To determine which pair of functions has one consistently growing at a faster rate than the other over the interval [tex]\(0 \leq x \leq 5\)[/tex], we will compare the exponential function [tex]\(f(x) = e^x\)[/tex] and the quadratic function [tex]\(g(x) = x^2\)[/tex] at the integer points within this interval.
### Step-by-Step Solution:
1. Define the interval:
- The interval we are looking at is [tex]\(0 \leq x \leq 5\)[/tex].
2. Calculate the values of the functions:
- Exponential Function [tex]\( f(x) = e^x \)[/tex]:
- When [tex]\( x = 0\)[/tex]: [tex]\( e^0 = 1 \)[/tex]
- When [tex]\( x = 1\)[/tex]: [tex]\( e^1 = 2.71828183 \)[/tex]
- When [tex]\( x = 2\)[/tex]: [tex]\( e^2 = 7.3890561 \)[/tex]
- When [tex]\( x = 3\)[/tex]: [tex]\( e^3 = 20.08553692 \)[/tex]
- When [tex]\( x = 4\)[/tex]: [tex]\( e^4 = 54.59815003 \)[/tex]
- When [tex]\( x = 5\)[/tex]: [tex]\( e^5 = 148.4131591 \)[/tex]
- Quadratic Function [tex]\( g(x) = x^2 \)[/tex]:
- When [tex]\( x = 0\)[/tex]: [tex]\( 0^2 = 0 \)[/tex]
- When [tex]\( x = 1\)[/tex]: [tex]\( 1^2 = 1 \)[/tex]
- When [tex]\( x = 2\)[/tex]: [tex]\( 2^2 = 4 \)[/tex]
- When [tex]\( x = 3\)[/tex]: [tex]\( 3^2 = 9 \)[/tex]
- When [tex]\( x = 4\)[/tex]: [tex]\( 4^2 = 16 \)[/tex]
- When [tex]\( x = 5\)[/tex]: [tex]\( 5^2 = 25 \)[/tex]
3. Compare the values at each point:
- At [tex]\( x = 0 \)[/tex]: [tex]\( e^0 = 1 \)[/tex] and [tex]\( 0^2 = 0 \)[/tex]; hence, [tex]\( 1 > 0 \)[/tex]
- At [tex]\( x = 1 \)[/tex]: [tex]\( e^1 = 2.71828183 \)[/tex] and [tex]\( 1^2 = 1 \)[/tex]; hence, [tex]\( 2.71828183 > 1 \)[/tex]
- At [tex]\( x = 2 \)[/tex]: [tex]\( e^2 = 7.3890561 \)[/tex] and [tex]\( 2^2 = 4 \)[/tex]; hence, [tex]\( 7.3890561 > 4 \)[/tex]
- At [tex]\( x = 3 \)[/tex]: [tex]\( e^3 = 20.08553692 \)[/tex] and [tex]\( 3^2 = 9 \)[/tex]; hence, [tex]\( 20.08553692 > 9 \)[/tex]
- At [tex]\( x = 4 \)[/tex]: [tex]\( e^4 = 54.59815003 \)[/tex] and [tex]\( 4^2 = 16 \)[/tex]; hence, [tex]\( 54.59815003 > 16 \)[/tex]
- At [tex]\( x = 5 \)[/tex]: [tex]\( e^5 = 148.4131591 \)[/tex] and [tex]\( 5^2 = 25 \)[/tex]; hence, [tex]\( 148.4131591 > 25 \)[/tex]
### Conclusion:
Based on these calculations, we observe that for every point in the interval [tex]\(0 \leq x \leq 5\)[/tex], the exponential function [tex]\(e^x\)[/tex] has a greater value than the quadratic function [tex]\(x^2\)[/tex]. This indicates that the exponential function is consistently growing at a faster rate than the quadratic function over the given interval. Thus, the pair of functions for which the exponential is consistently growing at a faster rate than the quadratic is [tex]\(f(x) = e^x\)[/tex] and [tex]\(g(x) = x^2\)[/tex].
