Part 5/7:
For ( f_1'(x) = 2^x \cdot \lim_{h \to 0} \frac{2^h - 1}{h} ), we find that ( f_1'(0) \approx 0.69 ).
For ( f_2'(x) = 3^x \cdot \lim_{h \to 0} \frac{3^h - 1}{h} ), this yields ( f_2'(0) \approx 1.1 ).
Subsequently, we observe that the derivative function for ( e^x ) sits in between those of ( 2^x ) and ( 3^x ), effectively confirming that:
[
0.69 < e < 1.1
]
Graphical Interpretation
Visually, the graphs of ( e^x ), ( 2^x ), and ( 3^x ) reveal interesting patterns. When plotted, the slopes at the origin can be compared:
The slope of ( 3^x ) at ( x = 0 ) is approximately ( 1.1 ).
The slope of ( 2^x ) at ( x = 0 ) is approximately ( 0.69 ).