Course Description
MATH 1120 is an introductory calculus course for science students. The course includes limits, continuity, and the differentiation of algebraic, trigonometric, inverse trigonometric, exponential and logarithmic functions. Differentiation techniques are applied to graphing, extrema, related rates, and rectilinear motion, as well as to parametric and polar equations. This course is taught using a graphing calculator.
Course Content
- Limits and Continuity
- calculations of limits
- limit theorems
- continuity at a point and on an interval
- essential and removable discontinuities
- Intermediate Value Theorem
- The Derivative
- rates of change and tangent lines
- differentiation from definition
- differentiation formulas and rules
- chain rule
- implicit differentiation
- higher derivatives
- the differential and differential approximations
- linear approximations
- applications to related rates
- Inverse Functions: Exponential, Logarithmic and Inverse Trigonometric Functions
- definitions, properties, and graphs
- differentiation of logarithmic and exponential functions (any base)
- logarithmic differentiation
- differentiation of inverse trigonometric functions
- applications to related rates
- limits involving combinations of exponential, logarithmic, trigonometric, and inverse trigonometric functions
- L'Hôpital's rule
- Graphing and Algebraic Functions
- increasing and decreasing functions
- local extrema
- Rolle's Theorem and Mean Value Theorem
- curve sketching
- concavity; inflection points
- asymptotic behaviour; limits at infinity; infinite limits
- applied maximum and minimum problems
- antidifferentiation
- rectilinear motion
- Parametric Equations and Polar Coordinates
- parametric representation of curves in R²
- derivatives and tangent lines of functions in parametric form
- tangent lines to graphs in polar form
- definitions and relationships between polar and Cartesian coordinates
- graphing of r = f(?)
- Optional Topics (included at the discretion of the instructor).
- a formal limit proof (using epsilonics)
- application of the absolute value and greatest integer functions
- proofs of the rules of differentiation (differentiation formulas) for algebraic functions
- proofs of the differentiation formulas for trigonometric functions from the definition of derivative
- a proof of L'Hôpital's rule for the case of "0/0"
- Newton’s Method
Learning Activities
Lectures, problem sessions and assignments
Means of Assessment
Evaluation will be carried out in accordance with Douglas College policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on some of the following criteria:
| Weekly quizzes |
0-40% |
| Tests |
20-70% |
| Assignments |
0-15% |
| Attendance |
0-5% |
| Class participation |
0-5% |
| Tutorials |
0-10% |
| Final examination |
30-40% |
Note: All sections of a course with a common final examination will have the same weight given to that examination.
Learning Outcomes
MATH 1120 is a first course in calculus. The four-semester sequence of MATH 1120, 1220, 2321, and 2421 provides the foundation for continued studies in science, engineering, computer science, or a major in mathematics.
At the conclusion of this course, the student should be able to:
- find limits involving algebraic, exponential, logarithmic, trigonometric, and inverse trigonometric functions by inspection as well as by limit laws
- calculate infinite limits and limits at infinity
- apply L'Hôpital's rule to evaluating limits of the types: 0/0, 8/8, 8 - 8, 00, 80, 18
- determine intervals of continuity for a given function
- calculate a derivative from the definition
- differentiate algebraic, trigonometric and inverse trigonometric functions as well as exponential and logarithmic functions of any base using differentiation formulas and the chain rule
- differentiate functions by logarithmic differentiation
- apply the above differentiation methods to problems involving implicit functions, curve sketching, applied extrema, related rates, and growth and decay problems
- use differentials to estimate the value of a function in the neighbourhood of a given point, and to estimate errors
- apply derivatives to solve problems in velocity and acceleration, related rates, and functional extrema
- interpret and solve optimisation problems
- sketch graphs of functions including rational, trigonometric, logarithmic and exponential functions, identifying intercepts, asymptotes, extrema, intervals of increase and decrease, and concavity
- compute simple antiderivatives, and apply to velocity and acceleration
- recognise and apply the Mean Value Theorem and the Intermediate Value Theorem
- be able to convert between parametric and Cartesian forms for simple cases
- use parametric forms to determine first and second derivatives of a function
- sketch graphs of parametric equations and find the slope of a line tangent to the graph at a specified point
- sketch the graph of a polar equation r = f(?), and be able to find intercepts and points of intersection
- find the slope of a line tangent to the graph of a polar equation at a point (r,?)
Textbook Materials
- James Stewart, Calculus: Early Transcendentals, Current Edition, Brooks/Cole.
- A graphing calculator is also required.