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Solution to Approximating Definite Integrals (IN2) #772

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Solution to Approximating Definite Integrals (IN2)
mlamich1 May 21, 2025
38ce96f
Merge remote-tracking branch 'origin/main' into Answer-to-4.2-Approxi…
mlamich1 May 23, 2025
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140 changes: 128 additions & 12 deletions source/calculus/source/04-IN/02.ptx
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Expand Up @@ -21,6 +21,9 @@
</introduction>

<task permid="IYe">
<statement>


<p permid="NHX">
On the left-hand axes provided in <xref ref="F-4-2-IN1">Figure</xref>,
sketch a labeled graph of the velocity function <m>v(t) = 3</m>.
Expand Down Expand Up @@ -59,37 +62,78 @@
the units on the right-hand axes differ from those on the left.
The right-hand axes will be used in question (d).
</p>
</statement>
<answer>
<p>
The velocity function <m> v(t)= 3</m> is a horizontal line at
<m>y = 3</m> since the velocity is constant.
</p>
</answer>
</task>

<task permid="pfn">
<statement>

<p permid="Gdy">
How far did the person travel during the two hours?
How is this distance related to the area of a certain region under the graph of <m>y = v(t)</m>?
</p>
</statement>
<answer>
<p>
6
</p>
</answer>
</task>

<task permid="Vmw">
<statement>

<p permid="mkH">
Find an algebraic formula, <m>s(t)</m>,
for the position of the person at time <m>t</m>,
assuming that <m>s(0) = 0</m>.
Explain your thinking.
</p>
</statement>
<answer>
<p>
<m> S(t) = 3t </m>
</p>
</answer>
</task>

<task permid="BtF">
<statement>

<p permid="SrQ">
On the right-hand axes provided in <xref ref="F-4-2-IN1" />,
sketch a labeled graph of the position function <m>y = s(t)</m>.
</p>
</statement>
<answer>
<p>
It is a line with constant slope of 3
</p>
</answer>
</task>

<task permid="hAO">
<statement>

<p permid="yyZ">
For what values of <m>t</m> is the position function <m>s</m> increasing?
Explain why this is the case using relevant information about the velocity function <m>v</m>.
</p>
</statement>
<answer>
<p>
<m> S(t) </m> is increasing for all <m> t \geq 0</m>
</p>
</answer>
</task>


</activity>

<activity xml:id="act-4-1-1" permid="gon">
Expand Down Expand Up @@ -175,38 +219,67 @@
</sidebyside>
</introduction>

<task permid="JWp">
<task permid="JWp"> <statement>

<p permid="Ehd">
Using the grid, graph,
and given data appropriately,
estimate the distance traveled by the walker during the two hour interval from <m>t = 0</m> to <m>t = 2</m>.
You should use time intervals of width <m>\Delta t = 0.5</m>,
choosing a way to use the function consistently to determine the height of each rectangle in order to approximate distance traveled.
</p>
</statement>
<answer>
<p>
Distance <m> \approx </m> 3.7505 miles.
</p>
</answer>
</task>

<task permid="qdy">
<task permid="qdy"> <statement>

<p permid="kom">
How could you get a better approximation of the distance traveled on <m>[0,2]</m>?
Explain, and then find this new estimate.
</p>
</statement>
<answer>
<p>
Use smaller intervals with <m> \Delta t = 0.25</m> instead of <m> 0.5</m> and new the estimate will be <m> 3.87575</m>.
</p>
</answer>
</task>

<task permid="WkH">
<task permid="WkH"> <statement>

<p permid="Qvv">
Now suppose that you know that <m>v</m> is given by <m>v(t) = 0.5t^3-1.5t^2+1.5t+1.5</m>.
Remember that <m>v</m> is the derivative of the walker's position function,
<m>s</m>.
Find a formula for <m>s</m> so that <m>s' = v</m>.
</p>
</statement>
<answer>
<p>
<m> S(t) = \frac{1}{8}t^4 -\frac{1}{2}t^3 + \frac{3}{4}t^2+ \frac{3}{2}t + C </m>
</p>
</answer>
</task>

<task permid="CrQ">
<task permid="CrQ"> <statement>

<p permid="wCE">
Based on your work in (c),
what is the value of <m>s(2) - s(0)</m>?
What is the meaning of this quantity?
</p>
</statement>
<answer>
<p>
<m> s(2) - s(0) = 2 </m>. It means The walker traveled exactly 2 miles between time <m> t = 2 </m> and <m> t = 0 </m> hrs.

</p>
</answer>
</task>
</activity>

Expand Down Expand Up @@ -246,7 +319,7 @@
</statement>
<solution>
<p>
Some of the values <m>f(s_i)</m> are negative.
C. Some of the values <m>f(s_i)</m> are negative.
</p>
</solution>
</activity>
Expand Down Expand Up @@ -330,7 +403,7 @@
</statement>
<solution>
<p>
Some of the values <m>f(s_i)</m> are negative.
C. Some of the values <m>f(s_i)</m> are negative.
</p>
</solution>
</activity>
Expand All @@ -351,31 +424,74 @@
on the interval <m>[2, 4]</m> with 3 subintervals.
</p>
</introduction>
<task>
<task> <statement>

<p>What are <m>a</m> and <m>b</m> in this case?</p>
</statement>
<answer>
<p>
<m> a = 2 </m> and <m> b = 4 </m>.
</p>
</answer>
</task>
<task>
<task> <statement>

<p>
What is the value of <m>n</m>?
</p>
</statement>
<answer>
<p>
<m> n = 3 </m>.
</p>
</answer>
</task>
<task>
<task> <statement>

<p>
What are the values of the <m>x_i</m>?
</p>
</statement>
<answer>
<p>
<m> x_0 = 2 </m>, <m> x_1= \frac{8}{3} </m>, <m> x_2= \frac{10}{3} </m>, and <m> x_3= 4 </m>
</p>
</answer>
</task>
<task>
<task> <statement>

<p>
What are the values of the <m>s_i</m>?
</p>
</statement>
<answer>
<p>
<m> s_1= x_0 = 2 </m>, <m> s_2 = x_1= \frac{8}{3} </m>, and <m> s_3= x_2= \frac{10}{3} </m>
</p>
</answer>
</task>
<task>
<task> <statement>

<p>What do you notice about the subinterval widths <m>x_{i} - x_{i-1}</m>?</p>
</statement>
<answer>
<p>
Each subinterval has same width of <m> \frac{2}{3} </m>
</p>
</answer>

</task>
<task>
<task> <statement>

<p>
What is the value of the left Riemann sum?
</p>
</statement>
<answer>
<p>
The left Riemann sum is approximately <m> 3.995 </m>
</p>
</answer>
</task>
</activity>

Expand Down