Let's consider the pair of functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = e^x \)[/tex] over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
We want to determine the points at which the exponential function [tex]\( g(x) = e^x \)[/tex] grows faster than the quadratic function [tex]\( f(x) = x^2 \)[/tex]. In other words, we need to find the points where [tex]\( e^x \)[/tex] is greater than [tex]\( x^2 \)[/tex].
First, we need to evaluate both functions over the given interval. By examining the values of [tex]\( x \)[/tex] from [tex]\( 0 \)[/tex] to [tex]\( 5 \)[/tex]:
- At [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0^2 = 0 \quad \text{and} \quad g(0) = e^0 = 1
\][/tex]
Here, [tex]\( 1 > 0 \)[/tex].
- At [tex]\( x = 0.05 \)[/tex]:
[tex]\[
f(0.05) \approx 0.00255 \quad \text{and} \quad g(0.05) \approx 1.05
\][/tex]
Here, [tex]\( 1.05 > 0.00255 \)[/tex].
- At [tex]\( x = 0.10 \)[/tex]:
[tex]\[
f(0.10) \approx 0.0102 \quad \text{and} \quad g(0.10) \approx 1.11
\][/tex]
Here, [tex]\( 1.11 > 0.0102 \)[/tex].
Continuing this evaluation for several points within the interval, we observe:
- At [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 1 \quad \text{and} \quad g(1) = e \approx 2.72
\][/tex]
Here, [tex]\( 2.72 > 1 \)[/tex].
- At [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 4 \quad \text{and} \quad g(2) \approx 7.39
\][/tex]
Here, [tex]\( 7.39 > 4 \)[/tex].
- At [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 9 \quad \text{and} \quad g(3) \approx 20.09
\][/tex]
Here, [tex]\( 20.09 > 9 \)[/tex].
- At [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 16 \quad \text{and} \quad g(4) \approx 54.60
\][/tex]
Here, [tex]\( 54.60 > 16 \)[/tex].
- At [tex]\( x = 5 \)[/tex]:
[tex]\[
f(5) = 25 \quad \text{and} \quad g(5) \approx 148.41
\][/tex]
Here, [tex]\( 148.41 > 25 \)[/tex].
By examining these values, we see a consistent pattern:
For all [tex]\( x \)[/tex] in the interval [tex]\( 0 \leq x \leq 5 \)[/tex], the values of [tex]\( g(x) = e^x \)[/tex] are greater than the values of [tex]\( f(x) = x^2 \)[/tex]. Hence, the exponential function [tex]\( g(x) = e^x \)[/tex] is consistently growing at a faster rate than the quadratic function [tex]\( f(x) = x^2 \)[/tex] over the entire interval [tex]\( 0 \leq x \leq 5 \)[/tex].
