In this lab, we will explore how to determine which functions have derivative equal to itself. In other words, for which functions \( f(x) \) do we have that \( f'(x) = f(x) \)?
One such function is the constant function \( f(x) = 0 \), since \( f'(x) = 0 \). Are there others?
In the previous lab, we learned that we can use tangent lines to approximate a function. Suppose that \( f(0) = 1 \) and \( f'(x) = f(x) \). Then, under that assumption, \( f'(0) = f(0) = 1 \). Therefore, the tangent line to this mysterious function is \( y – 1 = 1(x – 0) \) or \( y = x + 1 \). Since this line is approximately equal to \( f(x) \) near \( x = 0 \), we can estimate \( f(a) \) for \( a \) near \( 0 \) with it. For example, using this rough approximation, we can estimate that \( f(1) \approx 1 + 1 = 2 \).
However, one could argue that \( 1 \) is not particularly close to \( 0 \). We could instead use \( \frac{1}{2} \). We get then that \( f\left(\frac{1}{2}\right) \approx \frac{1}{2} + 1 = \frac{3}{2} \). If \( f'(x) = f(x) \), then, in particular, \( f’\left(\frac{1}{2}\right) \approx \frac{3}{2} \). Therefore, the tangent line at \( x = \frac{1}{2} \) is approximately \( y – \frac{3}{2} = \frac{3}{2}\left( x – \frac{1}{2} \right) \). Simplifying, we obtain \( y = \frac{3}{2}x + \frac{3}{4} \). Therefore, \( f(1) \approx \frac{3}{2} \cdot 1 + \frac{3}{4} = \frac{9}{4} = 2.25 \).
This is our “two step” approximation for \( f(1) \), where we first approximate \( f\left(\frac{1}{2}\right) \), then use that to approximate \( f(1) \). We can do this with any number of steps, though it becomes more difficult to do by hand when the number of steps gets large
1. Use the above method to approximate the value of \( f(1) \) for a function that satisfies the following conditions: \( f(0) = 1 \) and \( f'(x) = f(x) \). Do this using four steps. That is, first approximate \( f\left(\frac{1}{4}\right) \) using the tangent line at \( x = 0 \) (namely \( y = x + 1 \)), then approximate \( f\left(\frac{1}{2}\right) \) using the tangent line at \( x = \frac{1}{4} \), then approximate \( f\left(\frac{3}{4}\right) \) using the tangent line at \( x = \frac{1}{2} \), and finally approximate \( f(1) \) using the tangent line at \( x = \frac{3}{4} \).