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Lecture

# Exponential Growth, Max & Min Values, Fermat's Theorem, Closed Inteval Method

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University of Waterloo

Mathematics

MATH 137

Jennifer Nissen

Fall

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Lec 19 Exponential GrowthOctober2611831 AMExponential Growth Text 38Last Lecture We know that ifis a constant then Theorem page 237The only solution to the differential equation is Example 183Suppose we have a bacteria culture initially containing 100 cells and growing at a rate proportional to its size Let represent the of cells in the culture at time hoursAfter an hour there are 420 cells QuestionsHow many cells after 3 hrsaWhat is the growth rate at 3 hrsb cellshourWhen does the population reach 10000cMATH 137 Page 1Last LectureExponential Decay p239Radiation substances decay at a rate proportional to their mass Let be the mass at time Newtons Law of Cooling p240An object cools at a rate proportional to the difference in temperaturebetween the object and its surroundings Let be the temperature difference at time Related RatesProblems where were interested in the rate of change of a quantity ie Its derivative given the rate of change derivative of a related quantityEx 191 A cube has volumeside lengthThe side length expands as time passes and so increasesGiven are functions of timeFind an expression relating the rate at which the volume changes and the rate the side length changesa We knowDifferentiate with respect to If the sides of the cube expand at a rate of 10cmmin how fast is the volume increasing when the side length is b2cmGiven3If the volume is increasing at 100cmmin how fast is the surface area of the cube increasing when the side c MATH 137 Page 2

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