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To determine for which pairs of functions the exponential function grows at a faster rate than the quadratic function over the interval [tex]\(0 \leq x \leq 5\)[/tex], we will evaluate each function at the points [tex]\(0, 1, 2, 3, 4, 5\)[/tex] and compare their values.
We are considering two functions:
1. [tex]\( 2^x \)[/tex] (the exponential function)
2. [tex]\( x^2 \)[/tex] (the quadratic function)
Let's compute their values at each of these points:
### Step-by-Step Evaluation
1. At [tex]\( x = 0 \)[/tex]:
- Exponential function: [tex]\( 2^0 = 1 \)[/tex]
- Quadratic function: [tex]\( 0^2 = 0 \)[/tex]
Since [tex]\( 1 > 0 \)[/tex], the exponential function is greater.
2. At [tex]\( x = 1 \)[/tex]:
- Exponential function: [tex]\( 2^1 = 2 \)[/tex]
- Quadratic function: [tex]\( 1^2 = 1 \)[/tex]
Since [tex]\( 2 > 1 \)[/tex], the exponential function is greater.
3. At [tex]\( x = 2 \)[/tex]:
- Exponential function: [tex]\( 2^2 = 4 \)[/tex]
- Quadratic function: [tex]\( 2^2 = 4 \)[/tex]
Since [tex]\( 4 = 4 \)[/tex], the functions are equal.
4. At [tex]\( x = 3 \)[/tex]:
- Exponential function: [tex]\( 2^3 = 8 \)[/tex]
- Quadratic function: [tex]\( 3^2 = 9 \)[/tex]
Since [tex]\( 8 < 9 \)[/tex], the quadratic function is greater.
5. At [tex]\( x = 4 \)[/tex]:
- Exponential function: [tex]\( 2^4 = 16 \)[/tex]
- Quadratic function: [tex]\( 4^2 = 16 \)[/tex]
Since [tex]\( 16 = 16 \)[/tex], the functions are equal.
6. At [tex]\( x = 5 \)[/tex]:
- Exponential function: [tex]\( 2^5 = 32 \)[/tex]
- Quadratic function: [tex]\( 5^2 = 25 \)[/tex]
Since [tex]\( 32 > 25 \)[/tex], the exponential function is greater.
### Summary of Comparisons
- At [tex]\( x = 0 \)[/tex]: Exponential is greater.
- At [tex]\( x = 1 \)[/tex]: Exponential is greater.
- At [tex]\( x = 2 \)[/tex]: Both are equal.
- At [tex]\( x = 3 \)[/tex]: Quadratic is greater.
- At [tex]\( x = 4 \)[/tex]: Both are equal.
- At [tex]\( x = 5 \)[/tex]: Exponential is greater.
### Final Conclusion
The exponential function [tex]\( 2^x \)[/tex] grows faster than the quadratic function [tex]\( x^2 \)[/tex] at [tex]\( x = 0, 1 \)[/tex] and [tex]\( 5 \)[/tex]. Therefore, the exponential function is not consistently growing at a faster rate than the quadratic function over the interval [tex]\( 0 \leq x \leq 5 \)[/tex], as there are points where the quadratic function equals or exceeds the exponential function's growth.
Thus, the pairs of points where the exponential function grows faster are: [tex]\( (0, \text{True}), (1, \text{True}), (5, \text{True}) \)[/tex], but not consistently across the entire interval.
We are considering two functions:
1. [tex]\( 2^x \)[/tex] (the exponential function)
2. [tex]\( x^2 \)[/tex] (the quadratic function)
Let's compute their values at each of these points:
### Step-by-Step Evaluation
1. At [tex]\( x = 0 \)[/tex]:
- Exponential function: [tex]\( 2^0 = 1 \)[/tex]
- Quadratic function: [tex]\( 0^2 = 0 \)[/tex]
Since [tex]\( 1 > 0 \)[/tex], the exponential function is greater.
2. At [tex]\( x = 1 \)[/tex]:
- Exponential function: [tex]\( 2^1 = 2 \)[/tex]
- Quadratic function: [tex]\( 1^2 = 1 \)[/tex]
Since [tex]\( 2 > 1 \)[/tex], the exponential function is greater.
3. At [tex]\( x = 2 \)[/tex]:
- Exponential function: [tex]\( 2^2 = 4 \)[/tex]
- Quadratic function: [tex]\( 2^2 = 4 \)[/tex]
Since [tex]\( 4 = 4 \)[/tex], the functions are equal.
4. At [tex]\( x = 3 \)[/tex]:
- Exponential function: [tex]\( 2^3 = 8 \)[/tex]
- Quadratic function: [tex]\( 3^2 = 9 \)[/tex]
Since [tex]\( 8 < 9 \)[/tex], the quadratic function is greater.
5. At [tex]\( x = 4 \)[/tex]:
- Exponential function: [tex]\( 2^4 = 16 \)[/tex]
- Quadratic function: [tex]\( 4^2 = 16 \)[/tex]
Since [tex]\( 16 = 16 \)[/tex], the functions are equal.
6. At [tex]\( x = 5 \)[/tex]:
- Exponential function: [tex]\( 2^5 = 32 \)[/tex]
- Quadratic function: [tex]\( 5^2 = 25 \)[/tex]
Since [tex]\( 32 > 25 \)[/tex], the exponential function is greater.
### Summary of Comparisons
- At [tex]\( x = 0 \)[/tex]: Exponential is greater.
- At [tex]\( x = 1 \)[/tex]: Exponential is greater.
- At [tex]\( x = 2 \)[/tex]: Both are equal.
- At [tex]\( x = 3 \)[/tex]: Quadratic is greater.
- At [tex]\( x = 4 \)[/tex]: Both are equal.
- At [tex]\( x = 5 \)[/tex]: Exponential is greater.
### Final Conclusion
The exponential function [tex]\( 2^x \)[/tex] grows faster than the quadratic function [tex]\( x^2 \)[/tex] at [tex]\( x = 0, 1 \)[/tex] and [tex]\( 5 \)[/tex]. Therefore, the exponential function is not consistently growing at a faster rate than the quadratic function over the interval [tex]\( 0 \leq x \leq 5 \)[/tex], as there are points where the quadratic function equals or exceeds the exponential function's growth.
Thus, the pairs of points where the exponential function grows faster are: [tex]\( (0, \text{True}), (1, \text{True}), (5, \text{True}) \)[/tex], but not consistently across the entire interval.
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