#e^x

Old version

Key variables:

a = a(t)= amplitude

k= initial amp

c= damping constant

omega = h. stretch (or comp.)

Basic Example

#1: k=20, c=0.5, p=4, y=a(t)cos(omega*t)

Answer

Word Problems

Guitar

#2: A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2cm from its resting position. After 3 seconds, the displacement of string measures 1 cm. Find the damping constant.

Ans: ln(2)/3

Animal Population

#3: A deer population oscillates 19 above and below the average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 deer each year. Find a function that models population, P, in terms of months since January (t).

Ans: P(t)= -19cos([pi/6]t)+800 + 40t/3

Spring

#4 A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds and its amplitude decreases by 30% every second. Find an equation that models the position of the spring “t” seconds after geing released.

Ans: y= [-5e^(-0.357t)]cos([2pi/3]t)+10

#5

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Shortcut

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Euler’s Number ‘e’ and Natural Logarithm

Things grow in nature in a fashion which is related to their own numbers or size, or when decay, they decay with the relationship with their own number, see here.

Let say any Number of a natural event (such as birth or growth of any living thing) in a particular time period gets double. The time period is considered as a unit time period. 

In the beginning, the Number was 1. First unit time period