By Serge Lang
This 5th version of Lang's ebook covers the entire subject matters generally taught within the first-year calculus series. Divided into 5 elements, each one component of a primary direction IN CALCULUS comprises examples and functions in relation to the subject coated. moreover, the rear of the ebook comprises precise ideas to plenty of the routines, permitting them to be used as worked-out examples -- one of many major advancements over past versions.
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Extra info for A first course in calculus, Fifth Edition
11. An iron ingot, heated to a temperature of 300◦ C, is placed in a liquid bath held at a constant temperature of 90◦ C. If the ingot cools to 250◦ C in two minutes, what will its temperature be in 20 minutes? 12. A glass of ginger ale is left in a room. Initially, the ginger ale has a temperature of 45◦ F, but after one minute the temperature has increased to 50◦ F and after two minutes it has increased to 54◦ F. What is the temperature of the room? 4 13. In his book Liber Abaci (Book of the Abacus), Leonardo of Pisa, also know as Fibonacci, posed the following question: How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?
That is, the number of owls in any given year is equal to the number of owls in the previous year plus 2% of the number of owls in the previous year. 1) is an example of a first-order difference equation; it relates the number of owls in a given year with the number of owls in the previous year. Hence we know the value of a specific xn once we know the value of xn−1 . To get the sequence started we have to know the value of x0 . 243216. 1 Plot of (n, xn ), n = 0, 1, 2, . 02xn Thus we would expect about 108 owls in the population after 4 years.
Explain why f1 = 1 and f2 = 1. (b) Explain why fn+2 = fn+1 + fn for n = 1, 2, 3, . .. (c) Compute fn for n = 3, 4, 5, 6, 7, 8 by hand. (d) Compute fn for n = 1, 2, 3, . . , 100. (e) What is lim fn ? n→∞ (f) Compute rn = fn fn+1 for n = 1, 2, 3, . . , 100. Do you think lim rn exists? If so, what is a good approxn→∞ imation for this limit to five decimal places? (g) Show that 1 rn+1 = . 1 + rn (h) Using (g) and assuming that lim rn exists, show that n→∞ √ lim rn = n→∞ 5−1 , 2 the golden section ratio.
A first course in calculus, Fifth Edition by Serge Lang