BC Calculus Assignment
Summer 2013
Mrs. Heike
Name: _________________________
Grade: ________ Period: ________
Summer Assignment Grade: ________
1
WEST MORRIS CENTRAL HIGH SCHOOL
259 BARTLEY ROAD, CHESTER, NEW JERSEY 07930
Telephone: 908-879-5212
Fax: 908-879-2741
www.wmchs.org
STEVE RYAN, PRINCIPAL
KENT SCHILLING, SUPERVISOR OF ATHLETICS AND ACTIVITIES
ROBERT PERKINS, ASSISTANT PRINCIPAL
ANNE MEAGHER, ASSISTANT PRINCIPAL
19 June 2013
Dear BC Calculus Students,
Welcome! Congratulations on your decision to take BC Calculus. I affectionately call this course alphabet soup—have you
looked at the letters following our course title? AP/IB-HL…and now I’m going to add four more. I am pleased to announce
that West Morris Central High School has formed a partnership with NJIT to provide qualified high school students with the
opportunity to take college courses at WMC during the regular academic year. NJIT has certified me as an adjunct faculty
member in the Department of Mathematics and I will be providing you with the opportunity to take Calculus I and/or Calculus
II for college credit through NJIT.
Students who choose to participate will be registered at NJIT as non-matriculated students. There are no registration or
application fees. NJIT charges $125 per credit for participation in this program. (The regular cost per credit is $472.) Both
Calculus I and Calculus II are 4 credit courses. At the conclusion of the course, students will receive a transcript from NJIT
noting their grade and credit achievement. Participation in this program is optional, and all students will still have the option of
taking the AP Calculus BC exam and/or the IB Math Higher Level Exam.
So what does this mean for you? You have many options to earn college credit (up to 8 credits) by taking this course. They
are as follows:
(1) You are an IB student taking the HL Math exam. With an acceptable score (usually of 6 or 7) on the exam and
completion of your IB diploma you can earn up to 4 credits for Calculus I. IB students taking the SL Math exam
usually do not earn credit for Calculus I (there are exceptions to this, check with your college/university for
details on their policy).
(2) You choose to take the AP Calculus BC exam. All students enrolled may take this exam. With an acceptable
score (3, 4, or 5 depending upon the college) you can earn up to 8 credits for Calculus I and Calculus II. Please
note that not all colleges award AP credit. Again check with your college/university for details. The decision to
take this exam usually must be made by March with a cost of around $90.
(3) You can enroll at NJIT and take Calculus I, Calculus II, or both. These credits then can be transferred to the
college of your choice. (Again, please check with individual schools for their transfer policy). The decision to
enroll at NJIT for Calculus I must be made by the first week of September. The decision for Calculus II can be
made in December.
(4) Some combination of the above.
I am alerting you to the NJIT opportunity now, as registration for the Fall semester must take place immediately upon our
return in September. Take the summer to do some research into the colleges you plan to apply to and decide which option(s)
are best for you. If you have any questions, feel free to contact me at mheike@wmrhsd.org. I will be checking this email on a
non-regular basis throughout the summer. I look forward to seeing you all in September. Enjoy your summer.
Sincerely,
Melissa J Heike
Teacher of Mathematics
2
Please read carefully:
 This packet is designed to review (a) limits and continuity and (b) other key
PreCalculus topics required by the IB Curriculum for Mathematics HL, the AP
Curriculum for Calculus BC, and/or the NJIT curriculum for Calculus I. These
topics will be reviewed briefly at the beginning of the school year. You will be
assessed on any/all of the topics focused on in this packet.
 Problems in this packet will also introduce you to some new notation and what I like
to call “IB speak” (UK spellings of words, instructions and the like). Please see the
next page for some clarifications.
 You are encouraged to complete this packet on your own with minimal outside
help. It is intended to call your attention to any material you will need additional
help on over the course of next year. You may use the IB Formulae packet available
on my website (click on Calculus BC on left side of page). Please print out a copy
of this packet and use it as reference throughout the 2013-2014 school year. You
will also see a link to all of the notes from Pre-Calculus should you need
clarification on a topic from last year.
 The Limits and Continuity questions should be completed without a calculator. You
may use a TI-83/84 or equivalent calculator on the remainder of the assignment. If
you own a TI-89, we will be using these during the course of the year for the AP
topics only. If you use the 89 on this packet, do not utilize its algebraic
manipulation capabilities.
 YOU MUST HAVE THIS PACKET COMPELTED IN ITS ENTIRETY ON
THE FIRST FULL DAY OF CLASS! --Sept 9 or 10 (Not the very first day of
school. As of 6/18 I do not know what period we will meet, so the due date will
depend on the rotation). I will be grading it on your level of effort/completeness.
You will have an opportunity to ask questions on the packet before you are formally
assessed on the material.
 Answers will be posted on the course webpage by the end of August. Please
download the answer key and check your own work. Try to correct any of the
problems you get wrong. Make a list of questions you are unable to solve correctly.
 Pace yourself - It is lengthy! The idea is to complete a little bit each day/week so
your math brains are ready to go come September. Show all work and box your final
answer(s).
3
IB Speak, Terminology You May Have Forgotten and Alternative Notations:
Some Clarifications
Hence:
you must utilize your answer from the preceding part to solve, it should give you a
hint in how to solve the problem
Hence or otherwise: you should utilize your answer from the preceding part to solve (think of it
as a hint), but if you can solve it a different way that is okay too
Sketch:
this is a rough graph of the function, it should have the correct shape and key points
labeled but does not have to be to scale.
Graph:
this is a graph to scale
Show:
I give you the answer, now show me the steps to get there
Factorize:
factor
Singular Matrix:
a matrix whose determinant is 0
Mutually Exclusive: disjoint, the intersection of the two sets is empty
Alternative Notations:
f : x  x2
is equivalent to
f ( x)  x 2
 f  g x  f g x
n ( A) : the number of elements (items) in set A
Calculators
I recommend that you have both a TI-83/84 and a TI-89 calculator for use during the
school year. At minimum everyone needs to have access to a TI-83/84. Students planning
on taking the AP exam will find the TI-89 helpful. There are some great deals online on
both new and used TI-89s.
4
Limits and Continuity (Non-calculator)
Rates of Change and Tangents to Curves
1.
Find the average rate of change of the function over the given interval.
a. 𝑓(𝑥) = 𝑥 3 + 1
[2, 3]
b. 𝑓(𝑥) = 𝑥 3 + 1
[-1, 1]
c. 𝑅(𝜃) = √4𝜃 + 1
[0,2]
2. Let 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
a. Find the slope of the line through the points P(2, -3) and Q(2 + ℎ, 𝑓(2 + ℎ)).
b. Find the equation of the tangent line to 𝑓(𝑥) at P.
3. The accompanying graph shows the total distance s traveled by a bicyclist after t hours.
a. Estimate the bicyclist’s average speed over the time intervals [0, 1], [1, 2.5], and [2.5, 3.5].
1
b. Estimate the bicyclist’s instantaneous speed at the times 𝑡 = 2 , 𝑡 = 2, and 𝑡 = 3.
c. Estimate the bicyclist’s maximum speed and the specific time at which it occurs.
5
Limit of a Function and Limit Laws
4. For the function g(x) graphed here, find the following limits or explain why they do not exist.
a. lim g ( x)
x1
b. lim g ( x)
x 2
c. lim g ( x)
x3
d.
lim g ( x )
x 2.5
5. Which of the following statements about the function y = f (x) graphed here are true, and which are
false?
a. lim f ( x ) exists
x 0
b. lim f ( x)  0
x 0
c. lim f ( x)  1
x 0
d. lim f ( x)  1
x 1
e. lim f ( x)  0
x 1
f.
lim f ( x) exists at every point xo in (−1,1)
x x0
g. lim f ( x ) does not exist
x1
6. Calculate the limit.
a. lim (2 x  5)
x  7
b. lim
x2
x3
x6
c.
lim (5  x) 4 3
x3
d. lim
x 5
x5
x 2  25
6
e. lim
t 1
f.
t2  t  2
t 2 1
lim
x 9
g. lim
x 1
h. lim 2 sin x  1
x 0
x 3
x9
i.
lim
x  
x  4 cosx   
x 1
x3 2
7. Suppose lim f ( x)  5 and lim g ( x)  2 . Find
x c
a. lim f ( x) g ( x)
xc
b. lim 2 f ( x) g ( x)
xc
x c
c. lim  f ( x)  3 g ( x) 
x c
d. lim
x c
f ( x)
f ( x)  g ( x)
8. Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form
f ( x  h)  f ( x )
lim
h 0
h
occur frequently in calculus. Evaluate this limit for 𝑥 = 1 for the function 𝑓(𝑥) = 𝑥 2 .
7
9. If √5 − 2𝑥 2 ≤ 𝑓(𝑥) ≤ √5 − 𝑥 2 for −1 ≤ 𝑥 ≤ 1, find lim f ( x ) . Justify your answer. (Hint:
x 0
Squeeze Theorem).
10. If lim
f ( x)  5
 1 , find lim f ( x ) .
x 4
x2
11. If lim
f ( x)  5
 3 , find lim f ( x ) .
x 2
x2
12. If lim
f ( x)  5
 4 , find lim f ( x ) .
x 2
x2
x4
x2
x2
One-Sided Limits
13. Which of the following statements about the function 𝑦 = 𝑓(𝑥) graphed here are true, and which
are false?
a. lim  f ( x)  1
g. lim f ( x)  1
x 0
x  1
b.
c.
d.
lim f ( x)  0
h. lim f ( x)  1
lim f ( x)  1
i.
lim f ( x)  lim f ( x)
j.
x 0 
x 0 
x 0 
x 0
e. lim f ( x ) exists.
x 0
x 1
k.
lim f ( x)  0
x 1
lim f ( x)  2
x2
lim f ( x ) does not
x  1
exist.
f.
lim f ( x)  0
x 0
l.
lim f ( x)  0
x2
8
x0
0,

14. Let f ( x)   1
.
sin
,
x

0
 x
a. Does lim f ( x ) exist? If so, what is it? If not, why not?
x 0
b. Does lim f ( x ) exist? If so, what is it? If not, why not?
x 0
c. Does lim f ( x ) exist? If so, what is it? If not, why not?
x 0
In 15 and 16, graph 𝑓(𝑥). Then answer these questions.
a. What are the domain and range of f?
b. At what points c, if any, does lim f ( x ) exist?
x c
c. At what points does only the right-hand limit exist?
d. At what points does only the left-hand limit exist?
 1 x2 ,

15. f ( x)  1,
2,

0  x 1
1 x  2
x2
17. Calculate the limit,
 x  2 x  5 
a. lim  


x  2  x  1  x 2  x 
b.
lim
h 0
h 2  4h  5  5
h
 x,

16. f ( x)  1,
0,

c.
d.
 1  x  0, or 0  x  1
x0
x  1 or x  1
lim   x  3
x2
lim   x  3
x2
x  2
x  2
x2
x2
9
e. lim
 0
f.
lim
 0
g. lim  cos 
sin 2
 0
2
tan 2
h. lim

 0
tan 3
sin 8
Continuity
18. State whether the function graphed is continuous on [-1,3]. If not, where does it fail to be
continuous and why?
a.
c.
b.
d.
19. At what points are the functions continuous?
1
 3x
a. y 
x2
b.
cos x
y
x
c.
y  2 x  1
d.
 x2  x  6
,

f ( x)   x  3
5

13
x3
x3
10
20. Is the function continuous at the point being approached?


lim cos
t 0
 19  3 sec 2t



21. Define 𝑔(3) in a way that extends g ( x) 
 x 2  1,
22. For what value of a is f ( x)  
2ax,
x2  9
to be continuous at 𝑥 = 3.
x3
x3
x3
continuous at every x?
x  1
 2,

23. For what values of a and b is f ( x)  ax  b,  1  x  1 continuous at every x?
3,
x 1

24. Show that the equation 𝑥 3 − 15𝑥 + 1 = 0 has three solutions in the interval [−4,4]. (Do NOT
solve! Hint: The Intermediate Value Theorem).
Limits Involving Infinity; Asymptotes of Graphs
25. Calculate the limit.
2

a. lim   3 
x  x


2

b. lim   3 
x   x


  5  7 x  

c. lim 
x  3  1 x 2  


11
  5  7 x  

d. lim 
x  3  1 x 2  


 2  x  sin x 
e. lim 

x  
 x  cos x 
26. Find the limit of each rational function (a) as x   and (b) as x   .
a.
7x3
x 3  3x 2  6 x
f ( x) 
b.
f ( x) 
10 x 5  x 4  31
x6
27. Calculate the limit.
a. lim
x 
d. lim
x 7
8x 2  3
2x 2  x
4
 x  7 2
x 2  3x  2
28. Find lim 3
as
x  2x 2
a. x  0 
b. x  2 
3
b. lim
x  3
x 5 x
c. lim
x 5 x
x 
x3
4 x 2  25 x
e. lim  tan x
x  2 
c.
x  2
d. x  2
e. What, if anything, can be said about the limit as x  0 ?
12
 1

2
 as
29. Find lim  2 3 
23 
x


x

1



a. x  0
c.
b. x  0 
x  1
d. x  1
30. Graph the rational function. Label the x- and y-intercepts and the asymptotes. (No calculators!!)
x3
y
x2
31. Sketch a labeled graph of a function y  f (x) that satisfies the given conditions.
f (0)  0,
lim f ( x)  0,
x  
lim- f ( x)  lim  f ( x)  ,
x 1
x  1
lim f ( x)  lim - f ( x)  
x 1
32. Find lim
x 
x
x  1
2

 25  x 2  1 .
13
33. Graph the rational function. Label the x- and y-intercepts and the asymptotes. (No calculators!!)
y
x2  4
x 1
Algebra and Functions (Calculator Allowed)
34. Find the value of c for which the coefficient of x 4 in the expansion of 2 x  c 7 is 70.
35. Let f ( x)  x and g ( x)  4  x 2 .
a. Find: (i) f (g (2)) (ii) g ( f (2))
b. State all values of x for which f ( g ( x)) is defined.
c. State all values of x for which g ( f ( x)) is defined.
36. Solve each of the following equations for x, giving exact values in terms of the natural logarithm,
‘ln,’ or in terms of ‘e’ if necessary.
a. 3 x  6
b. log 3 x  17   2  log 3 2 x
14
37. Find the equation of the line which passes through both the intersection of x  y  2 and
2 x  3 y  8 and the point (0,0).
38. Find k such that the equation 2 x 2  kx  2k  0 has exactly one solution.
39. Consider the polynomial P( x)  2 x 3  x 2  13 x  6 .
a. Factorize P(x) .
b. Find x P( x)  0 (Read: Find x such that P( x)  0 )
40. Find a and b such that x 4  4 x 2  ax  b is exactly divisible by x 2  x  2 .
x  y  z  9

41. Solve the system of equations 3x  4 y  3z  2
4 x  5 y  3 z  5

15
42. Given that 2  i is a root of the equation x 3  6 x 2  13 x  10  0 find the other two roots.
43. The function f is given by f ( x)  x 2  6 x  13 for x  3 .
a. Write f (x) in the form of x  a 2  b .
b. Find the inverse function of f (x) .
c. State the domain of the inverse.
44. Find the three cube roots of 8i. Give your answers in the form x + yi. (Hint: Convert to trig
form and use DeMorivre’s Theorem).
16
Sequences and Series (Calculator Allowed)
45. Find the following sums
 n
6
a.
n 1
2
n

4
b.

5
3

  2  2 n  2 n 2 
n 1
46. Find the sum of the infinite geometric series:
2 4 8 16
 
 ...
3 9 27 81
47. In an arithmetic sequence, the first term is -2, the fourth term is 16 and the nth term is 11,998.
Find the common difference, d, and the value of n.
48. A geometric sequence has a first term 2 and a common ratio 1.5. Find the sum of the first 11
terms.
49. For what values of x will the sum 1  (1  x)  (1  x) 2  1  x 3  ... exist?
17
50. Prove by mathematical induction that the sum of the series
1  3  2  4  ...  n(n  2)  1 n(n  1)(2n  7)
6
18
Trigonometry (Calculator Allowed)
51. The lengths of the sides of a triangle ABC are x – 2, x, and x + 2. The largest angle is 120o.
a. Find the value of x.
b. Show that the area of the triangle is
15 3
.
4
c. Find sin A  sin B  sin C given your answer in the form
p q
r
where p, q, r   .
52. Given f ( x)  3 sin 2 x  11sin x  6 .
a. Factorize the expression.
b. Find the two values of sin x which satisfy the equation
c. Solve the equation for 0  x  180
19
Vectors (Calculator Allowed)
53. The vectors i and j are unit vectors along the x-axis and y-axis respectively.
a. Given the vectors u  i  j and v  3i  5j , find u  2 v in terms of i and j.
b. A vector w has the same direction as u  2 v , and has a magnitude of 26. Find w in terms
of i and j.
54. The line L passes through the origin and is parallel to the vector 2i+ 3j. Write down a vector
equation for L.
1
 
55. Find the size of the angle between the two vectors  1  and
 2
 
 2
 
 3  . Give your answer to the nearest
1
 
degree.
 2x 
 and
56. The vectors 
 x  3
 x  1

 are perpendicular for two values of x. Find the two values of x.
 5 
57. Find the equation of the plane that passes through (0,0,0), (1,2,3) and (-2,3,3).
20
Probability (Calculator Allowed)
58. In a survey of 200 people, 90 of whom were female, it was found that 60 people were
unemployed, including 20 males.
a. Use the information to complete the table below.
Male
Female
Totals
Unemployed
Employed
Totals
200
b. If a person is selected at random from this group of 200, find the probability that this
person is
i. An unemployed female
ii. A male, given that the person is employed.
59. The following Venn diagram shows a sample space U and events A and B where

n(U) =36, n(A) = 11, n(B) = 6 and n A  B   21 .
A
B

a. On the diagram, shade the region  A  B  .
b. Find n A  B .
c. Find p( A  B) .
d. Determine if the events are mutually exclusive. Explain your answer.
21
60. For the events A and B, p( A)  0.6 , p( B)  0.8 and p ( A  B )  1 . Find
a. p( A  B)
b.
p ( A  B)
61. Two fair 6-sided dice are rolled.
a. Complete the tree diagram by entering the probabilities.
?
?
?
?
?
?
6
?
?
Not 6
?
?
6
?
?
Not 6
6
Not 6
b. Find the probability of getting one or more sixes.
62. A factory makes staplers. Over a long period of time 4% of them are faulty. A random sample
of 200 staplers is tested.
a) Write down the expected number of faulty staplers in the sample.
b) Find the probability that exactly 5 staplers are faulty.
c) Find the probability that one or more are faulty.
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Item Analysis:
Please indicate below by a check mark on the line provided which of the questions you were unable to
solve and need further clarification on. Do this after you have checked your work and made at least
one additional attempt at the problem.
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