# graphs intersect

1 Section 9.1.3: 9) Consider the graphs of y = ππ and y = ππ (where x is a real variable). At what values of x do the 1 Section 9.1.3: 9) Consider the graphs of y = ππ and y = ππ (where x is a real variable). At what values of x do the graphs intersect? On what intervals is ππ > ππ and on what intervals is ππ > ππ ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = ππ and y = ππ , where m, n are positive integers. 2 Section 9.2.4: 1) In the text, we explored the graphs of f (x) = ππ where x is a positive integer. We now recall the graphs for negative integer exponents. a) By hand, graph the functions πβπ , πβπ , πβπ , and πβπ . Use the nonzero real numbers as the domain. b) Check your answers using a graphing calculator or computer software. c) Consider the function π(π) = ππ , where n is a negative integer. Make a conjecture on the values of π₯π’π¦ ππ , π₯π’π¦ ππ , π₯π’π¦+ ππ , πππ
π₯π’π¦β ππ . πββ πβββ πβπ πβπ d) What is the range of f ? e) How does this exercise help in understanding why ππ 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 { ππ } is an arithmetic sequence, then { πΆππ } is a geometric sequence. (Here Ξ± is any positive real number.) 3 6) Let f (x) = πΆπ 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 ππ is that it increases much more rapidly than any polynomial function as x β β . We illustrate this property with the exponential function ππ and the polynomial function ππ . a) Create a table of values for ππ , ππ , πππ
ππ β ππ , for integers 1 β€ k β€ 40. You can do this with a spreadsheet, or you can use Mathematica with the following commands: f[k_]:=kΛ5; g[k_]:=2Λk; t=Table[{f[k],g[k],g[k]-f[k]}, {k,1,40}]; TableForm[t,TableHeadings->{Automatic,{“kΛ5″,”2Λk”,”2Λk-kΛ5″}}] b) Which value (ππ ππ ππ ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x β¦ ππ and x βΌ ππ 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 ππ is eventually greater than ππ by graphing the two functions on the same axes and comparing the graphs. Try graphing the two functions on the same axes, for 1 β€ x β€ 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 ππ remains larger than ππ for x > 40 (whatβs to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that ππ > ππ when x > 40? 4 18) Let f (x) = π₯π¨π π π and g (x) = π₯π¨π π π . 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 (ο¬nd 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₯π¨π πΆ π and π₯π¨π π· π related to each other?

Purchase answer to see full attachment? On what intervals is ππ > ππ and on what intervals is ππ > ππ ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = ππ and y = ππ , where m, n are positive integers. 2 Section 9.2.4: 1) In the text, we explored the graphs of f (x) = ππ where x is a positive integer. We now recall the graphs for negative integer exponents. a) By hand, graph the functions πβπ , πβπ , πβπ , and πβπ . Use the nonzero real numbers as the domain. b) Check your answers using a graphing calculator or computer software. c) Consider the function π(π) = ππ , where n is a negative integer. Make a conjecture on the values of π₯π’π¦ ππ , π₯π’π¦ ππ , π₯π’π¦+ ππ , πππ
π₯π’π¦β ππ . πββ πβββ πβπ πβπ d) What is the range of f ? e) How does this exercise help in understanding why ππ 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 { ππ } is an arithmetic sequence, then { πΆππ } is a geometric sequence. (Here Ξ± is any positive real number.) 3 6) Let f (x) = πΆπ 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 ππ is that it increases much more rapidly than any polynomial function as x β β . We illustrate this property with the exponential function ππ and the polynomial function ππ . a) Create a table of values for ππ , ππ , πππ
ππ β ππ , for integers 1 β€ k β€ 40. You can do this with a spreadsheet, or you can use Mathematica with the following commands: f[k_]:=kΛ5; g[k_]:=2Λk; t=Table[{f[k],g[k],g[k]-f[k]}, {k,1,40}]; TableForm[t,TableHeadings->{Automatic,{“kΛ5″,”2Λk”,”2Λk-kΛ5″}}] b) Which value (ππ ππ ππ ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x β¦ ππ and x βΌ ππ 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 ππ is eventually greater than ππ by graphing the two functions on the same axes and comparing the graphs. Try graphing the two functions on the same axes, for 1 β€ x β€ 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 ππ remains larger than ππ for x > 40 (whatβs to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that ππ > ππ when x > 40? 4 18) Let f (x) = π₯π¨π π π and g (x) = π₯π¨π π π . 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 (ο¬nd 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₯π¨π πΆ π and π₯π¨π π· π related to each other?

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