Five Day Lesson
James Rahn
A Jet Tour Of
Calculus in Five Days
Introduce your students the
main concepts of Calculus
in the first 5 days.
Each lesson includes a one or two page lesson
and a two question assignment
A Jet Tour of Calculus
Day 1: Instantaneous Rate of Change
A homemade rocket is fired initially from a platform 400 feet above ground at a velocity of 300 ft/sec.
After 25 seconds, the rocket hits the ground. Make a sketch of the height of the rocket as a function of
time in Figure 1.
h
h








t
t



Figure 1



The rocket’s height can be modeled by the function
Make a sketch of the model in Figure 2.


Figure 2


h (t )  16t 2  300t  400 at any time t, t  0 .
How is your graph similar to Figure 1? How is it different?
How long was the rocket in the air? How can you tell?
What was the maximum height reached by the rocket? At what time did it reach that height?
At what height was the rocket at 1 second? Based upon the model, was the rocket on its way up or down
at t=1 second? How can you tell?
What was the average rate at which the rocket was traveling during the first second of flight? Explain how
you found this rate? Draw a line segment in Figure 2, whose slope represents this average rate of change.
Approximate the rate of change of the height of the rocket at 1 second, by first calculating the height of
the rocket at various times:
t
0.9
0.99
0.999
0.9999
0.99999
0.999999
1.0
h
Use these heights to calculate the average rate of change of the height over the following intervals? What
is the unit of measure on this rate of change?
Time Interval
Rate of Change
Rahn (c) 2014
[0.9,1]
[0.99,1]
[0.999,1]
[0.9999,1]
[0.99999,1]
[0.999999,1]
What do you notice about the rates of change of the height as the width of the intervals are decreased?
Use the table above to approximate the rate of change of the height of rocket at 1 second.
When is the rocket at the same height it was at 1 second?
Create a table of values to approximate the rate of change of the height at time when the rocket is at the
same height it was at 1 second?
Time Interval
[17.749,17.75]
[17.7499,17.75]
[17.74999,17.75]
[17.749999,17.75]
[17.7499999,17.75]
[17.74999999,17.75]
Rate of Change
Time Interval
Rate of Change
Use the table above to approximate the rate of change of the height of rocket at the moment it reached
the same height as it did at 1 second.
What do you notice about the average rate of change of the height at this time?
At what time did the rocket reach its maximum height? What do you believe the rate of change of the
height was at that time? Complete a chart to estimate the instantaneous rate of change of the height at
that time.
Time Interval
Rate of Change
Time Interval
Rate of Change
Rahn (c) 2014
A Jet Tour of Calculus
Day 1: Instantaneous Rate of Change
ANSWERS
A homemade rocket is fired initially from a platform 400 feet above ground at a velocity of 300 ft/sec.
After 25 seconds, the rocket hits the ground. Make a sketch of the height of the rocket as a function of
time in Figure 1.
h

Figure 1 sketches should be close
to the model graphed in Figure 2.



t

The rocket’s height is modeled by the function




h (t )  16t 2  300t  400 at any time t, t  0 .
Make a sketch of the model in Figure 2. How is your graph similar to Figure 1? How is it different?
How long was the rocket in the air? How can you tell? They should both resemble parabolas. They may
not begin and end at the same locations. They should both illustrate a maximum height at approximate
halfway through the trip.
What was the maximum height reached by the rocket? At what time did it reach that height? The
maximum height is 1806.25 feet at 9.375 minutes.
At what height was the rocket at 1 second? Based upon the model, was the rocket on its way up or down
at t=1 second? How can you tell? The rocket is at a height of 684 feet. The rocket is on its way up
because the heights are increasing when time is near t = 1 second.
What was the average rate at which the rocket was traveling during the first second of flight? Explain how
you found this rate? Draw a line segment in Figure 2, whose slope represents this average rate of change.
684  400
 284 ft
sec
1
Approximate the rate of change of the height of the rocket at 1 second, by first calculating the height of
the rocket at various times:
t
0.9
0.99
0.999
0.9999
0.99999
0.999999
1.0
h
657.04
681.318
683.731
683.973
683.997
683.9997
684
What was the average rate of change of the height over the following intervals? What is the unit of
measure on this rate of change?
Rahn (c) 2014
Time Interval
[0.9,1]
[0.99,1]
[0.999,1]
[0.9999,1]
[0.99999,1]
[0.999999,1]
Rate of Change
269.6
268.16
268.016
268.0016
268.00016
268.000016
What do you notice about the rates of change of the height as the width of the intervals are decreased?
The average rates of change appear to be approaching the number 268 ft/sec.
Use the table above to approximate the rate of change of the height of rocket at 1 second.
It appears that the rocket is traveling at 268 ft/sec when at the time 1 second.
When is the rocket at the same height it was at 1 second? The rocket is at the height of 684 feet when
t=17.75 seconds.
Create a table of values to approximate the rate of change of the height at time when the rocket is at the
same height it was at 1 second?
Time Interval
[17.749,17.75]
[17.7499,17.75]
[17.74999,17.75]
Rate of Change
-267.984
-267.9984
-267.99984
Time Interval
[17.749999,17.75]
[17.7499999,17.75]
[17.74999999,17.75]
Rate of Change
-267.999984
-267.9999984
-267.99999984
Use the table above to approximate the rate of change of the height of rocket at the moment it reached
the same height as it did at 1 second. It appears that the rate of change of the height is approaching -268
ft/sec when the rocket is at 684 ft from the ground on its return trip. This is the opposite of the rate at
time 1 second because the rocket is approaching the ground rather than moving away from the ground.
What do you notice about the average rate of change of the height at this time? It is opposite the rate at
time 1 second.
At what time did the rocket reach its maximum height? What do you believe the rate of change of the
height was at that time? Complete a chart to estimate the instantaneous rate of change of the height at
that time. The rocket reaches it maximum height of 1806.25 feet at 9.375 seconds. Intervals may vary.
Time Interval
[9.3749,9.375]
[9.37499,9.375]
[9.374999,9.375]
Rate of Change
0.0016
0.00016
0.000016
Time Interval
[9.3749999,9.375]
[9.37499999,9.375]
[9.374999999,9.375]
Rate of Change
0.0000016
0.00000016
0.000000016
The rocket’s rate of change of height at 9.375 seconds appears to be approaching 0 ft/sec.
Rahn (c) 2014
A Jet Tour of Calculus
Day 1: Instantaneous Rate of Change
Assignment
1.
A leaf is dropped from the observation deck of the city tower. The distance the leaf falls is given by
the formula d

1 2
gt
2
, where g is the force of gravity working on the leaf or g= 9.8 meter/sec2
and t is measured in seconds. Estimate the rate of change of the distance with respect to time the
leaf is falling at time 1 second and 2 seconds. Show work that supports your answer. Explain
why the rate of change of the distance with respect to time at 2 seconds makes sense based upon
the rate of change of distance with respect to time at 1 second.
2.
A cylindrical shaped oil spill has a constant height of .25 feet and a radius that is changing at the
rate of 6t +1 feet every hour. The volume of the oil spill is given by V
measured in hours.
Rahn (c) 2014
 .25 (6t  1) 2
Estimate the rate of change of the volume when t is 1 second.
where t is
A Jet Tour of Calculus
Day 1: Instantaneous Rate of Change
Assignment Answers
1.
A leaf is dropped from the observation deck of the city tower. The distance the leaf falls is given by
the formula
d
1 2
gt
2
, where g is the force of gravity working on the leaf or g= 9.8 meter/sec2
and t is measured in seconds. Estimate the rate of change of the distance with respect to time the
leaf is falling at time 1 second and 2 seconds. Show work that supports your answer. Explain
why the rate of change of the distance with respect to time at 2 seconds makes sense based upon
the rate of change of distance with respect to time at 1 second.
Time Interval
[0.9,1]
[0.99,1]
[0.999,1]
[0.9999,1]
[0.99999,1]
[0.999999,1]
Rate of Change
9.31
9.751
9.795
9.7995
9.7999
9.7999
Time Interval
[1.9,2]
[1.99,2]
[1.999,2]
[1.9999,2]
[1.99999,2]
[1.999999,2]
Rate of Change
19.11
19.55
19.595
19.59951
19.599951
19.5999951
The rate of change of the distance with respect to time should increase as the leaf falls. Therefore, the
rate of change of the distance should be larger at t=2 than t=1. The rate of change of distance with
respect to time is about 9.8 meters per second at time t=1 second and 19.6 meters per second at time t=2
seconds.
2.
A cylindrical shaped oil spill has a constant height of .25 feet and a radius that is changing at the
rate of 6t +1 feet every hour. The volume of the oil spill is given by V
measured in hours.
 .25 (6t  1) 2
where t is
Estimate the rate of change of the volume when t is 1 second.
Time Interval
[0.9,1]
[0.99,1]
[0.999,1]
[0.9999,1]
[0.99999,1]
[0.999999,1]
Rate of Change
63.146
65.690
65.945
65.970
65.973
65.973
The rate of change of the volume with respect to time at time t = 1 second is about 65.97 feet per second.
Rahn (c) 2014
A Jet Tour of Calculus
Day 2: Behavior of Functions
Make a sketch of each function in the given window. At x = 1 draw a tangent line that approximates the
steepness of the function at x = 1. Approximate the slope of the tangent line. Describe how this tangent
line describes the behavior of the graph at x = 1.

y







y
x
x







y  e x 1
y  ( x  2)2  1


y







y
x
x



y  ( x  1)3  1
Rahn (c) 2014




y  ( x  2)4  2

A Jet Tour of Calculus
Day 2: Behavior of Functions
ANSWERS
Make a sketch of each function in the given window. At x = 1 draw a tangent line that approximates the
steepness of the function at x = 1. Approximate the slope of the tangent line. Describe how this tangent
line describes the behavior of the graph at x = 1.

y







y
x
x








y  ( x  2) 2  1
y  e x 1
The tangent line to the graph at x=1
appears to have a slope of -2. The
negative slope on the tangent line
indicates that the graph is decreasing
at x = 1.
The tangent line to the graph at x = 1
appears to have a slope of 1. The
positive slope on the tangent line
indicates that the graph is increasing
at x = 1.

y







y
x
x








y  ( x  1)3  1
y  ( x  2) 4  2
The tangent line to the graph at x = 1
appears to have a slope of zero. The
zero slope on the tangent line
indicates that the graph has stopped
decreasing at x = 1.
The tangent to graph at x = 1 appears
to have a slope of about 4. The
positive slope on the tangent line
indicates the graph is increasing at x
= 1.
Rahn (c) 2014
A Jet Tour of Calculus
Day 2: Behavior of Functions
Assignment
1. Create a sketch of a function f that has a tangent line whose slope is positive at x = 1 and negative at x
= 3. What is the behavior of the function at each of these two x values?
y




x




2. Create a sketch of a function g that has a tangent line whose slope is zero at x = 1 and negative at x =
2. What is the behavior of the function at each of these two x values?
y




x

Rahn (c) 2014



A Jet Tour of Calculus
Day 2: Behavior of Functions
Assignment Answers
1. Create a sketch of a function f that has a tangent line whose slope is positive at x = 1 and negative at x
= 3. What is the behavior of the function at each of these two x values?
y




x




Answers will vary, but the graph
above has a positive slope at x = 1
and therefore, f is increasing. The
graph has a negative slope at x = 3
and therefore, is f is decreasing.
2. Create a sketch of a function g that has a tangent line whose slope is zero at x = 1 and negative at x =
2. What is the behavior of the function at each of these two x values?

y



x




Answers will vary, but the graph
above has a zero slope at x = 1 and
therefore, f is leveling off. The graph
has a negative slope at x = 3 and
therefore, is f is decreasing.
Rahn (c) 2014
A Jet Tour of Calculus
Day 3: What Can Area Represent?
As you pull out on the highway on your road bike you gradually increase your speed according the graph
below. Then you notice your speedometer approaching 465 ft per minute so you tap hand brake to slow
down your speed to a constant rate of 465 feet per minute.
feet per minute




minutes
        
Notice that the portion of the velocity graph between time t=60 minutes and t=100 minutes is constant.
The distance traveled during this time can be represented by what geometric shape? What are the units
of measure for the height of region? What are the units of measure for the length (or base) of this region?
Using correct units, what is the area of this region? Explain how you determined this unit of measure.
Each rectangular region on the graph represent what distance? Explain how you found your answer.
If the rectangle at the right has the same dimensions at those in the graph above, how many feet
does the shaded part of the rectangle represent?
Find an estimate for the distance traveled by the cyclist from t=0 to t = 60 minutes.
Find out how far your bicycle traveled in the first 100 minutes of the trip.
The distance you traveled on your road bike is represented by the bounded area under the velocity graph
and above the time axis and the two vertical lines t = 0 and t = 100. This area is called the definite
integral of the velocity from time t=0 to t= 100 minutes.
You have just found a geometric method to find an approximate value for the definite integral of the
velocity from t=0 to t= 60 minutes. It involved both estimating bounded area and using geometric area
formulas.
Rahn (c) 2014
The picture below illustrates a right circular cone sitting on its circular base. Suppose we slice the cone
parallel to the base at a point x centimeters from the base. What shape is each cross section?
The graph below shows the area of each cross-section as a function of height where we took the cross
section.
What is the largest cross-sectional area?
What is the cross-sectional area created at a point 1.75 cm from the base?
At what distance from the base was a 4 square centimeter circle cut?
Area (sq. cm.)
20
16
12
8
4
Centimeters
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.
What does each rectangle in this graph represent? Explain your answer.
The definite integral of the area from time x=0 to x=5 centimeters will determine the volume of the
cone.
Estimate the definite integral of y, the area, with respect to x for 0#x#5.
answer.
Rahn (c) 2014
Show work that leads to your
A Jet Tour of Calculus
Day 3: What Can Area Represent?
ANSWERS
As you pull out on the highway on your road bike you gradually increase your speed according the graph
below. Then you notice your speedometer approaching 465 ft per minute so you tap hand brake to slow
down your speed to a constant rate of 465 feet per minute.
feet per minute




minutes
        
Notice that the portion of the velocity graph between time t=60 minutes and t=100 minutes is constant.
The distance traveled during this time can be represented by what geometric shape? What are the units
of measure for the height of region? What are the units of measure for the length (or base) of this region?
Using correct units, what is the area of this region? Explain how you determined this unit of measure. The
region is a rectangle. The units for the height are feet per minute. The units for the length are minutes.
Therefore the units for each rectangle are
feet
•minute=feet .
minute
Each rectangular region on the graph represent what distance? Explain how you found your
answer. Each region represent
100
feet
•10 minutes = 1000 feet .
minute
If the rectangle at the right has the same dimensions at those in the graph above, how many feet does the
shaded part of the rectangle represent? It appears to represent about 0.4 x 1000 feet or 400 feet.
Find an estimate for the distance traveled by the cyclist from t=0 to t = 60 minutes. about 19,000 feet
Find out how far your bicycle traveled in the first 100 minutes of the trip. About 37,600 feet
The distance you traveled on your road bike is represented by the bounded area under the velocity graph
and above the time axis and the two vertical lines t = 0 and t = 100. This area is called the definite
integral of the velocity from time t=0 to t= 100 minutes.
You have just found a geometric method to find an approximate value for the definite integral of the
velocity from t=0 to t= 60 minutes. It involved both estimating bounded area and using geometric area
formulas.
Rahn (c) 2014
The picture below illustrates a right circular cone sitting on its circular base. Suppose we slice the cone
parallel to the base at a point x centimeters from the base. What shape is each cross section? Each shape
is a circle.
The graph below shows the area of each cross-section as a function of height where we took the cross
section.
What is the largest cross-sectional area? The largest area is about 20 square centimeters.
What is the cross-sectional area created at a point 1.75 cm from the base? About 8 square centimeters.
At what distance from the base was a 4 square centimeter circle cut? About 2.75 centimeters.
Area (sq. cm.)
20
16
12
8
4
Centimeters
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.
What does each rectangle in this graph represent? Explain your answer. Each rectangle represents
2square centimeters•0.25centimeters = 0.50 cubic centimeters .
The definite integral of the area from time x=0 to x=5 centimeters will determine the volume of the
cone.
Estimate the definite integral of y, the area, with respect to x for 0#x#5. Show work that leads to your
answer. About 32.5 cubic centimeters (Each square in the graph represents 0.5 cubic centimeters.)
Rahn (c) 2014
A Jet Tour of Calculus
Day 3: What Can Area Represent?
Assignment
1.
A hemisphere is placed on the table so it is sitting on a circular base. Each cross-section, parallel to
the base, is also a circle. The graph in Figure 1 represents the area of each cross-section at a
distance x units from the base. The definite integral from 0 to 5 represents the volume of the
hemisphere by calculating the area bounded by the graph. What does each rectangle in the graph
represent? Explain your answer. Estimate the volume of the hemisphere. Include units on your
answer.
square inches








inches











Figure 1
Area of each cross-section of a hemisphere
for 0  x  5
2.
The velocity of a caterpillar, traveling along a branch, is given in Figure 2. A definite integral from t
= 0 to t = 5 would represent the distance the caterpillar travels during the first five seconds. What
does each square in the graph represent. Support your reasoning. Estimate the distance the
caterpillar travels during the five seconds. If the velocity of the caterpillar is given by
1
v(t )  32  t 3 , estimate the rate of change of the velocity (acceleration) of the caterpillar at 2
2
minutes. Draw a tangent line that represents this
rate of change.
 in/min











min



Figure 2
Velocity of a caterpillar
Rahn (c) 2014

A Jet Tour of Calculus
Day 3: What Can Area Represent?
Assignment Answers
1.
A hemisphere is placed on the table so it is sitting on a circular base. Each cross-section, parallel to
the base, is also a circle. The graph in Figure 1 represents the area of each cross-section at a
distance x units from the base. The definite integral from 0 to 5 represents the volume of the
hemisphere by calculating the area bounded by the graph. What does each rectangle in the graph
represent? Explain your answer. Estimate the volume of the hemisphere. Include units on your
answer.
Each rectangle in the graph represents (10 square inches)(0.25 inches) or 2.5 cubic inches. The
hemisphere has a volume of about 260 cubic inches.
square inches








inches











Figure 1
Area of each cross-section of a hemisphere
for 0  x  5
2.
The velocity of a caterpillar, traveling along a branch, is given in Figure 2. A definite integral from t
= 0 to t = 5 would represent the distance the caterpillar travels during the first five seconds. What
does each square in the graph represent. Support your reasoning. Estimate the distance the
caterpillar travels during the five seconds. If the velocity of the caterpillar is given by
1
v(t )  32  t 3 , estimate the rate of change of the velocity (acceleration) of the caterpillar at 2
2
minutes. Draw a tangent line that represents this rate of change. Estimate the rate of change of
the velocity (acceleration) of the caterpillar at 2 minutes. Show all work that leads to your answer.
Draw a tangent line that represents this rate of change.
Each square in the graph represents (4 in/min)(0.5 min) or 2
inches. At the end of 5 minutes the caterpillar will have travel
about 110 inches.
Time Interval
[1.9,2]
[1.99,2]
[1.999,2]
Rate of Change
-5.705
-5.97005
-5.9970005
Time Interval
[1.9999,2]
[1.99999,2]
[1.999999,2]
Rate of Change
-5.9997
-5.99997
-5.999997
The rate of change of the velocity at 2 is about -6 in/min/min.
 in/min











min



Figure 2
Velocity of a caterpillar
Rahn (c) 2014

A Jet Tour of Calculus
Day 4: Determining a Definite Integral with Formulas
1.
Water is being pumped into a large storage tank at a rate, R (t )  ( x  2)  12 thousands of
3
gallons/day. Draw a sketch of R(t) in Figure 1 for time
0  t  4 days . The definite integral of R(t)
from t = 0 to t = 4 represents the thousands of gallons pumped into the tank during the four days.
1000 gallons/day





day





Figure 1
What does the area of each rectangular region represent in this problem?
Draw five vertical line segments to separate the time interval
0  t  4 days into four equal regions. Use
these five line segments to create rectangles that will approximate the area under the graph.
estimate for the number of gallons pumped into the tank during the four days.
Draw four additional vertical line segments to separate the time interval
Find an
0  t  4 days into eight equal
regions. Find a second estimate for the number of gallons pumped into the tank during the four days.
Explain how increasing the number of line segments will change your estimate for the number of gallons
being pumped into the tank.
Rahn (c) 2014
2.
The velocity of an inch worm is given by the function
v(t )  8 x  x 2 . Draw a sketch of this function
in figure 2.








mm/minutes
minutes





What does the area of each rectangle in the graph represent?
Draw vertical line segments to separate the time interval
0  t  8 seconds into four equal regions.
Use
these line segments, and points along the graph of v(t) to create triangles or trapezoids that will
approximate the area under the graph. Find an estimate for the distance traveled by the inch worm in 8
seconds.
Draw additional vertical line segments to separate the time interval
0  t  8 seconds
into more equal
regions. Use these new intervals and points along the graph of v(t) to find a second estimate for the
distance traveled by the inch worm in 8 seconds.
Explain how additional number of line segments will change your estimate for the distance traveled by the
inch worm in 8 seconds.
Rahn (c) 2014
A Jet Tour of Calculus
Day 4: Determining a Definite Integral with Formulas
ANSWERS
1.
Water is being pumped into a large storage tank at a rate, R (t )  ( x  2)  12 thousands of
3
gallons/day. Draw a sketch of R(t) in Figure 1 for time
0  t  4 days . The definite integral of R(t)
from t = 0 to t = 4 represents the thousands of gallons pumped into the tank during the four days.
Figure 1
What does the area of each rectangular region represent in this problem?
gallons because the dimensions are 4000 gallons/day by 0.5 days.
Draw five vertical line segments to separate the time interval
Each rectangle represent 2000
0  t  4 days into four equal regions. Use
these five line segments to create rectangles that will approximate the area under the graph. Find an
estimate for the number of gallons pumped into the tank during the four days.
Answers will vary. The figure shows four rectangles, whose height is drawn at the left hand endpoint of
each interval. The numbers in each rectangle are in 1000's of gallons. These rectangles have areas that
add up to 40,000 gallons. Students may use other types of rectangles and
have answers between 40,000 and 56,000 gallons.
Draw four additional vertical line segments to separate the time interval
0  t  4 days into eight equal regions. Find a second estimate for the
number of gallons pumped into the tank during the four days. Answers will
vary. As four additional line segments are added the area can range
between 44,000 and 56,000 gallons.
Explain how increasing the number of line segments will change your estimate
for the number of gallons being pumped into the tank.
As the number of rectangles are increased the answer for the number of gallons pumped into the tank
approach 48,000 gallons, the actual area.
Rahn (c) 2014
2.
The velocity of an inch worm is given by the function
v(t )  8 x  x 2 . Draw a sketch of this function
in figure 2.
Figure 2
What does the area of each rectangle in the graph represent? The area represents 2 mm because the
dimensions are 2 mm/minute by 1 minute.
Draw vertical line segments to separate the time interval
0  t  8 seconds into four equal regions.
Use
these line segments, and points along the graph of v(t) to create triangles or trapezoids that will
approximate the area under the graph. Find an estimate for the distance traveled by the inch worm in 8
minutes. The area of the two triangles and two trapezoids adds up to 80 mm. This is how far the inch
worm crawls in 8 minutes.
Draw additional vertical line segments to separate the time interval
0  t  8 seconds
into more equal regions.
Use these new intervals and
points along the graph of v(t) to find a second estimate for the distance
traveled by the inch worm in 8 seconds. Using eight trapezoids and/or
triangles the area will be 84 mm.
Explain how additional number of line segments will change your estimate for
the distance traveled by the inch worm in 8 seconds. Will these estimates
be an over or under estimate for the distance?
As additional trapezoids are added the area of all the triangles and trapezoids will approach the exact area
bounded under the velocity graph because the slanted sides better approximate the curvature of the
velocity graph. The area will approach the value of 85 1/3 mm. These estimates will be under estimates
since the graph of v(t) is concave down. The straight segments will be below the graph
Rahn (c) 2014
A Jet Tour of Calculus
Day 4: Determining a Definite Integral with Formulas
Assignment
1.
A region R is defined by the graph of
f ( x)  81  x 2 and the x-axis is graphed in Figure 1. What
does the area of each of the squares on the graph represent?
Estimate the bounded area by thinking about the full and partial squares contained in the region R.
Use 6 rectangles to approximate the area of region R.
Approximate area of the bounded region R using 6 trapezoids and/or triangles.
Explain why the three estimates differ from each other. Explain why one of the answer is a better
approximation.
y





    
Rahn (c) 2014
x
    
2.
A solid is sliced into cross sections whose area, A(x), is represented by the graph in Figure 2. The
definite integral of A(x) from x=0 to x = 4 will find the volume of the solid.
What does the area of each of the rectangles on the graph represent in the context of this problem?
Use two different estimation techniques to approximate volume of the solid. Compare the two
estimates to the actual volume of the solid.
square inches




inches



Figure 2
Rahn (c) 2014

A Jet Tour of Calculus
Day 4: Determining a Definite Integral with Formulas
Assignment Answers
1.
A region R is defined by the graph of
f ( x)  81  x 2 and the x-axis is graphed in Figure 1. What
does the area of each of the squares on the graph represent? Each rectangle represents 1 square
unit.
Estimate the bounded area by thinking about the full and partial squares contained in the region R.
The area of the bounded region is about 108 full and/or partial squares or about 108 cubic units.
Use 6 rectangles to approximate the area of region R. Using 6 rectangles the area can be between
92 square units and 145 square units.
Approximate area of the bounded region R using 6 trapezoids and/or triangles. Using 6 trapezoids
and/or triangles the area will be about 118.16 square units.
Explain why the three estimates differ from each other. Explain why one of the answer is a better
approximation. The estimate using trapezoids and/or triangles will be closer to the actual area
since the shapes fit closer to the actual shape, but less than the actual area. The actual region is a
semicircle with a radius of 9 so its area is
81
 or 127.234 square units. The rectangles extend
2
over the graph or under the graph, but the trapezoids fit tighter to the graph and all remain under
the graph.
y





    
Rahn (c) 2014
x
    
2.
A solid is sliced into cross sections whose area, A(x), is represented by the graph in Figure 2. The
definite integral of A(x) from x=0 to x = 4 will find the volume of the solid.
What does the area of each of the rectangles on the graph represent in the context of this problem?
Each rectangle represent 0.025 cubic inches of volume since the dimension are 0.1 square inches
by 0.25 inches.
Use two different estimation techniques to approximate volume of the solid. Compare the two
estimates to the actual volume of the solid.
Using 6 rectangles the approximate volume of the solid will be between 1.59 cubic inches and 2.647
cubic inches. Using 6 trapezoids the approximate volume is 2.123 cubic inches. The estimate
using 6 trapezoids should be a better approximation than some of the rectangular approximation
since the trapezoids fit the shape better than the rectangles. Rectangles that fit below the graph
will give an underestimate. Rectangles that fit above the graph will be an overestimate. Trapezoids
will give an overestimater.
square inches




inches



Figure 2
Rahn (c) 2014

A Jet Tour of Calculus
Day 5: What is a Limit?
Consider the graph of the function



f ( x) 
2 x 2  5 x  2
.
x2
Make a sketch of this function in Figure 1.
y





x






How does the graph differ from what you
expected to see?
What does your graph indicate about the value of
f(2)? Why is this? Algebraically calculate f(2).
Why is there no value for f(2)?
Figure 1
Let’s investigate how large the hole is in the function at x = 2 by looking at function values when x is close
to 2.
Complete this table of values for f(x) and then
sketch a new graph of f(x) in a window with those
dimensions:
x
1.9
f(x)
x
2.1
f(x)
Graph of f(x) in a smaller
window
What does this new graph illustrate about f(x)
when you are even closer to x = 2?
Complete this new table of values for f(x) and then sketch a new graph of f(x) in a window with those
dimensions. What does this new graph illustrate about f(x) when you are even closer to x = 2?
x
1.99
f(x)
x
f(x)
Rahn (c) 2014
2.01
What value is f(x) getting close to as x approaches 2?
Explain why you chose this value.
We call this function value the LIMIT of the function f(x) as x approaches 2. It is the number that all
the y values stay close to as x is very close to 2.
What is the limit of the function f(x) as x approaches 2?
Notice that as we got closer to x=2 the function values stayed very closed to your limit value. From the
last set of table values and the graph we can notice that all the function values for f(x) were within 0.02 of
-3 when the x values were within 0.01 of x=2.
Consider a new function g( x ) 
x2  x  6
. Consider the limit of this function as x approaches 3.
2x  6
Create table values for g(x) near x = 3. Complete the table and build a window to represent the numbers
in the shaded column:
x
2.9
2.99
3.1
3.01
g(x)
x
g(x)
Graph of g(x) in a smaller
window
As x got closer to x=3 the function values appears to be approaching what LIMIT value? Fill in the
following statement based on your table values and graphing window:
All the function values for f(x) were within ______ of g(x)=_____ when the x values were within 0.01 of
x=3.
Predict how close all function values for f(x) will be from the limit of g(x)=_____ when x values are kept
within 0.001 of x = 3. Show this on a new graphing window:
Graph of g(x) in a smaller
window
Rahn (c) 2014
A Jet Tour of Calculus
Day 5: What is a Limit?
ANSWERS
g ( x) 
Consider the graph of the function
y



x










2 x 2  5 x  2
.
x2
Make a sketch of this function in Figure 1.
How does the graph diffe r from what you
expected to see? This graph is a linear function
with a hole in it. Some students might expect to
see an asymptotic function.
What does your graph indicate about the value of
f(2)? Why is this? Algebraically calculate f(2).
Why is there no value for f(2)? There is no value
at x = 2. f (x)  (1  2x)(x  2)  (1  2x), x  2 . There is a

( x  2)
hole at x = 2. There is no value at x = 2 since
x 2
is not defined at x =2.
x 2
Figure 1
Let’s investigate how large the hole is in the function at x = 2 by looking at function values when x is close
to 2.
Complete this table of values for f(x) and then sketch a new graph of f(x) in a window with those
dimensions:
x
1.9
f(x)
-2.8
x
2.1
f(x)
-3.2
Graph of f(x) in a smaller
window
What does this new graph illustrate about f(x)
when you are even closer to x = 2? This graph
shows that all function values are staying very
close to f(x)=-3.
Complete this new table of values for f(x) and then sketch a new graph of f(x) in a window with those
dimensions. What does this new graph illustrate about f(x) when you are even closer to x = 2? The
function values are all staying very close to -3.
x
1.99
f(x)
-2.98
x
2.01
f(x)
-3.02
Rahn (c) 2014
What value is f(x) getting close to as x approaches 2? Explain why you chose this value. f(x) is getting
very close to -3 as x approaches 2. This is visible from the table values and the graph.
We call this function value the LIMIT of the function f(x) as x approaches 2. It is the number that all
the y values stay close to as x is very close to 2.
What is the limit of the function f(x) as x approaches 2? The limit of f(x) as x approaches 2 is -3.
Notice that as we got closer to x=2 the function values stayed very closed to your limit value. From the
last set of table values and the graph we can notice that all the function values for f(x) were within 0.02 of
-3 when the x values were within 0.01 of x=2.
Consider a new function g( x ) 
x2  x  6
. Consider the limit of this function as x approaches 3.
2x  6
Create table values for g(x) near x = 3. Complete the table and build a window to represent the numbers
in the shaded column:
x
2.9
2.99
g(x)
2.45
2.495
x
3.1
3.01
g(x)
2.55
2.505
The graph of g(x) in a smaller
window
As x got closer to x=3 the function values appears to be approaching what LIMIT value? The function
appears to be approaching 2.5. Fill in the following statement based on your table values and graphing
window:
All the function values for f(x) were within 0.005 of g(x)=2.5 when the x values were within 0.01 of x=3.
Predict how close all function values for f(x) will be from the limit of g(x)=2.5 when x values are kept
within 0.001 of x = 3. f(x) should be within 0.0005 of 2.5. Show this on a new graphing window:
The graph of f(x) in a smaller
window.
Rahn (c) 2014
A Jet Tour of Calculus
Day 5: What is a Limit?
Assignment
1.
3x 2  11x  6
Consider the function h( x ) 
.
( x  3)
What is the value of h(0)? Support your answer algebraically.
Determine the limit of this function as h approaches zero by considering the following table values.
x
2.9
2.99
2.999
2.9999
2.99999
3.1
3.01
3.001
3.0001
3.00001
h(x)
x
h(x)
If you chose a number in between 2.95  x  3.05 , complete the following statement about all function
values: ___  h(x )  _____ .
Show this visually on a graph of h(x) in a small window around (3,7)
2.
2 x  4, x  3

1
 2 x  2 , x  3
Consider the function k(x )   1
.
Notice this function is not defined at x = 3.
Determine if it appears that a limit as x approaches 3 by considering the following table values.
x
2.9
2.99
2.999
2.9999
2.99999
3.1
3.01
3.001
3.0001
3.00001
k(x)
x
k(x)
Show visually through a graph what is happening when
Rahn (c) 2014
2.9  x  3.1 and .9  k(x )  2.1 .
A Jet Tour of Calculus
Day 5: What is a Limit?
Assignment Answers
1.
Consider the function
h(x ) 
3x 2  11x  6
.
( x  3)
What is the value of h(0)? Support your answer algebraically. There is no value for h(0)
since
3x211x6 (3x2)(x3)
h()
x

3x2,x3
(x3)
(x3)
Determine the limit of this function as h approaches zero by considering the following table values.
x
2.9
2.99
2.999
2.9999
2.99999
h(x)
6.7
6.97
6.997
6.9997
6.99997
x
3.1
3.01
3.001
3.0001
3.00001
h(x)
7.3
7.03
7.003
7.0003
7.00003
If you chose a number in between 2.95  x  3.05 , complete the following
statement about all function values: 6.85  h( x )  7.15 .
Show this visually on a graph of h(x) in a small window around (3,7)
2.
2 x  4, x  3

1
 2 x  2 , x  3
Consider the function k(x )   1
.
Notice this function is not defined at x = 3.
Determine if it appears that a limit as x approaches 3 by considering the following table values.
x
2.9
2.99
2.999
2.9999
2.99999
k(x)
1.8
1.98
1.998
1.9998
1.99998
x
3.1
3.01
3.001
3.0001
3.00001
k(x)
1.05
1.005
1.0005
1.00005
1.000005
It appears that the function k approaches a limit of 2 from the left side of 3
and 1 from the right side of 3. Therefore, a limit does not exist at x = 3.
Show visually through a graph what is happening when
.9  k(x )  2.1 .
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2.9  x  3.1 and
Rahn (c) 2014