Curriculum Committee Approval: 11-4-14
CUYAMACA COLLEGE
COURSE OUTLINE OF RECORD
MATHEMATICS 180 – ANALYTIC GEOMETRY AND CALCULUS I
5 hours lecture, 5 units
Total contact hours: 80-90
Catalog Description
Graphic, numeric and analytic approaches to the study of analytic geometry, limits and continuity of functions, and introductory
differential and integral calculus. Applications involving analysis of algebraic, exponential, logarithmic, trigonometric and hyperbolic
functions from a variety of disciplines including science, business and engineering. First of three courses designed to provide serious
science students with a solid introduction to the theory and techniques of analysis.
Prerequisite
“C” grade or higher or “Pass” in MATH 170 and 175, or MATH 176 or equivalent
Entrance Skills
Without the following skills, competencies and/or knowledge, students entering this course will be highly unlikely to succeed:
1) Algebra
a. Knowledge and understanding of domain and range
b. Solving equations involving polynomials, logarithms, exponents, absolute value, and/or radicals
c. Function composition
d. Thorough understanding of the properties of rational exponents
e. Properties of real numbers
2) Trigonometry
a. Identifies and formulas
b. Standard angles
c. Converting between radian and degree measure
d. Solving trigonometric equations
3) Geometry
a. Analytic geometry
b. Formulas for standard geometric objects
c. Properties of geometric figures
d. Similarity
4) Mathematical Reasoning
a. Recognize mathematical arguments
b. Modeling
5) Graphing
a. Functions and their inverses
b. Conic sections
c. Graph interpretation
Course Content
1) Review of basic functions: Linear, exponential, power and trigonometric functions. Polynomial and rational functions. Inverse
functions, logarithms, compound interest and trigonometric functions.
2) Limits: The tangent and velocity problems to motivate limits. The limit of a function and calculating limits using the limit laws. The
precise definition of a limit. The squeeze theorem. Continuity.
3) The Concept of the Derivative: The notion of the derivative at a point defined graphically as the slope of the tangent line to the
graph of a function and defined numerically as a limit of difference quotients. The derivative function. Second derivative and
concavity of a function. Applications involving velocity and acceleration.
4) Short Cuts to Differentiation: Formulas for derivatives of some common functions including power, exponential, trigonometric, and
inverse trigonometric functions. Rules for differentiation including the product and quotient rules and the chain rule. Implicit
differentiation.
5) Applications of Derivatives: Finding local maxima and minima of a function. Finding global maxima and minima of functions. Using
critical points and points of inflection to graph functions. The Intermediate and Mean Value Theorems. Applications of the derivative
from a variety of disciplines such as business, science and engineering, and including the effect of parameters on graphs of families
of curves, economic applications and optimization.
6) Indeterminate forms and L’Hospital’s Rule.
7) The Concept of the Definite Integral: Definition of the definite integral as the limit of Riemann sums. Interpretations in terms of area.
Applications involving distance traveled and average value of a function. Statement of the fundamental theorem of calculus.
8) Reconstructing a Function from its Derivative and Methods of Finding Antiderivatives: Graphical, numeric and algebraic construction
of antiderivatives based on the fundamental theorem of calculus. Integration of power functions. Antiderivatives for some basic
functions. Method of substitution.
9) Historical contributions of number and mathematical theories and concepts from diverse cultures.
Course Objectives
Students will be able to:
1) Evaluate:
MATH 180
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a. finite limits as x approaches a,
b. infinite limits,
c. limits at infinity and
d. limits using L’Hospital’s Rule.
Apply the definition of continuity of a function to:
a. find intervals of continuity
b. find points of discontinuity
Define and evaluate the derivative
a. graphically as the slope of the tangent line at a point of a function, and
b. analytically as a limit of difference quotients.
Use the methods of calculus to graph a function.
Determine if a function is differentiable.
Use differentiation to solve application problems involving velocity and/or acceleration.
Apply the derivative of power, exponential, trigonometric, inverse trigonometric, logarithmic, or hyperbolic functions while using the
following:
a. the product rule,
b. the quotient rule,
c. the chain rule,
d. implicit differentiation, and
e. logarithmic differentiation.
Find and interpret:
a. local maximum and minimum values,
b. intervals of increase and/or decrease,
c. intervals of concavity, and
d. inflection points.
Solve application problems requiring the use of the derivative for
a. optimization, and
b. related rates.
Evaluate:
a. definite integrals, and
b. indefinite integrals.
Identify the definition of the definite integral as the limit of Riemann sums.
Evaluate and interpret the integral as:
a. the area under a curve, and
b. the area between two curves.
Use the definite integral to solve application problems involving distance traveled.
Method of Evaluation
A grading system will be established by the instructor and implemented uniformly. Grades will be based on demonstrated proficiency in
subject matter determined by multiple measurements for evaluation, one of which must be essay exams, skills demonstration or, where
appropriate, the symbol system.
1) Quizzes which measure the student’s ability to: define and evaluate derivative as the slope of the tangent line to the graph of a
function and numerically as a limit of difference quotients; identify local extrema of a function using critical points; use critical points
and points of inflection to graph a function; identify the definition of the definite integral as the limit of Riemann sums and interpret
the definite integral in terms of area.
2) Exams which measure the student’s ability to: evaluate the second derivative and use it to interpret the concavity of a function;
identify and apply formulas for derivatives of common functions; identify and apply rules of differentiation; reconstruct a function
from its derivative and employ various methods including integration of power functions and substitution to find antiderivatives.
3) Individual exploration activities that measure the student’s ability to: use differentiation to solve application problems involving
velocity and acceleration; solve application problems such as economic applications and optimization; select and apply appropriate
technology to model, analyze and interpret a collection of real-world data.
4) Group-work, class activities, homework, and/or quizzes which measure the student’s ability to select and apply appropriate
technology including but not limited to computer programs and graphing utilities to model, analyze and interpret a collection of data
or to solve real-world application problems requiring the use of analytic geometry and calculus.
Special Materials Required of Student
Graphing utility, portfolio
Minimum Instructional Facilities
Smart classroom with whiteboards covering three walls, graphing utility overhead viewing panels, projection screen
Method of Instruction
1) Lecture and discussion
2) Teamwork
Out-of-Class Assignments
1) Problem sets
2) Exploratory activities and/or projects
3) Reading and/or writing assignments
Texts and References
1) Required (representative example): Stewart, James. Calculus, Early Transcendentals. 8th edition. Cengage, 2015.
2) Supplemental: None
MATH 180
Exit Skills
Students having successfully completed this course exit with the following skills, competencies and/or knowledge:
1) Essential vocabulary and concepts
a. Limits
b. Continuity
c. Differentiation
d. Integration
2) Evaluating limits: algebraic, trigonometric, logarithmic and exponential functions
3) Limit Calculations
a. Using L'Hopital's Rule
b. Solving limits with indeterminate forms
4) Evaluating derivatives
a. Implicitly
b. Algebraic, trigonometric, logarithmic and exponential functions
5) Evaluating integrals: algebraic, trigonometric, logarithmic and exponential functions
6) Applying the Fundamental Theorem of Calculus.
7) Graphing: interpreting function behavior from derivatives
8) Modeling and applications
a. Related rates
b. Relative extrema
c. Area between curves
Student Learning Outcomes
Upon successful completion of this course, students will be able to:
1) Compute the limit of a function at a real number.
2) Determine if a function is continuous at a real number.
3) Find the derivative of a function as a limit.
4) Find the equation of a tangent line to a function.
5) Compute derivatives including implicit differentiation.
6) Use differentiation to solve applications such as related rate problems and optimization problems.
7) Graph functions using methods of calculus.
8) Evaluate a definite and indefinite integrals.
9) Apply integration to find area.
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