# graphs intersect

1 Section 9.1.3: 9) Consider the graphs of y = ππ and y = ππ (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 and y = ππ (where x is a real variable). At what values of x do the graphs intersect? On what intervals is ππ > ππ and on what intervals is ππ > ππ ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = ππ and y = ππ , where m, n are positive integers. 2 Section 9.2.4: 1) In the text, we explored the graphs of f (x) = ππ where x is a positive integer. We now recall the graphs for negative integer exponents. a) By hand, graph the functions πβπ , πβπ , πβπ , and πβπ . 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) = ππ , where n is a negative integer. Make a conjecture on the values of π₯π’π¦ ππ , π₯π’π¦ ππ , π₯π’π¦+ ππ , πππ π₯π’π¦β ππ . πββ πβββ πβπ πβπ d) What is the range of f ? e) How does this exercise help in understanding why ππ 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 } is an arithmetic sequence, then { πΆππ } is a geometric sequence. (Here Ξ± is any positive real number.) 3 6) Let f (x) = πΆπ 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 ππ is that it increases much more rapidly than any polynomial function as x β β . We illustrate this property with the exponential function ππ and the polynomial function ππ . a) Create a table of values for ππ , ππ , πππ ππ β ππ , 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_]:=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 ππ ππ ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x β¦ ππ and x βΌ π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π is eventually greater than ππ 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 ππ remains larger than ππ for x > 40 (whatβs to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that ππ > ππ 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  πΆ π and π₯π¨π  π· π related to each other?
Purchase answer to see full attachment? On what intervals is ππ > ππ and on what intervals is ππ > ππ ? Give convincing explanations of your answers. 10) Generalize the last exercise to the graphs of y = ππ and y = ππ , where m, n are positive integers. 2 Section 9.2.4: 1) In the text, we explored the graphs of f (x) = ππ where x is a positive integer. We now recall the graphs for negative integer exponents. a) By hand, graph the functions πβπ , πβπ , πβπ , and πβπ . 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) = ππ , where n is a negative integer. Make a conjecture on the values of π₯π’π¦ ππ , π₯π’π¦ ππ , π₯π’π¦+ ππ , πππ π₯π’π¦β ππ . πββ πβββ πβπ πβπ d) What is the range of f ? e) How does this exercise help in understanding why ππ 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 } is an arithmetic sequence, then { πΆππ } is a geometric sequence. (Here Ξ± is any positive real number.) 3 6) Let f (x) = πΆπ 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 ππ is that it increases much more rapidly than any polynomial function as x β β . We illustrate this property with the exponential function ππ and the polynomial function ππ . a) Create a table of values for ππ , ππ , πππ ππ β ππ , 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_]:=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 ππ ππ ) is greatest when x = 40? c) From the spreadsheet, it is clear that the graphs of x β¦ ππ and x βΌ π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π is eventually greater than ππ 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 ππ remains larger than ππ for x > 40 (whatβs to keep them from crossing one more time?). What mathematically convincing argument could you give to prove that ππ > ππ 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  πΆ π and π₯π¨π  π· π related to each other?
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