graphs intersect

1 Section 9.1.3: 9) Consider the graphs of y = π’™πŸ’ and y = π’™πŸ” (where x is a real variable). At what values of x do the 1 Section 9.1.3: 9) Consider the graphs of y = π’™πŸ’ and y = π’™πŸ” (where x is a real variable). At what values of x do the graphs intersect? On what intervals is π’™πŸ’ > π’™πŸ” and on what intervals is π’™πŸ” > π’™πŸ’ ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = 𝒙𝒏 and y = π’™π’Ž , 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 π’™βˆ’πŸ , π’™βˆ’πŸ , π’™βˆ’πŸ‘ , and π’™βˆ’πŸ’ . 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 π₯𝐒𝐦 𝒙𝒏 , π₯𝐒𝐦 𝒙𝒏 , π₯𝐒𝐦+ 𝒙𝒏 , 𝒂𝒏𝒅 π₯π’π¦βˆ’ 𝒙𝒏 . π’™β†’βˆž π’™β†’βˆ’βˆž π’™β†’πŸŽ π’™β†’πŸŽ d) What is the range of f ? e) How does this exercise help in understanding why πŸŽπ’ 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 { π’™π’Š } is an arithmetic sequence, then { πœΆπ’™π’Š } is a geometric sequence. (Here Ξ± is any positive real number.) 3 6) Let f (x) = πœΆπ’™ for some unknown Ξ±. Suppose that the value of f is known for one value of x (say, f (c) = Ξ»). Show that this information completely determines f. (Solve for Ξ± in terms of c and Ξ» .) 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 β†’ ∞ . We illustrate this property with the exponential function πŸπ’™ and the polynomial function π’™πŸ“ . a) Create a table of values for π’™πŸ“ , πŸπ’™ , 𝒂𝒏𝒅 πŸπ’™ βˆ’ π’™πŸ“ , for integers 1 ≀ k ≀ 40. You can do this with a spreadsheet, or you can use Mathematica with the following commands: f[k_]:=kΛ†5; g[k_]:=2Λ†k; t=Table[{f[k],g[k],g[k]-f[k]}, {k,1,40}]; TableForm[t,TableHeadings->{Automatic,{“kΛ†5″,”2Λ†k”,”2Λ†k-kΛ†5″}}] b) Which value (π’™πŸ“ 𝒐𝒓 πŸπ’™ ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x ↦ π’™πŸ“ and x ⟼ πŸπ’™ 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 πŸπ’™ is eventually greater than π’™πŸ“ 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 πŸπ’™ remains larger than π’™πŸ“ for x > 40 (what’s to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that πŸπ’™ > π’™πŸ“ when x > 40? 4 18) Let f (x) = π₯𝐨𝐠 πŸ‘ 𝒙 and g (x) = π₯𝐨𝐠 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 π₯𝐨𝐠 𝜢 𝒙 and π₯𝐨𝐠 𝜷 𝒙 related to each other?
Purchase answer to see full attachment? On what intervals is π’™πŸ’ > π’™πŸ” and on what intervals is π’™πŸ” > π’™πŸ’ ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = 𝒙𝒏 and y = π’™π’Ž , 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 π’™βˆ’πŸ , π’™βˆ’πŸ , π’™βˆ’πŸ‘ , and π’™βˆ’πŸ’ . 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 π₯𝐒𝐦 𝒙𝒏 , π₯𝐒𝐦 𝒙𝒏 , π₯𝐒𝐦+ 𝒙𝒏 , 𝒂𝒏𝒅 π₯π’π¦βˆ’ 𝒙𝒏 . π’™β†’βˆž π’™β†’βˆ’βˆž π’™β†’πŸŽ π’™β†’πŸŽ d) What is the range of f ? e) How does this exercise help in understanding why πŸŽπ’ 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 { π’™π’Š } is an arithmetic sequence, then { πœΆπ’™π’Š } is a geometric sequence. (Here Ξ± is any positive real number.) 3 6) Let f (x) = πœΆπ’™ for some unknown Ξ±. Suppose that the value of f is known for one value of x (say, f (c) = Ξ»). Show that this information completely determines f. (Solve for Ξ± in terms of c and Ξ» .) 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 β†’ ∞ . We illustrate this property with the exponential function πŸπ’™ and the polynomial function π’™πŸ“ . a) Create a table of values for π’™πŸ“ , πŸπ’™ , 𝒂𝒏𝒅 πŸπ’™ βˆ’ π’™πŸ“ , for integers 1 ≀ k ≀ 40. You can do this with a spreadsheet, or you can use Mathematica with the following commands: f[k_]:=kΛ†5; g[k_]:=2Λ†k; t=Table[{f[k],g[k],g[k]-f[k]}, {k,1,40}]; TableForm[t,TableHeadings->{Automatic,{“kΛ†5″,”2Λ†k”,”2Λ†k-kΛ†5″}}] b) Which value (π’™πŸ“ 𝒐𝒓 πŸπ’™ ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x ↦ π’™πŸ“ and x ⟼ πŸπ’™ 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 πŸπ’™ is eventually greater than π’™πŸ“ 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 πŸπ’™ remains larger than π’™πŸ“ for x > 40 (what’s to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that πŸπ’™ > π’™πŸ“ when x > 40? 4 18) Let f (x) = π₯𝐨𝐠 πŸ‘ 𝒙 and g (x) = π₯𝐨𝐠 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 π₯𝐨𝐠 𝜢 𝒙 and π₯𝐨𝐠 𝜷 𝒙 related to each other?
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