Limits

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 sin πt + 4 cos πt, where t is measured in seconds. (Round your answers to two decimal places.)

i. [1,2]

$$\\s(1)=2sin(\pi *1)+4cos(\pi *1)\\ s(1)=2sin(\pi)+4cos(\pi)\\ s(1)=0+4*-1\\ s(1)=-4\\\\ \\s(2)=2sin(\pi *2)+4cos(\pi *2)\\ s(2)=2sin(2\pi)+4cos(2\pi)\\ s(2)=0+4*+1\\ s(2)=4\\\\ Average\; velocity=\frac{distance}{time}\\ Average\; velocity=\frac{4--4}{2-1}\\ Average\; velocity=\frac{8}{1}\\ Average\; velocity=8 \;cm/sec$$

ii. [1,1.1]

iii. [1.1.01]

iv. [1.1.001]

(b) Find the instaneous velocity of the particle when t=1

$$\\s=2sin(\pi t)+4cos(\pi t)\\ v(t)=\frac{ds}{dt}=2\pi cos(\pi t)+-4\pi sin(\pi t)\\ v(t)=\frac{ds}{dt}=2\pi cos(\pi t)-4\pi sin(\pi t)\\ v(1)=2\pi cos(\pi*1 )-4\pi sin(\pi *1)\\ v(1)=2\pi *-1 \quad -4\pi *0\\ v(1)=-2\pi \;\;cm/sec\\$$