '09
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