The Campbell’s Soup Lab Argument Writing
AP Calculus AB

Follow-Up: Suppose you are the owner of Saucy Soup Company. You need to present an argument to your board of directors as to what shape soup can your company should sell. Some things to keep in mind:

• ECONOMIC REQUIREMENTS: The product must be cost efficient.

• FUNCTIONAL REQUIREMENTS: The product must also be easy for retailers to store and stock on the shelves or the floor, and simple to process at a check-out counter. There is limited shelving in the grocery store for your product and standard roll out soup holders can only hold the “standard” shaped can.

• SALES REQUIREMENTS: In addition to functional and economic requirements, product packaging must be designed in a way that will appeal to buyers. The less attractive can, the fewer cans you will sell. Attractiveness of cans is frequently based on the golden ratio (which is where [pic]). New cans can be advertised as being “environmentally friendly.”

Argument Writing Task:
Choose which soup can to present to the board of Saucy Soup Company. Defend your claim using evidence from the lab or given information. Present the counter-claim with evidence, then refute it. Be sure to identify all assumptions. Finally, summarize your position.

This should be typed using 12 point-font and double-spaced. It should take about ¾ of a page to fully present this argument.

This assignment should be coherent and free of spelling and grammar mistakes.

...Portfolio in Calculus
Submitted by:
Chloe Regina C. Paderanga
Submitted to:
Sir Ferdinand Corpuz
Journal for the Month of June
WHAT I LEARNED?
I learned many things this month. It was good that our teacher repeated the topics in basic math to strengthen our foundation. Even if we had a hard time, I don’t see any reason why we should complain because I understand that our wanted to master these topics to be able to move to a higher math. The topics tackled this month are namely:
Inequalities
Rational Inequalities
Circles
Distances
Slopes
Angles of Incidence
WHAT IS THE HARDEST TOPIC?
For me, the hardest topic to master was the inequalities, which I know I should master to be able to understand the next topics.
HOW DID I LEARN?
I reviewed my wrong answers in our summative tests because I don’t want to be left behind with the topics.
REFLECTION
When Sir Corpuz said that we are going to have a double program in Math, I was excited because we are not just advancing but are reviewing in the same time.
APPLICATION TO LIFE
A lot of advance technologies are product of such very simple concepts in math as long as it is utilized in a very good way. For example the distance formula, this is not just used in Math but also in Physics, Science and many other fields.
Journal for the Month of July
WHAT I LEARNED?
This month, I learned that there are also ‘other’ versions of circles. Namely:
Parabola
Ellipse...

...
Calculus in Medicine
Calculus in Medicine
Calculus is the mathematical study of changes (Definition). Calculus is also used as a method of calculation of highly systematic methods that treat problems through specialized notations such as those used in differential and integral calculus. Calculus is used on a variety of levels such as the field of banking, data analysis, and as I will explain, in the field of medicine. Medicine is defined as the science and/or practice of the prevention, diagnosis, and treatment of physical or mental illness (Definition). The term medicine can also mean a compound or a preparation applied in treatment or control of diseases, mostly in form of a drug that is usually taken orally (Definition). Calculus has been widely used in the medical field in order to better the outcomes of both the science of medicine as well as the use of medicine as treatment. (Luchko, Mainardi & Rogosin, 2011). There has been a strong movement towards the inclusion of additional mathematical training throughout the world for future researchers in biology and medicine. It can be hard to develop new courses as well as alter major requirements, but institutions should consider the importance of a clear understanding of the function of mathematics in science. However, scientists who have not had the level of mathematical training needed to work in...

...ACCT 410 ProjectProject ID: 21
xxxxx xxxxxxxxx
ACCT 410
Dr. xxxx xxx
November 29, 2012
Article Summary:
Does Professor Quality Matter? Evidence from Random Assignment of Students to Professors.
It is a common perception that better education outcomes are achieved through higher-quality teachers. Teacher evaluation measures in education are different at secondary and post secondary levels. Authors argue that these measures can be influenced impacting actual student learning. At secondary and elementary level, teachers usually teach with the focus on ‘test’ and in postsecondary level, professors can reduce academic curriculum to enhance student evaluation or in some circumstances can even directly inflate the grades. The moot question then remains how the teacher evaluation measures can impact the desired outcomes of students learning.
Various studies have been conducted time to time to find the relationship between student achievements at the secondary and elementary levels vis. a vis. teacher contribution and the evidences available in this respect are somewhat mixed in nature. The clarity is much lesser when the question comes of measuring student outcomes at the postsecondary level with respect to the quality of instruction provided by the teachers. The reason is that standardized tests are not used at the postsecondary level and moreover, students select their own professors and their own...

...1. 7.5/8
The height in metres of a ball dropped from the top of the CN Tower is given by h(t)= -4.9t2+450, where t is time elapsed in seconds.
(a) Draw the graph of h with respect to time
(b) Find the average velocity for the first 2 seconds after the ball was dropped
h(0)=(0,450), h(2)=(2,430.4)
= (430.4-450)/(2-0)
= -9.8m/s √
(c) Find the average velocity for the following time intervals
(1) 1 ≤ t ≤ 4
h(1)=(1,445.1) h(4)=(4,371.6)
= (371.6-445.1)/(4-1)
= -24.5m/s √
(2) 1 ≤ t ≤ 2
h(1)=(1,445.1) h(2)=(2,430.4)
= (430.4-445.1)/(2-1)
= -14.7m/s √
(3) 1 ≤ t ≤ 1.5
h(1)=(1,445.1) h(1.5)=(1.5, 438.98)
= (438.98-445.1)/(1.5-1)
= -12.25m/s √
(d) Use the secant method to approximate the instantaneous velocity at t=1
h(0.5) = (0.5, 448.78) h(1) = (1, 445.1)
= (445.1-448.78)/(1-0.5)
= -7.35m/s
(You must choose a point very close to (1, 445.1) in order to get an accurate answer. Try to choose values within 0.1 of the t-value given. Therefore, t = 0.9, or the point (0.9,446.031) is a good choice, and will result in an IROC of -9.31 m/s. This is very close to the true value of -9.8 m/s. As you can see, when t = 0.5, the answer is not as close to the true value.)
2. 6/7
The mass M in grams of undissolved sugar left in a teacup after t seconds is given by M =10.5-0.4t2
(a) When will all the sugar dissolve?
0=10.5-0.4t2
0.4t2=10.5
t2=26.25
t=5.12 sec
(b) Find the average rate of change in the interval 0 ≤ t ≤ 1
M(0) = (0, 10.5); M(1)= (1, 10.1)
= (10.1-10.5) / (1-0)
= -0.4...

...No 1. 2. 3. 4. 5. 6. 7. 8. Code: UCCM1153 Status: Credit Hours: 3 Semester and Year Taught:
Information on Every Subject Name of Subject: Introduction to Calculus and Applications
Pre-requisite (if applicable): None Mode of Delivery: Lecture and Tutorial Valuation: Course Work Final Examination 40% 60%
9. 10.
Teaching Staff: Objective(s) of Subject: • Review the notion of function and its basic properties. • Understand the concepts of derivatives. • Understand linear approximations. • Understand the relationship between integration and differentiation and continuity. Learning Outcomes: After completing this unit, students will be able to: 1. describe the basic ideas concerning functions, their graphs, and ways of transforming and combining them; 2. use the concepts of derivatives to solve problems involving rates of change and approximation of functions; 3. apply the differential calculus to solve optimization problems; 4. relate the integral to the derivative; 5. use the integral to solve problems concerning areas.
11.
12.
Subject Synopsis: This unit covers topics on Functions and Models, Limits and Derivatives, Differentiation Rules, Applications of Differentiation and Integrals.
13.
Subject Outline and Notional Hours: Topic Learning Outcomes 1 L 4 T 1.5 P SL 6.25 TLT 11.75
Topic 1: Functions and Models
• • • • • • Functions Models and curve fitting Transformations, combinations, composition and graphs of...

...Calculus
is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integralcalculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamental theorem ofcalculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-definedlimit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot.
Calculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If quantities are continually changing, we need calculus to study what is going on.
BRANCHES OF CALCULUSCalculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If...

...INDETERMINATE FORMS AND IMPROPER
INTEGRALS
Deﬁnition. If f and g are two functions such that
lim f (x) = 0
x→a
and
lim g(x) = 0
x→a
then f (x)/g(x) has the indeterminate form 0/0 at a.
sin t
x2 − 9
Illustration.
has the indeterminate form 0/0 at 0 and
has the
t
x−3
indeterminate form 0/0 at 3.
Theorem. (L’Hopital’s Rule) Let f and g be functions diﬀerentiable on
an open interval I, except possibly at the number a on I. Suppose that for
Chapter 2: INDETERMINATE FORMS AND IMPROPER INTEGRALS
1
all x = a in I, g (x) = 0. If lim f (x) = 0 and lim g(x) = 0, and
x→a
f (x)
if lim
=L
x→a g (x)
x→a
then
f (x)
lim
=L
x→a g(x)
The theorem is valid if all the limits are right-hand limits or all the limits
are left-hand limits.
Illustrations.
1. Because lim sin t = 0 and lim t = 0, we can apply L’Hopitals rule and
t→0
t→0
obtain
sin t
lim
t→0 t
cos t
= lim
t→0 1
= 1
Chapter 2: INDETERMINATE FORMS AND IMPROPER INTEGRALS
2
2. Because lim (x2 − 9) = 0 and lim (x − 3) = 0, we can apply L’Hopitals
x→3
x→3
rule and obtain
x2 − 9
lim
x→3 x − 3
2x
x→3 1
= 6
=
Chapter 2: INDETERMINATE FORMS AND IMPROPER INTEGRALS
lim
3
Examples.
1. Given f (x) =
x
, ﬁnd the lim f (x).
x−1
x→0
e
We can use L’Hopital’s rule since lim x = 0 and lim (ex − 1) = 0
x→0
x
x→0 ex − 1
lim
x→0
1
x→0 ex
= 1
=
lim
x3 − 3x + 2
2. Given f (x) =
evaluate lim f (x) if it exists.
x→1...

...CHALLENGING PROBLEMS FOR CALCULUS STUDENTS
MOHAMMAD A. RAMMAHA
1. Introduction In what follows I will post some challenging problems for students who have had some calculus, preferably at least one calculus course. All problems require a proof. They are not easy but not impossible. I hope you will ﬁnd them stimulating and challenging. 2. Problems (1) Prove that eπ > π e . (2.1) Hint: Take the natural log of both sides and try to deﬁne a suitable function that has the essential properties that yield inequality 2.1. 1 4 1 2 (2) Note that = but = . Prove that there exists inﬁnitely many 4 2 pairs of positive real numbers α and β such that α = β; but αα = β β . Also, ﬁnd all such pairs. Hint: Consider the function f (x) = xx for x > 0. In particular, focus your attention on the interval (0, 1]. Proving the existence of such pairs is fairly easy. But ﬁnding all such pairs is not so easy. Although such solution pairs are well known in the literature, here is a neat way of ﬁnding them: look at an article written by Jeﬀ Bomberger1, who was a freshman at UNL enrolled in my calculus courses 106 and 107, during the academic year 1991-92.
1 4 1 ; 2
1 1
(3) Let a0 , a1 , ..., an be real numbers with the property that a1 a2 an a0 + + + ... + = 0. 2 3 n+1 Prove that the equation a0 + a1 x + a2 x2 + ...an xn = 0
1Jeﬀrey
Bomberger, On the solutions of aa = bb , Pi Mu Epsilon Journal, Volume 9(9)(1993),
1
571-572....