21,23 and 29 please EXERCISES 1.5Is the function, whose graph is drawn in Fig. 7, continuous at thefollowing values of x?-5 -4 -3 -2Figure 71. x = 04. x = .00115. f(x) =16. f(x) =[x³18. f(x) =^Is the function, whose graph is drawn in Fig. 7, differentiable at thefollowing values of x?7. x = 010. x = .00117. f(x) = {1ƒ(x) = { 2(x+2 for-1 ≤x≤1(3xfor 1 < x≤ 5Determine whether each of the following functions is continuousand/or differentiable at x = 1.13. f(x) = x²19. f(x)=x2. x=-35. x = -2[2x-1 for 0≤x≤1for 1 < x20. f(x) = 18. x = -311. x = -2for 0≤x<1for 1 ≤ x ≤ 2x-1for x 1for x = 1(2x-21214. f(x)==for x1for x = 1for 0 ≤ x < 1for x = 1for x>13 43. x = 36. x = 29. x = 312. x = 2+x5XThe functions in Exercises 21-26 are defined for all x except forone value of x. If possible, define f(x) at the exceptional point in away that makes f(x) continuous for all x.21. f(x)==22. f(x)=23. f(x)24. f(x)=-25. f(x) =x²7x+10x-5x5x²+x-12x +4x3 - 5x2 +4x²x² +25X-5x5-,x-4(6 + x)² - 36x0x0X26. f(x) =√9+x-√9x0X27. Computing Income Tax The tax that you pay to the federalgovernment is a percentage of your taxable income, which iswhat remains of your gross income after you subtract yourallowed deductions. In a recent year, there were five rates orbrackets for a single taxpayer, as shown in Table 1.Table 1 Single Taxpayer RatesBut Not OverAmount OverSO$27,050$27,050$65,550$65,550$136,750$136,750$297,350$297,350Tax Rate15%27.5%30.5%35.5%39.1%So, if you are single and your taxable income was lessthan $27,050, your tax is your taxable income times 15%(.15). The maximum amount of tax that you will pay onyour income in this first bracket is 15% of $27,050, or(15) X 27,050 4057.50 dollars. If your taxable incomeis more than $27,050 but less than $65,550, your tax is$4057.50 plus 27.5% of the amount in excess of $27,050. So,for example, if your taxable income is $50,000, your tax is4057.5+275(50,000 -27,050) = 4057.5+275 x 22,950 =$10,368.75. Let x denote your taxable income and 7(x)your tax. (a) Find a formula for 7(x) if x is not over $136,750.(b) Plot the graph of 7(x) for 0≤x≤ 136,750.(c) Find the maximum amount of tax that you will pay onthe portion of your income in the second tax bracket..Express this amount as a difference between two valuesof 7(x).28. Refer to Exercise 27.(a) Find a formula for 7(x) for all taxable income x.(b) Plot 7(x).(c) Determine the maximum amount of tax that you will payon the portion of your income in the fourth tax bracket.29. Revenue from Sales The owner of a photocopy store charges7 cents per copy for the first 100 copies and 4 cents per copyfor each copy exceeding 100. In addition, there is a setup feeof $2.50 for each photocopying job.(a) Determine R(x), the revenue from selling x copies.(b) If it costs the store owner 3 cents per copy, what is theprofit from selling x copies? (Recall that profit is revenueminus cost.)30. Do Exercise 29 if it costs 10 cents per copy for the first 50copies and 5 cents per copy for each copy exceeding 50, andthere is no setup fee.31. Department Store Sales The graph in Fig. 8 shows the totalsales in thousands of dollars in a department store during atypical 24-hour period.(a) Estimate the rate of sales during the period between8 A.M. and 10 A.(b) Which 2-hour interval in the day sees the highest rate ofsales and what is this rate?Solutions to Check Your Understanding 1.51. The function f(x) is defined at x = 3, that is, f(3) = 4. Whencomputing lim f(x), we exclude consideration of x = 3;therefore, we can simplify the expression for f(x) as follows:x-3(x-3)(x + 2)x-3x²-x-6f(x)x-3lim /(x) = lim (x + 2) = 5.=32. Refer to Exercise 31.(a) From midnight to noon, which 2-hour time intervals havethe same rate of sales and what is this rate?=x+2(b) What is the total amount of sales between midnight and8 A.M.? Compare this amount to the total sales in theperiod between 8 A.M. and 10 A.M.In Exercises 33 and 34, determine the value of a that makes thefunction f(x) continuous at x = 0.[1 for x ≥ 033. f(x)=x+a for x < 0Since lim f(x) = 5*4=f(3), f(x) is not continuous atx-x = 3.[2(x-a)34. f(x)=x²+1Sales in Thousands of Dollars2420161.6 Some Rules for Differentiation 97128402 A.M.Figure 8for x 20for x < 06 A.M.12 noon6 PM. 10 PM.2. There is no need to compute any limits to answer this ques-tion. By Theorem I, since f(x) is not continuous at x = 3, itcannot possibly be differentiable there.

us at the +x 5 ble at the The functions in Exercises 21-26 are defined for all x except for one value of x. If possible, define f(x) at the exceptional point in a way that makes f(x) continuous for all .x. x²7x+10 21. f(x) = x-5 x²+x-12 x+4 x3 - 5x2 +4 x² x² +25 X-5+X5 (6 + x)² - 36 X √9+x-√9 26. f(x) = x 0 X 27. Computing Income Tax The tax that you pay to the federal government is a percentage of your taxable income, which is what remains of your gross income after you subtract your 22. f(x) = 23. f(x) = 24. f(x) = x5 25. f(x)= x = -4 **0 **0

us at the +x 5 ble at the The functions in Exercises 21-26 are defined for all x except for one value of x. If possible, define f(x) at the exceptional point in a way that makes f(x) continuous for all .x. x²7x+10 21. f(x) = x-5 x²+x-12 x+4 x3 - 5x2 +4 x² x² +25 X-5+X5 (6 + x)² - 36 X √9+x-√9 26. f(x) = x 0 X 27. Computing Income Tax The tax that you pay to the federal government is a percentage of your taxable income, which is what remains of your gross income after you subtract your 22. f(x) = 23. f(x) = 24. f(x) = x5 25. f(x)= x = -4 0 0

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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