Respuesta :
Answer:
The instantaneous velocity for [tex]s(t) = 8t + 19[/tex] when t = 4 is [tex]v(t)=8 \:\frac{ft}{s}[/tex].
Step-by-step explanation:
The average rate of change of function f over the interval is given by this expression:
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
The average velocity is the average rate of change of distance with respect to time
[tex]average \:velocity=\frac{distance \:traveled}{time \:elapsed} =\frac{\Delta s}{\Delta t}[/tex]
The instantaneous rate of change is defined to be the result of computing the average rate of change over smaller and smaller intervals.
The derivative of f with respect to x, is the instantaneous rate of change of f with respect to x and is thus given by the formula
[tex]f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]
For any equation of motion s(t), we define what we call the instantaneous velocity at time t to be the limit of the average velocity, between t and t + Δt, as Δt approaches 0.
[tex]v(t)=\lim_{\Delta t \to 0} \frac{s(t+\Delta t)-s(t)}{\Delta t}[/tex]
We know the equation of motion [tex]s(t) = 8t + 19[/tex] and we want to find the instantaneous velocity for the given value of t = 4.
Applying the definition of instantaneous velocity, we get
[tex]v(t)=\lim_{\Delta t \to 0} \frac{s(t+\Delta t)-s(t)}{\Delta t}\\\\v(4)=\lim_{\Delta t \to 0} \frac{8(4+\Delta t)+19-(8(4)+19)}{\Delta t}\\\\v(t)=\lim_{\Delta t \to 0} \frac{32+8\Delta t+19-32-19}{\Delta t}\\\\v(t)=\lim_{\Delta t \to 0} \frac{8\Delta t}{\Delta t}\\\\v(t)=8 \:\frac{ft}{s}[/tex]
The instantaneous velocity for [tex]s(t) = 8t + 19[/tex] when t = 4 is [tex]v(t)=8 \:\frac{ft}{s}[/tex].
