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To determine whether the exponential function consistently grows at a faster rate than the quadratic function over the interval [tex]\( 0 \leq x \leq 5 \)[/tex], we need to compare their values and growth rates at each point within this interval.
Let's consider the exponential function [tex]\( f(x) = 2^x \)[/tex] and the quadratic function [tex]\( g(x) = x^2 \)[/tex]. We will calculate the values of each function at integer points from 0 to 5, and then analyze the growth rates.
1. Calculate function values over the interval [tex]\( 0 \leq x \leq 5 \)[/tex]:
For the exponential function [tex]\( f(x) = 2^x \)[/tex]:
- [tex]\( f(0) = 2^0 = 1 \)[/tex]
- [tex]\( f(1) = 2^1 = 2 \)[/tex]
- [tex]\( f(2) = 2^2 = 4 \)[/tex]
- [tex]\( f(3) = 2^3 = 8 \)[/tex]
- [tex]\( f(4) = 2^4 = 16 \)[/tex]
- [tex]\( f(5) = 2^5 = 32 \)[/tex]
So, the exponential function values are: [tex]\([1, 2, 4, 8, 16, 32]\)[/tex].
For the quadratic function [tex]\( g(x) = x^2 \)[/tex]:
- [tex]\( g(0) = 0^2 = 0 \)[/tex]
- [tex]\( g(1) = 1^2 = 1 \)[/tex]
- [tex]\( g(2) = 2^2 = 4 \)[/tex]
- [tex]\( g(3) = 3^2 = 9 \)[/tex]
- [tex]\( g(4) = 4^2 = 16 \)[/tex]
- [tex]\( g(5) = 5^2 = 25 \)[/tex]
So, the quadratic function values are: [tex]\([0, 1, 4, 9, 16, 25]\)[/tex].
2. Determine growth rates for each function:
The growth rate is the difference between consecutive function values.
For the exponential function [tex]\( f(x) = 2^x \)[/tex]:
- Growth from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( 2 - 1 = 1 \)[/tex]
- Growth from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( 4 - 2 = 2 \)[/tex]
- Growth from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( 8 - 4 = 4 \)[/tex]
- Growth from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]: [tex]\( 16 - 8 = 8 \)[/tex]
- Growth from [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]: [tex]\( 32 - 16 = 16 \)[/tex]
Growth rates for the exponential function: [tex]\([1, 2, 4, 8, 16]\)[/tex].
For the quadratic function [tex]\( g(x) = x^2 \)[/tex]:
- Growth from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( 1 - 0 = 1 \)[/tex]
- Growth from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( 4 - 1 = 3 \)[/tex]
- Growth from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( 9 - 4 = 5 \)[/tex]
- Growth from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]: [tex]\( 16 - 9 = 7 \)[/tex]
- Growth from [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]: [tex]\( 25 - 16 = 9 \)[/tex]
Growth rates for the quadratic function: [tex]\([1, 3, 5, 7, 9]\)[/tex].
3. Compare growth rates:
We now compare the growth rates of both functions at each step:
- [tex]\( \text{At } x = 1: \)[/tex] Exponential growth rate [tex]\(1\)[/tex] vs. Quadratic growth rate [tex]\(1\)[/tex]
- [tex]\( \text{At } x = 2: \)[/tex] Exponential growth rate [tex]\(2\)[/tex] vs. Quadratic growth rate [tex]\(3\)[/tex]
- [tex]\( \text{At } x = 3: \)[/tex] Exponential growth rate [tex]\(4\)[/tex] vs. Quadratic growth rate [tex]\(5\)[/tex]
- [tex]\( \text{At } x = 4: \)[/tex] Exponential growth rate [tex]\(8\)[/tex] vs. Quadratic growth rate [tex]\(7\)[/tex]
- [tex]\( \text{At } x = 5: \)[/tex] Exponential growth rate [tex]\(16\)[/tex] vs. Quadratic growth rate [tex]\(9\)[/tex]
For the exponential function to consistently grow at a faster rate than the quadratic function, its growth rate must always be greater than the quadratic function's growth rate. Observing the growth rates:
- At [tex]\( x = 1 \)[/tex], the growth rates are equal.
- At [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex], the quadratic function's growth rate is higher.
- At [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex], the exponential function's growth rate is higher.
Thus, the exponential function does not consistently grow at a faster rate than the quadratic function over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
Let's consider the exponential function [tex]\( f(x) = 2^x \)[/tex] and the quadratic function [tex]\( g(x) = x^2 \)[/tex]. We will calculate the values of each function at integer points from 0 to 5, and then analyze the growth rates.
1. Calculate function values over the interval [tex]\( 0 \leq x \leq 5 \)[/tex]:
For the exponential function [tex]\( f(x) = 2^x \)[/tex]:
- [tex]\( f(0) = 2^0 = 1 \)[/tex]
- [tex]\( f(1) = 2^1 = 2 \)[/tex]
- [tex]\( f(2) = 2^2 = 4 \)[/tex]
- [tex]\( f(3) = 2^3 = 8 \)[/tex]
- [tex]\( f(4) = 2^4 = 16 \)[/tex]
- [tex]\( f(5) = 2^5 = 32 \)[/tex]
So, the exponential function values are: [tex]\([1, 2, 4, 8, 16, 32]\)[/tex].
For the quadratic function [tex]\( g(x) = x^2 \)[/tex]:
- [tex]\( g(0) = 0^2 = 0 \)[/tex]
- [tex]\( g(1) = 1^2 = 1 \)[/tex]
- [tex]\( g(2) = 2^2 = 4 \)[/tex]
- [tex]\( g(3) = 3^2 = 9 \)[/tex]
- [tex]\( g(4) = 4^2 = 16 \)[/tex]
- [tex]\( g(5) = 5^2 = 25 \)[/tex]
So, the quadratic function values are: [tex]\([0, 1, 4, 9, 16, 25]\)[/tex].
2. Determine growth rates for each function:
The growth rate is the difference between consecutive function values.
For the exponential function [tex]\( f(x) = 2^x \)[/tex]:
- Growth from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( 2 - 1 = 1 \)[/tex]
- Growth from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( 4 - 2 = 2 \)[/tex]
- Growth from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( 8 - 4 = 4 \)[/tex]
- Growth from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]: [tex]\( 16 - 8 = 8 \)[/tex]
- Growth from [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]: [tex]\( 32 - 16 = 16 \)[/tex]
Growth rates for the exponential function: [tex]\([1, 2, 4, 8, 16]\)[/tex].
For the quadratic function [tex]\( g(x) = x^2 \)[/tex]:
- Growth from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( 1 - 0 = 1 \)[/tex]
- Growth from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( 4 - 1 = 3 \)[/tex]
- Growth from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( 9 - 4 = 5 \)[/tex]
- Growth from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]: [tex]\( 16 - 9 = 7 \)[/tex]
- Growth from [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]: [tex]\( 25 - 16 = 9 \)[/tex]
Growth rates for the quadratic function: [tex]\([1, 3, 5, 7, 9]\)[/tex].
3. Compare growth rates:
We now compare the growth rates of both functions at each step:
- [tex]\( \text{At } x = 1: \)[/tex] Exponential growth rate [tex]\(1\)[/tex] vs. Quadratic growth rate [tex]\(1\)[/tex]
- [tex]\( \text{At } x = 2: \)[/tex] Exponential growth rate [tex]\(2\)[/tex] vs. Quadratic growth rate [tex]\(3\)[/tex]
- [tex]\( \text{At } x = 3: \)[/tex] Exponential growth rate [tex]\(4\)[/tex] vs. Quadratic growth rate [tex]\(5\)[/tex]
- [tex]\( \text{At } x = 4: \)[/tex] Exponential growth rate [tex]\(8\)[/tex] vs. Quadratic growth rate [tex]\(7\)[/tex]
- [tex]\( \text{At } x = 5: \)[/tex] Exponential growth rate [tex]\(16\)[/tex] vs. Quadratic growth rate [tex]\(9\)[/tex]
For the exponential function to consistently grow at a faster rate than the quadratic function, its growth rate must always be greater than the quadratic function's growth rate. Observing the growth rates:
- At [tex]\( x = 1 \)[/tex], the growth rates are equal.
- At [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex], the quadratic function's growth rate is higher.
- At [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex], the exponential function's growth rate is higher.
Thus, the exponential function does not consistently grow at a faster rate than the quadratic function over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
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