Question

An exponential function and a quadratic function are graphed below. Which of the following is true of the growth rate of the functions over the interval
a.The exponential grows at half the rate of the quadratic.
b.The exponential grows at the same rate as the quadratic.
c.The exponential grows at twice the rate of the quadratic.
d.The exponential grows at four times the rate of the quadratic.

Answer 1

Just took the test and the answer is a.

Answer 2

Given an exponential function, say f(x), such that f(0) = 1 and f(1) = 2 and a quadratic finction, say g(x), such that g(0) = 0 and g(1) = 1.

The rate of change of a function f(x) over an interval
$a \leq x \leq b$
is given by
$\frac{f(b)-f(a)}{b-a}$

Thus, the rate of change (growth rate) of the exponential function, f(x) over the interval
$0 \leq x \leq 1$
is given by
$\frac{f(1)-f(0)}{1-0} = \frac{2-1}{1} =1$

Similarly, the rate of change (growth rate) of the quadratic function, g(x) over the interval
$0 \leq x \leq 1$
is given by
$\frac{g(1)-g(0)}{1-0} = \frac{1-0}{1} =1$

Therefore, the exponential grows at the same rate as the quadratic in the interval $0 \leq x \leq 1$.