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CALCULUS AB
WORKSHEET ON CONTINUITY AND INTERMEDIATE VALUE THEOREM
Work the following on notebook paper.
On problems 1 – 4, sketch the graph of a function f that satisfies the stated conditions.
1. f has a limit at x = 3, but it is not continuous at x = 3.
2. f is not continuous at x = 3, but if its value at x = 3 is changed from 3 1f
to 3 0f , f becomes continuous at x = 3.
3. f has a removable discontinuity at x = c for which f c is undefined.
4. f has a removable discontinuity at x = c for which f c is defined.
________________________________________________________________________________
On problems 5 – 7, use the definition of continuity to prove that the function is discontinuous at the
given value of a. Sketch the graph of the function.
________________________________________________________________________________
On problems 8 – 9, use the definition of continuity to find the values of k and/or m that will make the
function continuous everywhere.
2
2
3
5, 2, 2
8. 9. ( 3) , 1 22 , 2
2 7, 1
x xkx x
f x g x m x k xx k x
x x x
________________________________________________________________________________
On problems 10 – 12, a function f and a closed interval [a, b] are given. Show whether the
conditions of the Intermediate Value Theorem hold for the given value of k. If the conditions hold,
find a number c such that .f c k If the theorem does not hold, give the reason. Whether the
theorem holds or not, sketch the curve and the line y = k.
2 2 110. 2 11. 25 12.
2
, 0, 3 , 4.5, 3 , 3, 5
51 3
6
f x x x f x x f xx
a b a b a b
k k k
________________________________________________________________________________
13. Given the function 2 2 5.f x x x
(a) Does 7f x somewhere on the interval 1, 3 ? Use the Intermediate Value Theorem to
show why or why not.
(b) Must 12f x somewhere on the interval 1, 3 ? Use the Intermediate Value Theorem to
show why or why not.
TURN->>>
22
22
3 if 05 4 if 3
5. , 1 6. 3 7. 091 if 0
1 if 3
xx xe xx x x
f x a g x a h x axx x x
x
14. Use the Intermediate Value Theorem to show that 3 0x x has a root in the interval 1, 2 .
15. One night in January, the outside temperature at midnight was 42°F. At 10 AM the next
morning, the temperature was 57°F.
(a) Must there have been a time between midnight and 10 AM when the temperature was 50°F?
Explain how you know.
(b) Must there have been a time between midnight and 10 AM when the temperature was 40°F?
Explain how you know.
(c) Could there have been a time between midnight and 10 AM when the temperature was 40°F?
Explain how you know.
16. One afternoon you were driving on Hwy 290, headed for College Station. At 2 PM, you were
driving 60 miles per hour. At 3 PM, you were driving 55 miles per hour.
(a) Must there have been a time between 2 PM and 3PM when you were driving 57 miles per hour?
Explain how you know.
(b) Must there have been a time between 2 PM and 3PM when you were driving 45 miles per hour?
Explain how you know.
(c) Could there have been a time between 2 PM and 3PM when you were driving 45 miles per hour?
Explain how you know.
CALCULUS AB
WORKSHEET 1 ON 2.1
Work the following on notebook paper. Give decimal answers correct to three decimal places.
1. With an initial deposit of $1000, the balance in a bank account after t years is 1000 1.03t
f t
dollars.
(a) What are the units of the rate of change of ?f t
(b) Compute the average rate of change of f t on the following time intervals.
Time interval 1.9, 2 1.99, 2 1.999, 2 2, 2.1 2, 2.01 2, 2.001
Average
rate of
change
(c) Use your answers to (b) to estimate the instantaneous rate of change of f t at t = 2 years.
__________________________________________________________________________________
2. The figure on the right shows the estimated number N of
internet users in Chile, based on data from the United Nations
Statistics Division.
(a) Estimate the rate of change of N at t = 2003.5.
(b) Does the rate of change increase or decrease as t increases?
Explain.
(c) Compute the average rate of change over [2001, 2005].
(d) Is the rate of change at t = 2002 greater than or less than
the average rate of change you found in (c)?
__________________________________________________________________________________
3. Given 2 5 .f x x x
(a) Sketch the function and the tangent line to the graph at the point where x = 1.
(b) Based on your sketch, do you expect 1f to be positive or negative?
(c) Compute 1f three different ways:
1) by using the definition of the derivative
2) by using the alternative form of the definition of the derivative
3) by using the shortcut that you learned last year.
(d) Write the equation of the tangent line to the graph of y f x at x = 1. Leave your equation in
point-slope form, 1 1 .y y m x x
TURN->>>
4. Use the figure on the right to answer the following.
(a) Find the slope of the secant line through 2, 2f
and 2.5, 2.5 .f Is it larger or smaller than 2 ?f
Explain.
(b) Estimate 2 2f h f
h
for 0.5.h What does
this quantity represent? Is it larger or smaller than 2 ?f
(c) Estimate 1f and 2f .
(d) Find a value of h for which 2 2
0.f h f
h
__________________________________________________________________________________
5. Use the figure on the right to answer the following.
(a) Determine f a for a = 1, 2, 4, 7.
(b) For which values of x is 0?f x
(c) Which is larger, 5.5 or 6.5 ?f f
(d) Do you think that 3f exists? Explain.
_________________________________________________________________________________
6. Sketch the graph of sinf x x on 0, 2 , and guess the value of .2
f
Then calculate the
difference quotient at 2
x
for two small positive and negative values of h. Are these calculations
consistent with your guess?
__________________________________________________________________________________
7. Use the graph of the function on the right to determine
the intervals along the x-axis on which the derivative
is positive.
__________________________________________________________________________________
Use the definition of the derivative,
0limh
f x h f xf x
h
, to find f x on problems 8 – 10.
8. 232 5f x x x
9. 1
3f x
x
10. 4f x x
Answers to Worksheet 1 on 2.1
1. (a) dollars
year
(b) 31.313, 31.354, 31.358, 31.405, 31.364, 31.359
(c) About 31.359 dollars per year
2. (a) About 0.3 million internet users per year
(b) As t increases, the slope of the tangent line decreases so the rate of change decreases.
(c) About 0.35 million internet uses per year
(d) Greater. The tangent line at t = 2002 has a great slope than the secant line that passes
through (2001, N(2001) and (2005, N(2005).
3. The derivative is positive for 1 2.5 and 3.5 4.5.x x
4. (a) 1
1 0, 2 0, 4 , 7 02
f f f f
(b) 7 9x
(c) 5.5f is larger because the slope of the tangent line at x = 5.5 is greater than the slope of
the tangent line at x = 6.5
(d) No. As x approaches 3 from the left, the slope is 0, but as x approaches 3 from the right,
the slope is 1
.2
Since the slopes are not approaching the same value, 3f does not exist.
5. (a) The secant line through 2, 2 and 2.5, 2.5f f is steeper than the tangent line at x = 2
so the secant line has a larger slope than the tangent line.
(b) The secant line through (2, 2) and (2.5, 2.5) has a slope of 1.
(c) 2.5 1.5 2.4 1.6 4
2 0.8 or 2.5 1.5 2.5 1.5 5
f ff
6. (a) Sketch
(b) negative
(c) 1) –3 2) – 3 3) – 3
(d) 4 3 1y x
7. 26 10x x
8.
2
1
3x
9. 1
2 4x
CALCULUS
WORKSHEET ON DERIVATIVES
Work problems 6, 7, and 8 on notebook paper.
1. For each of the graphs shown below, list the x-values for which the function appears to be:
(i) not continuous (ii) not differentiable
__________________________________________________________________________________
2. Match the points labeled on the curve with the given slopes.
__________________________________________________________________________________
3. For the graph y f x shown in the figure, arrange the following numbers in ascending order:
The slope of the graph at A
The slope of the graph at B
The slope of the graph at C
The slope of the line AB
The number 0
The number 1
__________________________________________________________________________________
4. For the function shown below, at what labeled points is the slope of the graph:
positive?
negative?
At which labeled point does the graph have the:
greatest (i.e., most positive) slope?
the least slope (i.e., negative and with the largest magnitude)?
TURN>>>
Slope Point
3
1
0
1
2
1
2
Estimate the slope of the graph at the points 1 1 2 2, and ,x y x y .
5. (a)
__________________________________________________________________________________
Work problems 6, 7, and 8 on notebook paper.
6. Let f be a function which satisfies the property 2f x y f x f y xy for all real
numbers x and y, and suppose that
07lim
h
f h
h . Use the definition of the derivative
to find f x .
__________________________________________________________________________________
7. Let f be a function which satisfies 2 31 1 3 4 5f h f h h h for all real numbers h.
Find 1f .
__________________________________________________________________________________
8. Suppose f is a function for which
2
20
2limx
f x f
x
. Which of the following questions
MUST be true, MIGHT be true, or can NEVER be true? Explain your answers.
(a) f x is continuous at x 2.
(b) f x is continuous at x 0.
(c) 2 0f
(d) 2
2limx
f x f
.
(e) 2 2f
Answers to Worksheet on Derivatives
1. (i) 1 (ii) 1, 2, 3
(i) None (ii) 2, 4
2. F, C, E, A, B, D
3. 0 < slope at C < slope at B < slope of AB < 1 < slope at A
4. positive? A and D
negative? C and F
greatest? A
least? F
5. (a) 0, 2.5 (b) – 2.5, 2
6. 7 2x
7. 3
8. (a) Must
(b) Might
(c) Might
(d) Must
(e) Never
CALCULUS AB
WORKSHEET ON DERIVATIVES USING DATA
Work the following on notebook paper. Give decimal answers correct to three decimal places.
1. A roast turkey is taken from an oven and placed on a counter to cool. The table shows the temperature of
the turkey at various times over a three hour period.
t (minutes) 0 30 60 90 120 150 180
T t (° F) 185 150 130 110 95 88 79
(a) Use the data in the table to find 90T . Using appropriate units, explain the meaning of your answer.
(b) Use the data in the table to find 1 150T . Using appropriate units, explain the meaning of your answer.
(c) For 0 180t , must there be a time t when the temperature of the turkey is 125° F? Justify your answer.
(d) Use the data in the table to find an approximation for 75T . Show the computations that lead to your
answer. Using appropriate units, explain the meaning of your answer.
2. The number of locations of a popular coffeehouse chain is modeled by a differentiable function N for
0 8,t where t is the number of years since 1996 and N t is the number of locations. Values of N t
at selected values of t are shown in the table below.
t (years) 0 2 4 6 8
N t locations 1015 1886 3300 4617 5824
(a) Use the data in the table to find 2N Using appropriate units, explain the meaning of your answer.
(b) Use the data in the table to find 14617N
. Using appropriate units, explain the meaning of your answer.
(c) Use the data in the table to find an approximation for 5N . Show the computations that lead to your
answer. Using appropriate units, explain the meaning of your answer.
3. A hot cup of coffee is taken into a classroom and set on a desk to cool. The temperature of the coffee is
modeled by a differentiable function T for 0 12,t where t is measure in minutes and T t is measured
in degrees Fahrenheit. Values of T t at selected values of t are shown in the table below.
t (minutes) 0 2 5 6 8 10 12
T t (° F) 113 103 95 94 92 91 90
(a) For 0 12t , must there be a time t when the temperature of the coffee is 99° F? Justify your answer.
(b) Use the data in the table to find an approximation for 3T . Show the computations that lead to your
answer. Using appropriate units, explain the meaning of your answer.
(c) Use the data in the table to find the average rate of change of T t on the time period 5 12t minutes.
Show the computations that lead to your answer.
(d) A model for the temperature is given by 0.31270 70
3
ty t e
, where y t is measured in degrees
Fahrenheit and t is measured in minutes. Find 3y .
(e) Use the model given in part (d) to find the average rate of change of y t on the time period
5 12t minutes.
TURN->>>
4. The temperature of the water in a pond is modeled by a differentiable function T for 0 15,t
where t is measure in days and W t is measured in degrees Celsius. Values of W t at selected
values of t are shown in the table below.
t (days) 0 3 6 9 12 15
W t (°C) 20 31 28 24 22 20
(a) Use data from the table to find 9W . Using appropriate units, explain the meaning of your answer.
(b) Use data from the table to find 131W
. Using appropriate units, explain the meaning of your answer.
(c) Use data from the table to find an approximation for 7W . Show the computations that lead to your
answer.
(d) Based on values in the table, what is the smallest number of instances at which the temperature of the pond
could equal 23°C on the open interval 0 < t < 15? Justify your answer.
(e) A student proposes the function P, given by 320 10
t
P t te
, as a model for the temperature of the
water in the pond at time t, where t is measured in days and P t is measured in degrees Celsius. Find
7P . Using appropriate units, explain the meaning of your answer.
5. A car travels on a straight road. During the time interval 0 60t the car’s velocity v t , measured in feet
per second, is a twice-differentiable function. The table below shows selected values of this function.
t (sec) 0 14 24 33 47 60
v t (ft/sec) 18 25 30 21 15 17
(a) Use data from the table to find an approximation for 17v . Show the computations that
lead to your answer. Using appropriate units, explain the meaning of your answer.
(b) Based on the values in the table, what is the smallest number of instances at which the velocity of
the car could be 20 feet per second on the open interval 0 60t ? Justify your answer.
6. A test plane flies in a straight line with positive velocity v t , in miles per minute at time t minutes, where
v is a differentiable function of t. Selected values of v t for 0 40t are shown in the table below.
t (min) 0 5 10 15 20 25 30 35 40
v t (mpm) 7. 0 9.2 9.5 7. 0 4.5 2.4 2.4 4.3 7.2
(a) Use data from the table to find an approximation for 18v . Show the computations that lead to your
answer. Using appropriate units, explain the meaning of your answer.
(b) Use data from the table to find the average acceleration of the particle over the time interval
5 25t minutes. (Hint: The average acceleration is the average rate of change of the velocity.)
(c) Based on the values in the table, what is the smallest number of instances at which the velocity of
the plane could be 8 miles per minute on the open interval 0 40t ? Justify your answer.
(d) The function f, defined by 7
6 cos 3sin10 40
t tf t
, is used to model the velocity of the
plane, in miles per minute, for 0 40t . According to this model, what is the acceleration of
the plane at t = 23? Indicate units of measure.
Answers to Derivatives using Data
1. (a) 90T = 110° F. When t = 90 minutes, the temperature of the turkey is 110° F.
(b) 1 150T = 30 minutes. When the temperature of the turkey is 150° F, the time is 30 minutes.
(c) Yes. Since T is a continuous function and 60 110, 90 110, and 110 125 130,T T ,
the Intermediate Value Theorem guarantees a t in (60, 90) so that 125T t .
(d) 110 130 2
75 F / min.90 60 3
T
When t = 75 minutes, the temperature of the turkey is
dropping at a rate of 2
F / min.3
2. (a) 2N = 1886 locations. When t = 2 years (in 1998), the number of locations is 1886.
(b) 1 4617N = 6 years. When the number of locations is 4617, the time is 6 years (in 2002).
(c) 4617 3300
5 658.5 locations / year.6 4
N
When t = 5 years, the number of locations is
increasing at a rate of 658.5 locations / year .
3. (a) Yes. Since T is a continuous function and 2 103, 5 95, and 95 99 103,T T , the
Intermediate Value Theorem guarantees a t in (2, 5) so that 99T t .
(b) 95 103 8
3 F / min.5 2 3
T
When t = 3 minutes, the temperature of the coffee is
dropping at a rate of 8
F / min.3
(c) Average rate of change = 90 95 5
F / min12 5 7
(d) 3 2.846y F/ min.
(e) 0.653 F / min
4. (a) 24 28 4
7 C / day.9 6 3
W
When t = 7 days, the temperature of the pond is
dropping at a rate of 4
C / day.3
(b) Two times. Since W is a continuous function and 0 20, 3 31, and 20 23 31,W W the
Intermediate Value Theorem guarantees a t in (0, 3) where 23T t . Since
9 24, 12 22, and 22 23 24,W W the Intermediate Value Theorem guarantees a t in
(9, 3) where 23T t
(c) One time. Since W is a continuous and differentiable function, the Mean Value Theorem
guarantees a value of t somewhere in the interval (0, 15) where 0W t .
(d) 7 1.293 C/dayP . When t = 7 days, the temperature of the pond is dropping at a rate of
1.293 C/day .
5. (a) 24.5 718 0.5mi / min
20 15v
. When t = 18 minutes, the acceleration of the plane is
20.5mi / min (or the velocity of the plane is decreasing at a rate of 20.5mi / min ).
(b) Average acceleration = 22.4 9.20.34mi / min
25 5
(c) Two times. Since v is a continuous function and 0 7.0, 5 9.2, and 7 8 9.2v v , the
Intermediate Value Theorem guarantees a t in (0, 5) where 8v t . Since
10 9.5, 15 7.0, and 7 8 9.5v v the Intermediate Value Theorem guarantees a t
in (10, 15) where 8v t .
(d) Two times. Since v is a continuous and differentiable function, the Mean Value Theorem
guarantees a value of t somewhere in the interval (0, 15) and somewhere in the interval
(25, 30) where 0v t .
(e) 223 0.408 mi / minf
(f) Average acceleration = 225 5
0.341mi / min25 5
f f
6. (a) 230 25 517 ft / sec
20 12 8v
. When t = 17 minutes, the velocity of the car is
25ft / sec
8 (or the velocity of the car is decreasing at a rate of 25
ft / sec8
).
(b) Two times. Since v is a continuous function and
0 18 20 25 14 and 47 15 20 21 33v v v v , the Intermediate Value
Theorem guarantees a t in (0, 14) where 20v t and a t in (33, 47) where 20v t .
(c) Two times. Since v is differentiable on [0, 60], v must also be continuous there.
Since 14 24 and 24 33 ,v v v v v attains a maximum value on (14, 33).
Similarly, since 33 47 and 47 60 ,v v v v v attains a minimum on (33, 60).
v is differentiable so 0v t at each relative extreme point on (0, 60). Therefore
0v t for at least two values of t in [0, 60].
CALCULUS AB
REVIEW WORKSHEET ON 1.4 & 2.1
Work the following on notebook paper except for problems 2 – 3. Use your calculator only on problem 1, and
give decimal answers correct to three decimal places.
1. A hot cup of coffee is poured and set on a table to cool. The table shows the temperature of
the coffee at various times over a fifteen minute period.
t (minutes) 0 4 7 12 15
T t (° F) 128 117 113 110 109
(a) Use data from the table to find an approximation for 5T . Show the computations that lead to your
answer. Using appropriate units, explain the meaning of your answer.
(b) For 0 15,t must there be a time t when the temperature of the coffee is 125° F? Justify your answer.
(c) A model for the temperature is given by 0.21324 60
3
ty t e
, where y t is measured in
degrees Fahrenheit and t is measured in minutes. Use the model to find 5 .y Using appropriate units,
explain the meaning of your answer.
(d) Use the model given in (c) to find 5y . Using appropriate units, explain the meaning of your answer.
__________________________________________________________________________________________
No calculator.
Draw tangent lines on the graph of f at five points, and record the slopes of the lines in the table below. Then
sketch the graph of the derivative on the grid on the right.
2. Graph of f x Graph of f x
3.
x f x
x f x
Use the alternative form of the derivative to find the derivative at x = c if it exists. If the derivative does not
exist, use the alternative form to show why. All work must be shown.
2
1, 24. 2
4 5, 2
x xf x c
x x
Does 2f exist? ______ If yes, then 2f = _______
What happens on the graph of f at x = 2?
Choices are: smooth curve, sharp turn, cusp, vertical tangent __________________
2
4 , 25. 2
6 10, 2
x xf x c
x x x
Does 2f exist? ______ If yes, then 2f = _______
What happens on the graph of f at x = 2?
Choices are: smooth curve, sharp turn, cusp, vertical tangent __________________
__________________________________________________________________________________________
(a) For what value(s) of x is f not continuous?
(b) Classify the discontinuities as point, jump, or asymptotic.
(c) Classify the discontinuities as removable or nonremovable.
6. 2 1, 0
2 1, 0
x xf x
x x
7.
__________________________________________________________________________________
8. Use the definition of continuity to find k so that f will be continuous everywhere given
2
, 1
12 , 1
kx xf x
k x x
9. Use the definition of continuity to find a and b so that f will be continuous everywhere given
17 if 3
if 3 1
3 if 1
x
f x ax b x
x
_________________________________________________________________________________
10. Given 2 6, [ , ] [ 4,1], 1f x x x a b k .
(a) Does the IVT hold? Show why or why not.
(b) If the theorem holds, find the c which the theorem guarantees.
(c) Sketch , the line , and the point ( , )y f x y k c k that you found.
__________________________________________________________________________________
11. Given 1 3
, [ , ] [0,2],1 4
f x a b kx
.
(a) Does the IVT hold? Show why or why not.
(b) If the theorem holds, find the c which the theorem guarantees.
(c) Sketch , the line , and the point ( , )y f x y k c k that you found.
__________________________________________________________________________________
Use the definition of the derivative to find the derivative.
12. 1
3f x
x
13. 2f x x
2
2
6
9
x xf x
x
