Appendix: The Mathematics of Exponential Growth

The appendix explains the mathematics behind exponential growth and contrasts it with linear growth. The text explores why some processes, like lithium-ion battery energy density, appear linear even when plotted logarithmically. Using a Taylor series expansion, it shows that when kCx (where k depends on the base) is very small, higher-order terms are negligible, making exponential growth approximate linear growth. This explains why early stages of exponential processes can resemble linear increases. The same reasoning applies to exponential decay, with adjustments for sign.