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To find an exponential function that grows at a faster rate than the quadratic function [tex]\( y = 3x^2 \)[/tex] within the interval [tex]\( 0 < x < 3 \)[/tex], we'll compare the values of both functions at specific points within the interval.
First, let's evaluate the quadratic function [tex]\( y = 3x^2 \)[/tex] at [tex]\( x = 0, 1, 2, \)[/tex] and [tex]\( 3 \)[/tex].
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0)^2 = 0 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1)^2 = 3 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 3(2)^2 = 12 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 3(3)^2 = 27 \][/tex]
So, the quadratic values at [tex]\( x = 0, 1, 2, \)[/tex] and [tex]\( 3 \)[/tex] are [tex]\( 0, 3, 12, \)[/tex] and [tex]\( 27 \)[/tex] respectively.
Now, let's evaluate the exponential function [tex]\( y = e^x \)[/tex] at the same points.
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = e^0 = 1.0 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = e^1 \approx 2.718 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = e^2 \approx 7.389 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = e^3 \approx 20.086 \][/tex]
So, the exponential values at [tex]\( x = 0, 1, 2, \, \text{and} \, 3 \)[/tex] are approximately [tex]\( 1.0, 2.718, 7.389, \, \text{and} \, 20.086 \)[/tex] respectively.
Comparing the values of the quadratic function [tex]\( y = 3x^2 \)[/tex] and the exponential function [tex]\( y = e^x \)[/tex] for [tex]\( x \)[/tex] in the interval [tex]\( 0 < x < 3 \)[/tex]:
- At [tex]\( x = 1 \)[/tex]:
- Quadratic: [tex]\( 3 \)[/tex]
- Exponential: [tex]\( \approx 2.718 \)[/tex]
The quadratic is slightly higher.
- At [tex]\( x = 2 \)[/tex]:
- Quadratic: [tex]\( 12 \)[/tex]
- Exponential: [tex]\( \approx 7.389 \)[/tex]
The quadratic is higher.
- At [tex]\( x = 3 \)[/tex]:
- Quadratic: [tex]\( 27 \)[/tex]
- Exponential: [tex]\( \approx 20.086 \)[/tex]
The quadratic is still higher, but the exponential is catching up.
Looking closely, we can see that for [tex]\( x < 3 \)[/tex], the exponential values start lower but approach the quadratic values. Exponential functions generally have the property of surpassing polynomial functions beyond a certain point because their growth rate is faster.
Thus, the exponential function [tex]\( y = e^x \)[/tex] grows at a faster rate than the quadratic function [tex]\( y = 3x^2 \)[/tex] starting slightly after [tex]\( x = 3 \)[/tex], hence it is a candidate function which will surpass the quadratic growth slightly beyond [tex]\( x = 3 \)[/tex]. For [tex]\( x \)[/tex] between 0 and 3, while the quadratic may still be slightly higher, the exponential function [tex]\( y = e^x \)[/tex] is definitely on the verge of growing faster and surpassing it.
First, let's evaluate the quadratic function [tex]\( y = 3x^2 \)[/tex] at [tex]\( x = 0, 1, 2, \)[/tex] and [tex]\( 3 \)[/tex].
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0)^2 = 0 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1)^2 = 3 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 3(2)^2 = 12 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 3(3)^2 = 27 \][/tex]
So, the quadratic values at [tex]\( x = 0, 1, 2, \)[/tex] and [tex]\( 3 \)[/tex] are [tex]\( 0, 3, 12, \)[/tex] and [tex]\( 27 \)[/tex] respectively.
Now, let's evaluate the exponential function [tex]\( y = e^x \)[/tex] at the same points.
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = e^0 = 1.0 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = e^1 \approx 2.718 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = e^2 \approx 7.389 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = e^3 \approx 20.086 \][/tex]
So, the exponential values at [tex]\( x = 0, 1, 2, \, \text{and} \, 3 \)[/tex] are approximately [tex]\( 1.0, 2.718, 7.389, \, \text{and} \, 20.086 \)[/tex] respectively.
Comparing the values of the quadratic function [tex]\( y = 3x^2 \)[/tex] and the exponential function [tex]\( y = e^x \)[/tex] for [tex]\( x \)[/tex] in the interval [tex]\( 0 < x < 3 \)[/tex]:
- At [tex]\( x = 1 \)[/tex]:
- Quadratic: [tex]\( 3 \)[/tex]
- Exponential: [tex]\( \approx 2.718 \)[/tex]
The quadratic is slightly higher.
- At [tex]\( x = 2 \)[/tex]:
- Quadratic: [tex]\( 12 \)[/tex]
- Exponential: [tex]\( \approx 7.389 \)[/tex]
The quadratic is higher.
- At [tex]\( x = 3 \)[/tex]:
- Quadratic: [tex]\( 27 \)[/tex]
- Exponential: [tex]\( \approx 20.086 \)[/tex]
The quadratic is still higher, but the exponential is catching up.
Looking closely, we can see that for [tex]\( x < 3 \)[/tex], the exponential values start lower but approach the quadratic values. Exponential functions generally have the property of surpassing polynomial functions beyond a certain point because their growth rate is faster.
Thus, the exponential function [tex]\( y = e^x \)[/tex] grows at a faster rate than the quadratic function [tex]\( y = 3x^2 \)[/tex] starting slightly after [tex]\( x = 3 \)[/tex], hence it is a candidate function which will surpass the quadratic growth slightly beyond [tex]\( x = 3 \)[/tex]. For [tex]\( x \)[/tex] between 0 and 3, while the quadratic may still be slightly higher, the exponential function [tex]\( y = e^x \)[/tex] is definitely on the verge of growing faster and surpassing it.
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