A weight is attached to a spring. Suppose the position (in meters) of the weight above the floor t seconds after it is released is given by $P ( t ) = 0.5 \sin \left( \pi t + \frac { \pi } { 2 } \right) + 1.2$ . What is the average rate of change of the position of the weight (in m/s) over the time interval [3, 5]?
A)-1.7
B)-1.0
C)-0.5
D)0
E)0.5
F)1.0
G)1.7
H)Cannot be determined

Question

Quizplus · Accepted Answer

The average rate of change of a function over an interval [a, b] is given by $\frac{f(b) - f(a)}{b - a}$. Here, $f(t) = 0.5 \sin(\pi t + \frac{\pi}{2}) + 1.2$, and we need to find the average rate of change over [3, 5]. So, we calculate $f(5)$ and $f(3)$ and then find the difference divided by $5 - 3$. The sine function has a period of $2\pi$, so the values of $f(3)$ and $f(5)$ will be the same because they are one period apart. Therefore, the average rate of change is $0$.

A Weight Is Attached to a Spring P(t)=0.5sin⁡(πt+π2)+1.2P ( t ) = 0.5 \sin \left( \pi t + \frac { \pi } { 2 } \right) + 1.2P(t)=0.5sin(πt+2π​)+1.2

A Weight Is Attached to a Spring $P ( t ) = 0.5 \sin \left( \pi t + \frac { \pi } { 2 } \right) + 1.2$