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Unformatted text preview: Math 1a: The chain rule
October 16, 2009
1. Differentiate following functions. 1. ((x - 1)(x + 3))11 2. ex 2 3. f (x) = 2x tan x 4. x+ x+ x 2. Some liquid is poured in a (conical) glass. Denote the volume of the liquid in the glass at time t by V (t). At what rate does the level of the liquid rise? Assume that the sides of the glass are slanted at an angle of /4. Express your answer in terms of V (t) and V (t). 1 3. 1 The population of large carnivores, C, on the African Savannah depends on the population of gazelles that are prey, P . The population of these gazelles, in turn, depends on the abundance of vegetation V , and this depends on the amount of rain in a given year, r. For simplicity, let us assume that the relations are given by C = P 2 , P = 2V, V = r1/2 . If there is a drought, and the rainfall changes, then there will be a change in the vegetation. This will result in a change in the gazelle population, which will eventually affect the population of carnivores on the d C in savanna. Compute the rate of change in the carnivores' population with respect to the rainfall dr terms of r. 4. Define f (x) by f (x) = a. Find f (x) for x = 0. x2 sin(1/x) 0 if x = 0 . if x = 0 b. Find f (0) using the definition of the derivative. c. Is f (x) continuous? 1 From http://www.ugrad.math.ubc.ca/coursedoc/math102/keshet.notes/chapter7Notes.pdf 2 ...
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