Limits, derivatives and integrals of algebraic, trigonometric, exponential and logarithmic functions. Differentials and applications of the derivative.
80.000-90.000 Total Hours
Total Hours
160.000-180.000 Total Hours
80.000-90.000 Total Hours
Prerequisite: MATH 138 or equivalent based on SBCC's Assessment Center placement via multiple measures.
Prerequisite or Corequisite: None Concurrent Corequisite: None Course Advisories: None Limitation on Enrollment: None
Course Objectives:
Compute the limit of a function at a real number
Determine if a function is continuous at a real number
Find the derivative of a function as a limit
Find the equation of a tangent line to a function
Compute derivatives using differentiation formulas
Use differentiation to solve applications such as related rate problems and optimization problems
Use implicit differentiation
Graph functions using methods of calculus
Evaluate the definite integral as a limit
Evaluate integrals using the Fundamental Theorem of Calculus
Apply integration to find area
Student Learning Outcomes
MATH150 SLO1 - Evaluate limits and use them to find derivatives and integrals
MATH150 SLO2 - Evaluate derivatives and analyze the connection between the derivative, the slope of the curve, and rates of change
MATH150 SLO3 - Determine the behavior of a function from its derivatives and use them to solve optimization problems and other applications
MATH150 SLO4 - Evaluate integrals and analyze the connection between integral, the area bounded by the curve, and total change
Limits
Intuitive and geometric definition of a limit
Calculation of limits using numerical, graphical, and algebraic methods.
One- and two- sided limits and limits at infinity
Continuity of functions.
Derivatives
Definition of the derivative of a function as a limit
Differentiability of a function.
Interpretation of derivative as slope of tangent line and rate of change
Computation of derivatives; constant functions, power, sum, product, quotient and chain rules and implicit differentiation.
Differentiation of inverse functions.
Derivatives of transcendental functions including trigonometric, exponential, and logarithmic functions.
Higher order derivatives.
Differentials
Applications of the derivative
Motion of a particle
Related Rates
Maximum and minimun values
Optimization.
Mean Value Theorem.
Graphing functions using first and second derivatives, concavity and asymptotes.
Economic Applications (optional)
Newton's Method
Indeterminant forms and L'Hopital's Rule
The definite integral
Riemann sums and the definition of the definite integral
Properties of the integral
The Fundamental Theorem of Calculus and evaluation of integrals
Antiderivatives and Indefinite integrals
Areas under and between curves
The substitution rule
Integration by parts
Inverse functions
The logarithm as an integral and the exponential as its inverse
Exponential growth and decay
The inverse trigonometric functions
Methods of Instruction
Directed Study
Lecture
Projects
Lecture is the primary activity, along with student problem-solving. Students are expected to work outside of class on supplemental reading from the text and on assigned exercises.
Find the general antiderivatives for the following functions: x sin(x), x cos(x), x tan(x), and x sec(x).
1. Appropriate Readings: Students are required to read assigned chapters in texts. Outside readings are generally not required.
2. Writing Assignments: Students must work assigned mathematical problems requiring the manipulation of abstract symbols.
3. Appropriate Outside Assignments: Students will be expected to spend a sufficient amount of time outside of class to practice techniques taught during class time, read assigned materials, and complete frequent homework assignments.
4. Appropriate Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve analyzing information, recognizing concepts in new contexts, and drawing analogies. They must also analyze logical arguments for validity and write proofs of their own using both inductive and deductive reasoning within a logical system.
A student's grade will be based on multiple measures of performance in the solving of problems, preparation and analysis of graphs, and analysis of logical arguments. Such measures will include at least three exams and a comprehensive final examination requiring demonstrations of problem-solving skills, In addition, instructors may make use of quizzes, written homework assignments, or other appropriate means to judge a student's dexterity with mathematical skills and familiarity with mathematical vocabulary. Calculator (or computer use) is incorporated in the course. Students are expected be able to perform differentiation and integration "by hand."
Calculus with Early TranscendentalsRogawski, Freeman Publishing, 2015
TI-84 Graphing Calculator, Maple, or equivalent computer algebra system