Given the results, let’s determine for which pair of functions the exponential consistently grows at a faster rate than the quadratic over the interval [tex]\(0 \leq x \leq 5\)[/tex].
1. Define the Functions:
We have two functions:
- Quadratic function: [tex]\( f(x) = ax^2 + bx + c \)[/tex]
- Exponential function: [tex]\( g(x) = ae^{bx} \)[/tex]
2. Identify Coefficients:
- For the quadratic function, we use [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], [tex]\( c = 0 \)[/tex], resulting in [tex]\( f(x) = x^2 \)[/tex].
- For the exponential function, we use [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], resulting in [tex]\( g(x) = e^x \)[/tex].
3. Evaluate the Functions over the Interval [tex]\( 0 \leq x \leq 5 \)[/tex]:
The interval is divided into 1000 points for a finer comparison. For simplicity, let's look at the beginning, middle, and end values.
- Quadratic [tex]\( f(x) = x^2 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0^2 = 0 \)[/tex]
- At [tex]\( x = 2.5 \)[/tex], [tex]\( f(2.5) = (2.5)^2 = 6.25 \)[/tex]
- At [tex]\( x = 5 \)[/tex], [tex]\( f(5) = (5)^2 = 25 \)[/tex]
- Exponential [tex]\( g(x) = e^x \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( g(0) = e^0 = 1 \)[/tex]
- At [tex]\( x = 2.5 \)[/tex], [tex]\( g(2.5) \approx e^{2.5} \approx 12.1825 \)[/tex]
- At [tex]\( x = 5 \)[/tex], [tex]\( g(5) \approx e^5 \approx 148.4132 \)[/tex]
From these values, we see a rapid growth in the exponential function compared to the quadratic function.
4. Compare the Functions Over the Interval:
We inspect the values at specific points:
- At [tex]\( x = 0 \)[/tex]:
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( g(0) = 1 \)[/tex]
- At [tex]\( x = 2.5 \)[/tex]:
- [tex]\( f(2.5) = 6.25 \)[/tex]
- [tex]\( g(2.5) \approx 12.1825 \)[/tex]
- At [tex]\( x = 5 \)[/tex]:
- [tex]\( f(5) = 25 \)[/tex]
- [tex]\( g(5) \approx 148.4132 \)[/tex]
Clearly, the exponential function [tex]\( g(x) = e^x \)[/tex] grows faster than the quadratic function [tex]\( f(x) = x^2 \)[/tex] at all inspected points within the interval.
5. Conclusion:
The exponential function grows at a consistently faster rate than the quadratic function over the interval [tex]\(0 \leq x \leq 5\)[/tex]:
The exponential function [tex]\( g(x) = e^x \)[/tex] grows faster than the quadratic function [tex]\( f(x) = x^2 \)[/tex] over the interval [tex]\( 0 \leq x \leq 5 \)[/tex]. This conclusion holds for all values in the interval, as the exponential function's rate of growth continuously exceeds that of the quadratic function.
