Let s(t)=t+19−t
 be the position of an object at time t
. Here s
 is measured in meters and t
 is measured in seconds.

(a)   Simplify the following difference quotient as much as possible.

s(t+h)−s(t)h=
 
functionsequation editor
(b)   Use your result from (a) and to calculate the instantaneous velocity of the object at time t
.

v(t)=s′(t)=limh→0s(t+h)−s(t)h=
 
functionsequation editor  m/s

(c)   What is the object's velocity at t=8
 seconds?

v(8)=
 
functionsequation editor  m/s

Let's tackle each part of the problem step by step.

### Part (a): Simplifying the difference quotient

We are given the position function \( s(t) = t + 19 - t = 19 \). This simplifies to a constant position function:
\[
s(t) = 19
\]

Now, let's calculate the difference quotient:
\[
\frac{s(t+h) - s(t)}{h}
\]

Since \( s(t) = 19 \) (a constant), we have:
\[
s(t+h) = 19 \quad \text{and} \quad s(t) = 19
\]

So,
\[
\frac{s(t+h) - s(t)}{h} = \frac{19 - 19}{h} = \frac{0}{h} = 0
\]

Thus, the simplified difference quotient is:
\[
\frac{s(t+h) - s(t)}{h} = 0
\]

### Part (b): Instantaneous velocity of the object at time \( t \)

The instantaneous velocity is given by:
\[
v(t) = s'(t) = \lim_{h \to 0} \frac{s(t+h) - s(t)}{h}
\]

From part (a), we already simplified the difference quotient to 0. Therefore, the instantaneous velocity is:
\[
v(t) = 0 \, \text{m/s}
\]

### Part (c): Object's velocity at \( t = 8 \) seconds

The velocity at any time \( t \) is the same, since \( v(t) = 0 \) for all \( t \). So, at \( t = 8 \) seconds, the velocity is:
\[
v(8) = 0 \, \text{m/s}
\]

### Summary:
- The difference quotient simplifies to 0.
- The instantaneous velocity of the object is 0 m/s at any time.
- At \( t = 8 \) seconds, the velocity is 0 m/s.

Would you like further details on any part of this? Here are 5 related questions to expand your understanding:

1. What is the interpretation of a constant position function like \( s(t) = 19 \)?
2. How does a constant velocity of 0 affect the object's motion?
3. What is the difference between average velocity and instantaneous velocity?
4. How can we verify that the object's velocity remains 0 for all \( t \)?
5. Can the velocity ever change if the position function is constant?

**Tip:** When the position function is constant, the object is not moving, meaning the velocity is always 0!

Let s(t) = t + 19 - t be the position of an object at time t. Here s is measured in meters and t is measured in seconds.

(a) Simplify the following difference quotient as much as possible: s(t+h) - s(t) / h.
(b) Use your result from (a) to calculate the instantaneous velocity of the object at time t.
(c) What is the object's velocity at t = 8 seconds?

Learn how to simplify a difference quotient and calculate instantaneous velocity for a constant position function s(t) = 19. This problem covers key calculus concepts such as the difference quotient and the limit definition of derivative, suitable for students in Grades 11-12.

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