For which pair of functions is the exponential consistently growing at a faster rate than the quadratic over the interval $0 \leq x \leq 5$?
Answer (1)
Analyze the growth rates of exponential and quadratic functions over the interval 0 ≤ x ≤ 5 .
Compare the values of a x and x 2 at different points within the interval.
Observe that x^2"> 3 x > x 2 for all x in the interval [ 0 , 5 ] , and the difference increases as x increases.
Conclude that the exponential function 3 x grows consistently faster than the quadratic function x 2 over the given interval, so the answer is 3 x and x 2 .
Explanation
Problem Analysis We are asked to determine which pair of functions has the exponential function consistently growing at a faster rate than the quadratic function over the interval 0 ≤ x ≤ 5 . This means we need to compare the values of the exponential and quadratic functions for each pair over the given interval and see if the exponential function's values are always greater than the quadratic function's values, and that the difference between their values increases as x increases.
General Approach Let's consider a general exponential function f ( x ) = a x and a general quadratic function g ( x ) = b x 2 + c x + d . We want to find a pair of functions such that g(x)"> f ( x ) > g ( x ) for all x in the interval [ 0 , 5 ] . We can test different pairs of exponential and quadratic functions to see which one satisfies this condition.
Testing the Pair (3^x, x^2) After testing several pairs, we find that the exponential function 3 x grows consistently faster than the quadratic function x 2 over the interval 0 ≤ x ≤ 5 . To verify this, we can compare their values at a few points:
At x = 0 : 3 0 = 1 and 0 2 = 0 . So, 0^2"> 3 0 > 0 2 .
At x = 1 : 3 1 = 3 and 1 2 = 1 . So, 1^2"> 3 1 > 1 2 .
At x = 2 : 3 2 = 9 and 2 2 = 4 . So, 2^2"> 3 2 > 2 2 .
At x = 3 : 3 3 = 27 and 3 2 = 9 . So, 3^2"> 3 3 > 3 2 .
At x = 4 : 3 4 = 81 and 4 2 = 16 . So, 4^2"> 3 4 > 4 2 .
At x = 5 : 3 5 = 243 and 5 2 = 25 . So, 5^2"> 3 5 > 5 2 .
In each case, the value of the exponential function 3 x is greater than the value of the quadratic function x 2 , and the difference between their values increases as x increases. This indicates that 3 x grows faster than x 2 over the interval [ 0 , 5 ] .
Conclusion Therefore, the pair of functions for which the exponential function is consistently growing at a faster rate than the quadratic function over the interval 0 ≤ x ≤ 5 is 3 x and x 2 .
Examples
In finance, understanding the growth rate of investments is crucial. Exponential growth, like compound interest, can significantly outpace quadratic growth, such as linearly increasing savings. For instance, consider a scenario where one investment grows exponentially (e.g., a high-yield stock) and another grows quadratically (e.g., a real estate investment with appreciation). Over time, the exponential investment will likely yield much higher returns, illustrating the importance of recognizing and leveraging exponential growth in financial planning.
