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To determine which function grows faster over the interval [tex]\( 0 \leq x \leq 5 \)[/tex], let's compare the exponential function [tex]\( f(x) = 2^x \)[/tex] with the quadratic function [tex]\( g(x) = x^2 + 3x + 2 \)[/tex].
Below is a breakdown of the comparison and the results:
1. Function Definitions:
- Exponential function: [tex]\( f(x) = 2^x \)[/tex]
- Quadratic function: [tex]\( g(x) = x^2 + 3x + 2 \)[/tex]
2. Interval:
- We are interested in the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
3. Calculations for the Interval:
- Consider different points within the interval and evaluate both functions at these points.
- The evaluated values for these functions over a range of points in the interval [tex]\( [0, 5] \)[/tex] are as follows:
[tex]\[ \begin{array}{c|c|c} x & 2^x & x^2 + 3x + 2 \\ \hline 0 & 1 & 2 \\ 0.5 & 1.03563 & 2.15407 \\ 1 & 2 & 2.31323 \\ 1.5 & 2.82843 & 2.47750 \\ 2 & 4 & 2.64687 \\ 2.5 & 5.65685 & 2.82134 \\ 3 & 8 & 3.00092 \\ 3.5 & 11.31371 & 3.18559 \\ 4 & 16 & 3.37537 \\ 4.5 & 22.62742 & 3.57025 \\ 5 & 32 & 3.77023 \\ \end{array} \][/tex]
4. Comparison:
- We observe that at each evaluated point in the interval from [tex]\( x = 0 \)[/tex] to [tex]\( x = 5 \)[/tex]:
- At [tex]\( x=0 \)[/tex]: [tex]\( 2^0 \)[/tex] is 1, which is less than [tex]\( 0^2 + 3(0) + 2 \)[/tex] which is 2.
- At [tex]\( x=0.5 \)[/tex]: [tex]\( 2^{0.5} \approx 1.03563 \)[/tex], less than [tex]\( (0.5)^2 + 3(0.5) + 2 \approx 2.15407 \)[/tex].
- At [tex]\( x=1 \)[/tex]: [tex]\( 2^1 = 2 \)[/tex], less than [tex]\( 1^2 + 3(1) + 2 = 6 \)[/tex].
- Continuing this process, we observe that [tex]\( 2^x \)[/tex] is consistently less than [tex]\( x^2 + 3x + 2 \)[/tex] for every [tex]\( x \)[/tex] in the interval [0, 5].
5. Concluding Statement:
- Based on this detailed evaluation, we see that the exponential function [tex]\( 2^x \)[/tex] does not consistently grow faster than the quadratic function [tex]\( x^2 + 3x + 2 \)[/tex] over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
- In fact, the quadratic function [tex]\( x^2 + 3x + 2 \)[/tex] grows faster throughout the entire interval.
Therefore, the exponential function [tex]\( 2^x \)[/tex] is not consistently growing at a faster rate than the quadratic function [tex]\( x^2 + 3x + 2 \)[/tex] over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
Below is a breakdown of the comparison and the results:
1. Function Definitions:
- Exponential function: [tex]\( f(x) = 2^x \)[/tex]
- Quadratic function: [tex]\( g(x) = x^2 + 3x + 2 \)[/tex]
2. Interval:
- We are interested in the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
3. Calculations for the Interval:
- Consider different points within the interval and evaluate both functions at these points.
- The evaluated values for these functions over a range of points in the interval [tex]\( [0, 5] \)[/tex] are as follows:
[tex]\[ \begin{array}{c|c|c} x & 2^x & x^2 + 3x + 2 \\ \hline 0 & 1 & 2 \\ 0.5 & 1.03563 & 2.15407 \\ 1 & 2 & 2.31323 \\ 1.5 & 2.82843 & 2.47750 \\ 2 & 4 & 2.64687 \\ 2.5 & 5.65685 & 2.82134 \\ 3 & 8 & 3.00092 \\ 3.5 & 11.31371 & 3.18559 \\ 4 & 16 & 3.37537 \\ 4.5 & 22.62742 & 3.57025 \\ 5 & 32 & 3.77023 \\ \end{array} \][/tex]
4. Comparison:
- We observe that at each evaluated point in the interval from [tex]\( x = 0 \)[/tex] to [tex]\( x = 5 \)[/tex]:
- At [tex]\( x=0 \)[/tex]: [tex]\( 2^0 \)[/tex] is 1, which is less than [tex]\( 0^2 + 3(0) + 2 \)[/tex] which is 2.
- At [tex]\( x=0.5 \)[/tex]: [tex]\( 2^{0.5} \approx 1.03563 \)[/tex], less than [tex]\( (0.5)^2 + 3(0.5) + 2 \approx 2.15407 \)[/tex].
- At [tex]\( x=1 \)[/tex]: [tex]\( 2^1 = 2 \)[/tex], less than [tex]\( 1^2 + 3(1) + 2 = 6 \)[/tex].
- Continuing this process, we observe that [tex]\( 2^x \)[/tex] is consistently less than [tex]\( x^2 + 3x + 2 \)[/tex] for every [tex]\( x \)[/tex] in the interval [0, 5].
5. Concluding Statement:
- Based on this detailed evaluation, we see that the exponential function [tex]\( 2^x \)[/tex] does not consistently grow faster than the quadratic function [tex]\( x^2 + 3x + 2 \)[/tex] over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
- In fact, the quadratic function [tex]\( x^2 + 3x + 2 \)[/tex] grows faster throughout the entire interval.
Therefore, the exponential function [tex]\( 2^x \)[/tex] is not consistently growing at a faster rate than the quadratic function [tex]\( x^2 + 3x + 2 \)[/tex] over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
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