Write an outline for sections 6.2-6.4. Work on the solutions to Section 6.1: exercises 21 and 36, Section 6.2: exercises 54 and 93, Section 6.3: exercises 44 and 64, and Section 6.4: exercises 26 and 74.

Submit a report in the comment box of this homework detailing your progress and status toward completion. The end of the report should contain a brief summary stating whether the outlines and exercises were given sufficient effort (whether the correct answer was derived or not).

I only want you to report on your homework activity (for each problem, whether it was found to be difficult, obvious, easy/hard, whether you at first had troubles but eventually found the/a solution, etc.) I would like you to document (i.e., report on) your homework experience.

Assignment 1: Questions from the 9th edition Calculus Textbook 6.1{21, 36}, 6.2{54, 93}, 6.3{44, 64}, 6.4{26, 74} Exercise 6.1 #21: The formula C = 3(F 32), when F 2 459.67, expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Q: Find a formula for the inverse function, f'l, and interpret it. What is the domain of the inverse function? Exercise 6.1 #36: Given f(:r) = \\3/1 3: a) Find g'l. How is it related to g? b) Graph 91. How do you explain your answer to part a? Exercise 6.2 # 54: Find an equation of the tangent line to the curve 3369 + ye" : 1 at the point (0, 1). Exercise 6.2 # 93: Find the average value of x) = 22219532 on the interval [0, 2]. Exercise 6.3 # 44: Find the domain of the function 90\"?) = 108:2 (1'2 '1' 333)- Exereise 6.3 # 64: A prime number is a positive integer that has no factors other than 1 and itself. The rst few primes are 2, 3, 5, 7, ll, 13, 17, We denote by (11.) the number of primes that are less than or equal to n. For instance, 7r(15) = 6 because there are six primes smaller than 15. a) b) c) Calculate the numbers (25) and (100) [Hintz To find 7r(100), first compile a list of the primes up to 100 using the sieve of Eratosthenes: Write the numbers from 2 to 100 and cross out all multiples of 2. Then cross out all multiples of 3. The next remaining number is 5, so cross out all remaining multiples of it, and so on.] By inspecting tables of prime numbers and tables of logarithms, the great mathe- matician K. F. Gauss made the guess in 1792 (when he was 15) that the number of primes up to n is approximately n/ ln(n) when 'n, is large. More precisely, he conjectured that _ 7r(7r) 11111 = 1 THUG n/ 111(n) This was nally proved, a hundred years later, by Jacques Hadamard and Charles de la Valle Poussin and is called the Prime Number Theorem. Provide evidence for the truth of this theorem by computing the ratio of 7r(n) to n/ ln(n) for n : 100,1000,104,105,106,107.. Use the following data: 7r(1000) = 168, (104) = 1229, 17(105) = 9592, (106) = 78498, (107) = 664579. Use the Prime Number Theorem to estimate the number of primes up to a billion. Exercise 6.4 #26: Q: Differentiate the function, y = iog2(:rlog5(2:)). Exercise 6.4 #74: The table gives the US population from 1790 to 1860. year population 1790 3,929,000 1800 5,308,000 1810 7,240,000 1820 9,639,000 1830 12,861,000 1840 17,063,000 1850 23,192,000 1860 31,443,000 a) Fit an exponential function to the data. Graph the data points and the exponential model. How good is the t? 1)) Estimate the rates of population growth in 1800 and 1850 by averaging slopes of secant lines. (I) Use the exponential model in part (a) to estimate the rates of growth in 1800 and 1850. Compare these estimates with the ones in part (b). d) Use the exponential model to predict the population in 1870. Compare with the actual population of . Can you explain the discrepancy