Santa Monica College
Mathematics Department Addendum
Math 7 – Calculus 1
Prerequisite Comparison Sheet – exit skills of Math 2 and entry skills for Math 7
Exit Skills for Math 2
Upon successful completion of Math 2, the student will be able to:
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
M.
Determine whether a relation represents a function. If it is a function, determine its
domain and range; determine whether it is odd or even or neither based on its
formula or its graph; and determine whether it is one-to-one, and if it is, determine
its inverse function and its domain and range.
Analyze and graph a given function, including but not limited to piecewise-defined,
polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric
functions, without the aid of graphing devices. Determine intercepts, coordinates of
holes, and equations of asymptotes. Determine intervals on which polynomial and
rational functions are positive and are negative.
Use transformation techniques including vertical and horizontal shifts, compression,
stretching, and reflection over the x- or y-axis to sketch the graph of a function.
Use the language and standard mathematical notation of the algebra of functions.
Determine algebraic combinations and compositions of functions and state their
domains. Write a given function as a composition of two non-identity functions.
Use techniques and facts including synthetic division, long division, the
Fundamental Theorem of Algebra and the Rational Zeros Theorem to find all
complex zeros of a polynomial function of degree three or higher, and write the
function in a completely factored form.
From memory, state and apply the definitions of the six trigonometric ratios of sides
of right triangles; the definitions of the six trigonometric functions of real numbers
using the unit circle; and the definitions, domains and ranges of the inverse sine,
inverse cosine, and inverse tangent functions.
Evaluate trigonometric functions at integer multiples of  / 6 and  / 4 , including
values outside of [0, 2 ] , without the use of notes or calculators. Evaluate
compositions of trigonometric functions and inverse trigonometric functions
including ones for which cancellation equations do not apply.
From memory, state and apply the fundamental reciprocal, quotient and
Pythagorean trigonometric identities and the sum, difference, double-angle and
half-angle identities for sine and cosine.
Write algebraic and trigonometric relationships to solve application problems,
including solution of right and oblique triangles by the Law of Sines and Law of
Cosines.
Prove trigonometric identities including those which require the use of sum,
difference, double-angle and half-angle identities.
Solve polynomial, rational, exponential, logarithmic, and trigonometric equations.
Given a quadratic equation in variables x and y, with no xy term, put it into a
standard form in order to classify its graph as one of the conic sections (circle,
ellipse, parabola and hyperbola). Determine the directrix, center, vertex points,
focus points, major/transverse axis, and minor/conjugate axis, if they exist, and
sketch the graph of the conic section.
Santa Monica College
Page 2 of 6
Exit Skills for Math 2
O.
Find terms of explicitly and recursively defined sequences. Find the nth term in a
sequence whose first several terms are given.
Evaluate, manipulate and interpret summation notation.
P.
Prove statements using mathematical induction.
Q.
Apply the binomial theorem to expand an integer power of a binomial and find a
required term.
Synthesize multiple skills and techniques in order to solve a complex, multi-step
problem.
N.
R.
Entry Skills for Math 7
Prior to enrolling in Math 7 students should be able to
1.
Determine whether a relation represents a function. If it is a function, determine its
domain and range; determine whether it is odd or even or neither based on its
formula or its graph; and determine whether it is one-to-one, and if it is, determine
its inverse function and its domain and range.
2.
Analyze and graph a given function, including but not limited to piecewise-defined,
polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric
functions, without the aid of graphing devices. Determine intercepts, coordinates of
holes, and equations of asymptotes. Determine intervals on which polynomial and
rational functions are positive and are negative.
Use transformation techniques including vertical and horizontal shifts, compression,
stretching, and reflection over the x- or y-axis to sketch the graph of a function.
Use the language and standard mathematical notation of the algebra of functions.
3.
4.
5.
6.
7.
8.
9.
Determine algebraic combinations and compositions of functions and state their
domains. Write a given function as a composition of two non-identity functions.
Use techniques and facts including synthetic division, long division, the
Fundamental Theorem of Algebra and the Rational Zeros Theorem to find all
complex zeros of a polynomial function of degree three or higher, and write the
function in a completely factored form.
From memory, state and apply the definitions of the six trigonometric ratios of
sides of right triangles; the definitions of the six trigonometric functions of real
numbers using the unit circle; and the definitions, domains and ranges of the
inverse sine, inverse cosine, and inverse tangent functions.
Evaluate trigonometric functions at integer multiples of  / 6 and  / 4 , including
values outside of [0, 2 ] , without the use of notes or calculators. Evaluate
compositions of trigonometric functions and inverse trigonometric functions
including ones for which cancellation equations do not apply.
From memory, state and apply the definitions of the six trigonometric ratios of
sides of right triangles; the definitions of the six trigonometric functions of real
numbers using the unit circle; and the definitions, domains and ranges of the
inverse sine, inverse cosine, and inverse tangent functions.
Santa Monica College
Page 3 of 6
Entry Skills for Math 7
10. Write algebraic and trigonometric relationships to solve application problems,
including solution of right and oblique triangles by the Law of Sines and Law of
Cosines.
11. Prove trigonometric identities including those which require the use of sum,
difference, double-angle and half-angle identities.
12. Solve polynomial, rational, exponential, logarithmic, and trigonometric equations.
13. Given a quadratic equation in variables x and y, with no xy term, put it into a
standard form in order to classify its graph as one of the conic sections (circle,
ellipse, parabola and hyperbola). Determine the directrix, center, vertex points,
focus points, major/transverse axis, and minor/conjugate axis, if they exist, and
sketch the graph of the conic section.
14. Find terms of explicitly and recursively defined sequences. Find the nth term in a
sequence whose first several terms are given.
15. Evaluate, manipulate and interpret summation notation.
16. Prove statements using mathematical induction.
17. Apply the binomial theorem to expand an integer power of a binomial and find a
required term.
18. Synthesize multiple skills and techniques in order to solve a complex, multi-step
problem.
Santa Monica College
Page 4 of 6
Santa Monica College
Student Learning Outcomes
Date: December 2009
Course Name and Number:
Math 7 Calculus 1
Student Learning Outcome(s):
 Individual faculty members will develop and reports on assessments for SLOs.
1.
Given an algebraic or trigonometric function, students will evaluate and apply limits and prove
basic limit statements.
2.
Given an algebraic or trigonometric function, students will differentiate the function and solve
application problems involving differentiation.
3.
Given an algebraic or trigonometric function, students will integrate the function and solve
application problems involving integration.
Demonstrate how this course supports/maps to at least one program and one institutional
learning outcome. Please include all that apply:
1.
Program Outcome(s):
The student will demonstrate an appreciation and understanding of mathematics in order to
develop creative and logical solutions to various abstract and practical problems.
As a result of studying calculus, students will be able to analyze and solve higher-level
abstract and practical problems.
2.
Institutional Outcome(s):
Through their experiences at SMC, students will obtain the knowledge and academic skills
necessary to access, evaluate, and interpret ideas, images, and information critically in order
to communicate effectively, reach conclusions, and solve problems.
As a result of knowledge gained through lectures, homework and exams, students will be
able to evaluate calculus-level problems critically and present solutions in a clear and logical
manner.
Santa Monica College
Page 5 of 6
Textbook: Swokowski, Earl, Calculus, The Classic Edition, Brooks/Cole Publishing, 1991
A Sample Schedule for Math 7
This schedule assumes a standard meeting schedule of 1 hr 5 min with 4 class meetings per week.
Session
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
Text Section/Activity
1.1 Precalculus review
1.2 Precalculus review
1.3 Precalculus review
Precalculus review/Quiz/Exam
2.1 Introduction to Limits
2.2 Definition of Limit
2.3 Techniques for Finding Limits
2.4 Limits Involving Infinity
2.5 Continuous Functions
2.5 Continuous Functions
Review
Exam 1 on Chapter 2
3.1 Tangent Lines and Rates of Change
3.2 Definition of Derivative
3.3 Techniques of Differentiation
3.4 Derivatives of the Trigonometric Functions
3.5 Increments and Differentials
3.6 The Chain Rule
3.6 The Chain Rule
3.7 Implicit Differentiation
3.8 Related Rates
3.8 Related Rates
Review
Exam 2 on Chapter 3
4.1 Extrema of Functions
4.2 The Mean Value Theorem
4.3 The First Derivative Test
4.4 Concavity and the Second Derivative Test
4.5 Summary of Graphical Methods
4.5 Summary of Graphical Methods
4.6 Optimization Problems
4.6 Optimization Problems
4.6 Optimization Problems
4.7 Rectilinear Motion and Other Applications
4.7 Rectilinear Motion and Other Applications
4.8 Newton’s Method
Review
Exam 4 on Chapter 4
5.1 Antiderivatives and Indefinite Integrals
5.2 Change of Variables and Indefinite Integrals
5.3 Summation Notation and Area
5.4 The Definite Integral
Santa Monica College
Page 6 of 6
Session
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
Text Section/Activity
5.5 Properties of the Definite Integral
5.6 The Fundamental Theorem of Calculus
5.6 The Fundamental Theorem of Calculus
5.7 Numerical Integration
Review
Exam 4 on Chapter 5
6.1 Area
6.2 Solids of Revolution (Volumes by disks & washers)
6.3 Volumes by Cylindrical Shells
6.4 Volumes by Cross Sections
6.5 Arc Lengths and Surfaces of Revolution
6.6 Work
Review
Exam 5 on Chapter 6
Review for Final
Review for Final