The graph below shows an exponential function and a quadratic function.
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How do the functions compare over the interval 0 H
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MrRoyal MrRoyal · Answer 1 · 2022-06-23T19:23:29+02:00

The exponential function grows at approximately half the rate of the quadratic function

The graph below shows an exponential function and a quadratic function. How do the functions compare over the interval 0 ≤ x ≤ 1?

At the interval 0 ≤ x ≤ 1, we have:

Exponential function Quadratic function

f(0) = 1 g(0) = 0

f(1) = 2 g(1) = 2

The average rate of change is calculated using:

[tex]m = \frac{y_2 -y_1}{x_2-x_1}[/tex]

So, we have:

[tex]f'(x) = \frac{2-1}{1-0}[/tex]

f'(x) = 1

[tex]g'(x) = \frac{2-0}{1-0}[/tex]

g'(x) = 2

By comparison;

g'(x) = 2 * f'(x)

This means that the exponential function grows at approximately half the rate of the quadratic function