ALG-20

Exponential growth

y = y0 e^{kt}. Continuous growth or decay.

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ExponentialExponential growth

Governing equation

y=y_0 e^{kt}

where

y_0: Initial y (—)
k: Rate k (1/s)
t: Time (s)
y: y(t) (—)

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-20 — Exponential growth) is the form associated with Exponential growth. Working symbols: $y_0$ , $k$ , $t$ $\rightarrow$ $y$ . The unique C¹ solution of y' = k y, y(0) = y0. k > 0 growth, k < 0 decay.

Purpose

Purpose: compute

y

from

y_0

k

t

in Algebra via

y=y_0 e^{kt}

y = y0 e^{kt}. Continuous growth or decay. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

y_0 = 10.000\,\mathrm{—}

k = 0.200\,\mathrm{1/s}

t = 4.000\,\mathrm{s}

, the governing relation

y=y_0 e^{kt}

yields

y = 22.2554\,\mathrm{—}

. Initial y0, rate k, time t. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Initial y y_010 —Rate k k0.2 1/sTime t4 s

Outputs

y(t) y22.2554 —

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Narration of this film

Initial y0, rate k, time t.

The unique C¹ solution of y' = k y, y(0) = y0. k > 0 growth, k < 0 decay.

Reading speed

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