Question

activity 1.1.4. for the function given by ( s(t) = 64 - 16(t - 1)^2 ) from preview activity 1.1.1, find the most simplified expression you can for the average velocity of the ball on the interval (2, 2 + h). use your result to compute the average velocity on (1.5, 2) and to estimate the instantaneous velocity at ( t = 2 ). finally, compare your earlier work in activity 1.1.3.

Explanation:

Step1: Recall average velocity formula

The average velocity \( v_{avg} \) on the interval \([a, a + h]\) for a position function \( s(t) \) is given by \( v_{avg}=\frac{s(a + h)-s(a)}{h} \). Here, \( a = 2 \), so we need to find \( s(2 + h) \) and \( s(2) \).

First, find \( s(2) \):
Substitute \( t = 2 \) into \( s(t)=64 - 16(t - 1)^2 \):
\( s(2)=64-16(2 - 1)^2=64 - 16(1)^2=64 - 16 = 48 \)

Next, find \( s(2 + h) \):
Substitute \( t = 2 + h \) into \( s(t) \):
\( s(2 + h)=64-16((2 + h)-1)^2=64-16(h + 1)^2 \)
Expand \( (h + 1)^2=h^2 + 2h + 1 \), so:
\( s(2 + h)=64-16(h^2 + 2h + 1)=64-16h^2-32h - 16=48-16h^2-32h \)

Step2: Compute the difference \( s(2 + h)-s(2) \)

\( s(2 + h)-s(2)=(48-16h^2-32h)-48=-16h^2-32h \)

Step3: Find the average velocity formula

Using the average velocity formula \( v_{avg}=\frac{s(2 + h)-s(2)}{h} \), substitute the difference:
\( v_{avg}=\frac{-16h^2-32h}{h} \), for \( h
eq0 \), we can factor out \( h \) in the numerator:
\( v_{avg}=\frac{h(-16h - 32)}{h}=-16h - 32 \)

Step4: Compute average velocity on \([1.5, 2]\)

First, note that the interval \([1.5, 2]\) can be written as \([2 - 0.5, 2]\), so \( a = 2 \), \( h=- 0.5 \) (since \( 2+h = 1.5\Rightarrow h=1.5 - 2=-0.5 \)).

Use the average velocity formula \( v_{avg}=-16h - 32 \), substitute \( h=-0.5 \):
\( v_{avg}=-16(-0.5)-32=8 - 32=-24 \)

Step5: Estimate instantaneous velocity at \( t = 2 \)

The instantaneous velocity at \( t = 2 \) is the limit of the average velocity as \( h
ightarrow0 \). So, take the limit of \( v_{avg}=-16h - 32 \) as \( h
ightarrow0 \):
\( \lim_{h
ightarrow0}(-16h - 32)=-32 \)

Part 1: Simplified average velocity on \([2, 2 + h]\)

Answer:

\( -16h - 32 \) (for \( h
eq0 \))

Part 2: Average velocity on \([1.5, 2]\)