# graphs intersect

1 Section 9.1.3: 9) Consider the graphs of y = 𝒙𝟒 and y = 𝒙𝟔 (where x is a real variable). At what values of x do the 1 Section 9.1.3: 9) Consider the graphs of y = 𝒙𝟒 and y = 𝒙𝟔 (where x is a real variable). At what values of x do the graphs intersect? On what intervals is 𝒙𝟒 > 𝒙𝟔 and on what intervals is 𝒙𝟔 > 𝒙𝟒 ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = 𝒙𝒏 and y = 𝒙𝒎 , where m, n are positive integers. 2 Section 9.2.4: 1) In the text, we explored the graphs of f (x) = 𝒙𝒏 where x is a positive integer. We now recall the graphs for negative integer exponents. a) By hand, graph the functions 𝒙−𝟏 , 𝒙−𝟐 , 𝒙−𝟑 , and 𝒙−𝟒 . Use the nonzero real numbers as the domain. b) Check your answers using a graphing calculator or computer software. c) Consider the function 𝒇(𝒙) = 𝒙𝒏 , where n is a negative integer. Make a conjecture on the values of 𝐥𝐢𝐦 𝒙𝒏 , 𝐥𝐢𝐦 𝒙𝒏 , 𝐥𝐢𝐦+ 𝒙𝒏 , 𝒂𝒏𝒅 𝐥𝐢𝐦− 𝒙𝒏 . 𝒙→∞ 𝒙→−∞ 𝒙→𝟎 𝒙→𝟎 d) What is the range of f ? e) How does this exercise help in understanding why 𝟎𝒏 is not defined if n is a negative integer? Section 9.4.2: 3) Recall that a sequence of real numbers is an arithmetic sequence if the difference between consecutive terms is constant, and a sequence of (nonzero) real numbers is a geometric sequence if the ratio of consecutive terms is constant. Prove that if { 𝒙𝒊 } is an arithmetic sequence, then { 𝜶𝒙𝒊 } is a geometric sequence. (Here α is any positive real number.) 3 6) Let f (x) = 𝜶𝒙 for some unknown α. Suppose that the value of f is known for one value of x (say, f (c) = λ). Show that this information completely determines f. (Solve for α in terms of c and λ .) 12) Let a > 1. One of the most important features of the exponential function 𝒂𝒙 is that it increases much more rapidly than any polynomial function as x → ∞ . We illustrate this property with the exponential function 𝟐𝒙 and the polynomial function 𝒙𝟓 . a) Create a table of values for 𝒙𝟓 , 𝟐𝒙 , 𝒂𝒏𝒅 𝟐𝒙 − 𝒙𝟓 , for integers 1 ≤ k ≤ 40. You can do this with a spreadsheet, or you can use Mathematica with the following commands: f[k_]:=kˆ5; g[k_]:=2ˆk; t=Table[{f[k],g[k],g[k]-f[k]}, {k,1,40}]; TableForm[t,TableHeadings->{Automatic,{“kˆ5″,”2ˆk”,”2ˆk-kˆ5″}}] b) Which value (𝒙𝟓 𝒐𝒓 𝟐𝒙 ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x ↦ 𝒙𝟓 and x ⟼ 𝟐𝒙 cross at (at least) two places between x = 1 and x = 40. Use graphing technology to generate clear graphs of these crossing points. d) One might try to argue that 𝟐𝒙 is eventually greater than 𝒙𝟓 by graphing the two functions on the same axes and comparing the graphs. Try graphing the two functions on the same axes, for 1 ≤ x ≤ 40. Does the graph provide better or worse information than the spreadsheet? e) Suppose that a student looked at the spreadsheet results and the graphs, but was not convinced that 𝟐𝒙 remains larger than 𝒙𝟓 for x > 40 (what’s to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that 𝟐𝒙 > 𝒙𝟓 when x > 40? 4 18) Let f (x) = 𝐥𝐨𝐠 𝟑 𝒙 and g (x) = 𝐥𝐨𝐠 𝟐 𝒙 . In this exercise, you will show that the graphs of f and g are related to each other by a graphing transformation. a) Using properties of logarithms, show that g (x) = c f (x) for some constant c (ﬁnd c ). This shows that the graph of g is obtained from the graph of f by a vertical stretch. b) Graph f and g on the same axes using a graphing utility. How could you convince another person that g (x) = c f (x), using only the graph? c) Based on your answer to (a), how do you expect the derivatives of g and f to be related to each other? d) In general, how are the graphs of 𝐥𝐨𝐠 𝜶 𝒙 and 𝐥𝐨𝐠 𝜷 𝒙 related to each other?
Purchase answer to see full attachment? On what intervals is 𝒙𝟒 > 𝒙𝟔 and on what intervals is 𝒙𝟔 > 𝒙𝟒 ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = 𝒙𝒏 and y = 𝒙𝒎 , where m, n are positive integers. 2 Section 9.2.4: 1) In the text, we explored the graphs of f (x) = 𝒙𝒏 where x is a positive integer. We now recall the graphs for negative integer exponents. a) By hand, graph the functions 𝒙−𝟏 , 𝒙−𝟐 , 𝒙−𝟑 , and 𝒙−𝟒 . Use the nonzero real numbers as the domain. b) Check your answers using a graphing calculator or computer software. c) Consider the function 𝒇(𝒙) = 𝒙𝒏 , where n is a negative integer. Make a conjecture on the values of 𝐥𝐢𝐦 𝒙𝒏 , 𝐥𝐢𝐦 𝒙𝒏 , 𝐥𝐢𝐦+ 𝒙𝒏 , 𝒂𝒏𝒅 𝐥𝐢𝐦− 𝒙𝒏 . 𝒙→∞ 𝒙→−∞ 𝒙→𝟎 𝒙→𝟎 d) What is the range of f ? e) How does this exercise help in understanding why 𝟎𝒏 is not defined if n is a negative integer? Section 9.4.2: 3) Recall that a sequence of real numbers is an arithmetic sequence if the difference between consecutive terms is constant, and a sequence of (nonzero) real numbers is a geometric sequence if the ratio of consecutive terms is constant. Prove that if { 𝒙𝒊 } is an arithmetic sequence, then { 𝜶𝒙𝒊 } is a geometric sequence. (Here α is any positive real number.) 3 6) Let f (x) = 𝜶𝒙 for some unknown α. Suppose that the value of f is known for one value of x (say, f (c) = λ). Show that this information completely determines f. (Solve for α in terms of c and λ .) 12) Let a > 1. One of the most important features of the exponential function 𝒂𝒙 is that it increases much more rapidly than any polynomial function as x → ∞ . We illustrate this property with the exponential function 𝟐𝒙 and the polynomial function 𝒙𝟓 . a) Create a table of values for 𝒙𝟓 , 𝟐𝒙 , 𝒂𝒏𝒅 𝟐𝒙 − 𝒙𝟓 , for integers 1 ≤ k ≤ 40. You can do this with a spreadsheet, or you can use Mathematica with the following commands: f[k_]:=kˆ5; g[k_]:=2ˆk; t=Table[{f[k],g[k],g[k]-f[k]}, {k,1,40}]; TableForm[t,TableHeadings->{Automatic,{“kˆ5″,”2ˆk”,”2ˆk-kˆ5″}}] b) Which value (𝒙𝟓 𝒐𝒓 𝟐𝒙 ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x ↦ 𝒙𝟓 and x ⟼ 𝟐𝒙 cross at (at least) two places between x = 1 and x = 40. Use graphing technology to generate clear graphs of these crossing points. d) One might try to argue that 𝟐𝒙 is eventually greater than 𝒙𝟓 by graphing the two functions on the same axes and comparing the graphs. Try graphing the two functions on the same axes, for 1 ≤ x ≤ 40. Does the graph provide better or worse information than the spreadsheet? e) Suppose that a student looked at the spreadsheet results and the graphs, but was not convinced that 𝟐𝒙 remains larger than 𝒙𝟓 for x > 40 (what’s to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that 𝟐𝒙 > 𝒙𝟓 when x > 40? 4 18) Let f (x) = 𝐥𝐨𝐠 𝟑 𝒙 and g (x) = 𝐥𝐨𝐠 𝟐 𝒙 . In this exercise, you will show that the graphs of f and g are related to each other by a graphing transformation. a) Using properties of logarithms, show that g (x) = c f (x) for some constant c (ﬁnd c ). This shows that the graph of g is obtained from the graph of f by a vertical stretch. b) Graph f and g on the same axes using a graphing utility. How could you convince another person that g (x) = c f (x), using only the graph? c) Based on your answer to (a), how do you expect the derivatives of g and f to be related to each other? d) In general, how are the graphs of 𝐥𝐨𝐠 𝜶 𝒙 and 𝐥𝐨𝐠 𝜷 𝒙 related to each other?
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