Comparing Growth of Different Types of Functions (Grades 11-12)

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Comparing Growth of Different Types of Functions

1. 
The parents of Stephanie and Julia have told them that they will give them a monthly allowance based on a math equation. Stephanie chooses the one which says [math]f(x)= 3x+20[/math] while Julia chooses the one which says [math]g(x)= 1.75^x[/math]. Which option below best describes what will happen?
  1. Julia will always make more.
  2. Julia will make more at first, but Stephanie will make more in the end.
  3. Stephanie will always make more.
  4. Stephanie will make more at first, but Julia will make more in the end.
2. 
The side of a skate board ramp could be created using either a quadratic equation or an exponential equation. Which one has the potential to create the steepest slope at the top?
  1. Quadratic
  2. Exponential
  3. They will both be the same
  4. Not enough information
3. 
The bank offers you two investments with very high returns. Investment A yields returns of [math]f(t)=1.8t^2+5.4t[/math] and investment B yields returns of [math]g(t)=1.081^t[/math]. Which is the better deal?
  1. Investment A is the better deal.
  2. Investment B is the better deal.
  3. Both investments yield the same return.
  4. It depends on how long the money stays invested.
4. 
A new type of bacteria has been developed and its growth can be controlled by certain chemicals. With chemical compound A, the bacteria will grow according to the function [math]f(t) = 400t^3 + 6t^2 + t + 10[/math], where [math]f[/math] measures the mass of bacteria in milligrams over [math]t[/math] days. With chemical compound B, the bacteria will grow according to the function [math]g(t) = 10e^(2t)[/math], where [math]g[/math] measures the mass of the bacteria in milligrams over [math]t[/math] days. Which chemical compound will result in greater bacterial growth over the long term?
  1. Eventually, they will both produce about the same growth.
  2. It is not possible to say, with the given amount of information.
  3. Chemical compound A.
  4. Chemical compound B.
5. 
The following table shows some of the values of functions [math]f(x)[/math], which is linear, and [math]g(x)[/math], which is exponential. What can be concluded from these values (and general properties of these functions), and why?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 2 [/math][math] 23 [/math][math] 4 [/math]
[math] 3 [/math][math] 33 [/math][math] 8 [/math]
[math] 4 [/math][math] 43 [/math][math] 16 [/math]
[math] 5 [/math][math] 53 [/math][math] 32 [/math]
[math] 6 [/math][math] 63 [/math][math] 64 [/math]
[math] 7 [/math][math] 73 [/math][math] 128 [/math]
  1. [math]g(x) > f(x)[/math] for all values of [math]x[/math]. This is because [math]f(x)[/math] is an exponential function, which must always be greater than a linear function.
  2. [math]g(x)>f(x)[/math] for [math]x>=6[/math]. Since the values of [math]g(x)[/math] continue to increase by ever greater amounts each unit interval, they will continue to be greater than the values of [math]f(x)[/math] which are only increasing by the same amount each unit interval.
  3. [math]g(x) > f(x)[/math] for [math]6 <= x <= 45[/math]. If a linear function intersects an exponential function, it must do so twice. Therefore, the two functions will intersect again at approximately [math]x~~2^5.5 ~~ 45.3[/math], and thereafter [math]f(x) > g(x)[/math].
  4. [math]g(x) > f(x)[/math] for [math]x>=6[/math] and [math]g(x) < f(x)[/math] for [math]x <= 5[/math]. Since [math]x~~5.5[/math] is the only point of intersection, and given the values in the table, [math]g(x)[/math] must be greater than [math]f(x)[/math] for values of [math]x[/math] greater than or equal to 6 and it must be less than [math]f(x)[/math] for values of [math]x[/math] less than or equal to 5.
6. 
The following table gives some of the values of [math]f(x)[/math], a quadratic function, and [math]g(x)[/math], an exponential function. What can be concluded from these values (and general properties of the functions), and why?


[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 8 [/math][math] 32.0 [/math][math] 1.5 [/math]
[math] 9 [/math][math] 40.5 [/math][math] 4.1 [/math]
[math] 10 [/math][math] 50.0 [/math][math] 11.0 [/math]
[math] 11 [/math][math] 60.5 [/math][math] 29.9 [/math]
[math] 12 [/math][math] 72.0 [/math][math] 81.4 [/math]
[math] 13 [/math][math] 84.5 [/math][math] 221.2 [/math]
  1. [math]g(x) > f(x)[/math] for [math]12 <= x <= 20[/math]. [math]f(x)[/math] is quadratic and has a minimum near [math]x =16[/math], as is indicated by the values in the table. After this point, [math]f(x)[/math] will start growing more rapidly than [math]g(x)[/math] and become greater than [math]g(x)[/math] for [math]x>20[/math].
  2. [math]g(x) > f(x)[/math] for [math]x >= 12[/math]. Since exponential functions and quadratic functions can only intersect at one point, the intersection point near [math]x=11.5[/math] means that [math] g(x) > f(x)[/math] for [math]x >= 12[/math] and [math]f(x) > g(x)[/math] for [math]x <= 11[/math].
  3. [math]g(x) > f(x)[/math] for all values of [math]x[/math]. Since [math]g(x)[/math] is exponential, it must therefore always be greater than a quadratic function.
  4. [math]g(x) > f(x)[/math] for [math]x>=12[/math]. Since [math]g(x)[/math] is exponential, the difference of values between each unit interval will continue to increase. Although the difference of quadratic values will also increase, they do so at a much slower rate.
7. 
Let [math]f(x)[/math] be an exponential function and [math]p(x)[/math] a 3rd degree polynomial function. Assume that [math]p(x)[/math] has 3 real zeros, which are all less than or equal to zero.
A. 
Below is a table of some of the values of [math]f(x)[/math] and [math]p(x)[/math], rounded to 4 decimal places if necessary. What conclusions can be seen from this information? Choose all correct answers.

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{p(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 1 [/math][math] 1.01 [/math][math] 5 [/math]
[math] 2 [/math][math] 1.0201 [/math][math] 40 [/math]
[math] 3 [/math][math] 1.0303 [/math][math] 135 [/math]
[math] 4 [/math][math] 1.0406 [/math][math] 320 [/math]
[math] 5 [/math][math] 1.0510 [/math][math] 625 [/math]
  1. That [math]p(x) > f(x)[/math] for all values of [math]x>1[/math].
  2. That, for these values, the average rate of change over a one unit interval of [math]p(x)[/math] is greater than that of [math]f(x)[/math].
  3. That [math]f(x)[/math] has a percent increase of about 1%.
  4. That the percent increase of [math]p(x)[/math] over a one unit interval is declining.
B. 
The following are some more values of [math]f(x)[/math] and [math]p(x)[/math], which have been rounded. Which of the following statements are supported by this information, and general properties of these functions? Choose all correct answers.

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{p(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 2521 [/math][math] 7.8376 xx 10^10 [/math][math] 8.0110 xx 10^10 [/math]
[math] 2522 [/math][math] 7.9160 xx 10^10 [/math][math] 8.0206 xx 10^10 [/math]
[math] 2523 [/math][math] 7.9951 xx 10^10 [/math][math] 8.0301 xx 10^10 [/math]
[math] 2524 [/math][math] 8.0751 xx 10^10 [/math][math] 8.0397 xx 10^10 [/math]
[math] 2525 [/math][math] 8.1558 xx 10^10 [/math][math] 8.0492 xx 10^10 [/math]
  1. [math]f(x) < p(x) [/math] for [math]x <= 2523[/math].
  2. The percent increase of [math]f(x)[/math] over a one unit interval continues to be about 1%.
  3. The percent increase of [math]p(x)[/math] over a one unit interval continues to decrease.
  4. The average rate of change of [math]p(x)[/math] over a one unit interval is greater than that of [math]f(x)[/math] for these values of [math]x[/math].
C. 
Given the information in the previous two questions and properties of both functions, choose which of the following statements is definitely true.
  1. [math]f(x) > p(x)[/math] for [math]x >= 2524[/math], since the steady percent increase of [math]f(x)[/math] means it will grow by ever greater amounts over a one unit interval. Although the amount that [math]p(x)[/math] increases over a one unit interval will also grow, it will do so more slowly.
  2. [math]f(x) > p(x)[/math] may be true for all values of [math]x[/math] greater than or equal to 2524, or it may only be greater for some interval before [math]p(x)[/math] becomes greater. Since the percent increase of [math]p(x)[/math] is always changing, it cannot be certain which function will be greater.
  3. [math]f(x) > p(x)[/math] for [math]2524 <= x < x_0[/math], where [math]x_0[/math] is some real value of [math]x[/math] greater than 2524. Because [math]f(x)[/math] is only increasing by 1% each unit interval, the amount it increases over a one unit interval will eventually slow down, and [math]p(x)[/math] will become larger.
  4. [math]f(x)[/math] will be equal to [math]p(x)[/math] for exactly one more value of x for [math]x > 2524[/math]. Let this point be [math]x_0[/math]. It cannot be determined with the information given which function will be greater for [math]x > x_0[/math].
D. 
For a general exponential function [math]g(x) = cb^x, c>0, b>0 " and " b!=1[/math] and a general polynomial function [math]P(x) = a_1 x^n + a_2 x^{n-1} + ... + a_n x + a_{n+1}, \ \ a_1 > 0[/math], which of the following conclusions is correct?
  1. Without knowing more about the specific functions, no conclusion can be made.
  2. That for an even degree polynomial, [math]P(x) > g(x)[/math], but for an odd degree polynomial [math]g(x) > P(x)[/math].
  3. That, as long as [math]P(x)[/math] has no maximum or minimums for [math]x>0[/math], [math]P(x) > g(x)[/math]. Otherwise, [math]g(x) > P(x)[/math].
  4. That there will always exist a value [math]x_0[/math] such that [math]g(x) > P(x)[/math] for [math]x > x_0[/math].

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