diff --git a/source/calculus/source/04-IN/02.ptx b/source/calculus/source/04-IN/02.ptx
index c2e8f7806..2a5fd4a9e 100644
--- a/source/calculus/source/04-IN/02.ptx
+++ b/source/calculus/source/04-IN/02.ptx
@@ -21,6 +21,9 @@
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On the left-hand axes provided in Figure,
sketch a labeled graph of the velocity function v(t) = 3.
@@ -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).
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+ The velocity function v(t)= 3 is a horizontal line at
+ y = 3 since the velocity is constant.
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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 y = v(t)?
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+ 6
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Find an algebraic formula, s(t),
for the position of the person at time t,
assuming that s(0) = 0.
Explain your thinking.
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+ S(t) = 3t
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+
On the right-hand axes provided in ,
sketch a labeled graph of the position function y = s(t).
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+ It is a line with constant slope of 3
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For what values of t is the position function s increasing?
Explain why this is the case using relevant information about the velocity function v.
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+ S(t) is increasing for all t \geq 0
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+
@@ -175,7 +219,8 @@
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Using the grid, graph,
and given data appropriately,
@@ -183,30 +228,58 @@
You should use time intervals of width \Delta t = 0.5,
choosing a way to use the function consistently to determine the height of each rectangle in order to approximate distance traveled.
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+ Distance \approx 3.7505 miles.
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How could you get a better approximation of the distance traveled on [0,2]?
Explain, and then find this new estimate.
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+ Use smaller intervals with \Delta t = 0.25 instead of 0.5 and new the estimate will be 3.87575.
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+
Now suppose that you know that v is given by v(t) = 0.5t^3-1.5t^2+1.5t+1.5.
Remember that v is the derivative of the walker's position function,
s.
Find a formula for s so that s' = v.
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+ S(t) = \frac{1}{8}t^4 -\frac{1}{2}t^3 + \frac{3}{4}t^2+ \frac{3}{2}t + C
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-
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+
Based on your work in (c),
what is the value of s(2) - s(0)?
What is the meaning of this quantity?
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+ s(2) - s(0) = 2 . It means The walker traveled exactly 2 miles between time t = 2 and t = 0 hrs.
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@@ -246,7 +319,7 @@
- Some of the values f(s_i) are negative.
+ C. Some of the values f(s_i) are negative.
@@ -330,7 +403,7 @@
- Some of the values f(s_i) are negative.
+ C. Some of the values f(s_i) are negative.
@@ -351,31 +424,74 @@
on the interval [2, 4] with 3 subintervals.
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What are a and b in this case?
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+ a = 2 and b = 4 .
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What is the value of n?
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+ n = 3 .
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-
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What are the values of the x_i?
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+ x_0 = 2 , x_1= \frac{8}{3} , x_2= \frac{10}{3} , and x_3= 4
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-
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+
What are the values of the s_i?
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+ s_1= x_0 = 2 , s_2 = x_1= \frac{8}{3} , and s_3= x_2= \frac{10}{3}
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-
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What do you notice about the subinterval widths x_{i} - x_{i-1}?
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+ Each subinterval has same width of \frac{2}{3}
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-
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What is the value of the left Riemann sum?
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+ The left Riemann sum is approximately 3.995
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+