Lectures, problem sessions and assignments
- General Functions
- Polynomial and Rational Functions
- Exponential and Logarithmic Functions
- Trigonometric Functions
- Analytic Trigonometry and Applications
Upon completion of MATH 1110 the student should be able to:
- understand the concept of function and be able to determine which relations are functions by an examination of the equation and/or the graph of the relation.
- find the domain of any function and the range of functions for which the inverse can be determined or for which the graph can be easily sketched.
- extract the functional rule from a 'word problem'.
- determine if a function is odd or even and understand the graphical implication of the property.
- sketch the graphs of the following functions:
- and the graphs of the following variations of the above functions:
y = x, y = x2, y = x3, y = |x|, y = vx, y = 1/x, y = 1/x2, y = va2 - x2, y = [|x|]
and the graphs of the following variations of the above functions:
y = f(x) + c, y = f (x + c), y = -f(x), y = cf(x)
- apply the above transformations to any given graph or function.
- sketch the graph of simple piece-wise defined functions.
- sketch the graph of any quadratic function and be able to determine all intercepts and the vertex using the quadratic formula and/or completing the square.
- determine the equation of a quadratic from its graphical properties.
- solve maximum-minimum 'word problems' involving a quadratic function.
- add, subtract, multiply and divide functions and be able to determine the domains of the resulting functions.
- determine the composite of several functions and its domain.
- determine the inverse of a given one-to-one function and the domain and range of the inverse function.
- prove that a given function is the inverse of another given function.
- sketch the graph of the inverse of a given one-to-one function when the inverse functional rule cannot be determined.
- understand the polar coordinate system and be able to graph a function written in polar coordinates. (optional)
- sketch the graph of a plane curve given by a set of parametric equations. (optional)
- find parametric equations of basic plane curves. (optional)
POLYNOMIAL AND RATIONAL FUNCTIONS
- find the quotient and remainder when a polynomial is divided by a second polynomial.
- use the remainder theorem.
- use the factor theorem to find the real roots of polynomial equations and the real zeros of polynomial functions.
- determine the multiplicity of zeros.
- use the rational root test to determine all possible rational roots.
- factor and graph any polynomial of degree n provided that the polynomial has at least n-2 rational roots.
- obtain the functional rule for a polynomial when given certain information about the roots and a value that satisfies the function and graph the function.
- sketch the graph of proper and improper rational functions that have a most one horizontal asymptote or an oblique asymptote.
- solve 'word problems' that involve polynomial or rational functions.
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
- find the exact value of logarithmic and exponential expressions.
- use a calculator to approximate the logarithm of a number to any base.
- use a calculator to approximate the solutions to exponential and logarithmic equations for all bases.
- find the inverse of a given exponential or logarithmic function and the domain and range of the inverse function.
- demonstrate an understanding of the rules of logarithms by rewriting given expressions.
- sketch the graph of exponential and logarithmic functions determining the value of all intercepts and the equation of the asymptote.
- solve 'word problems' which require the use of logarithms and/or exponentials; i.e. growth and decay problems and compound interest problems.
THE TRIGONOMETRIC FUNCTIONS
- convert radians to degrees, minutes and seconds and vice versa.
- solve problems that demonstrate an understanding of the relationship between the central angle, the arc length and the radius of a circle.
- solve problems that demonstrate an understanding of the relationship between the angular velocity, the linear velocity and the radius of a wheel or similar object.
- determine the area of a circular sector.
- demonstrate an understanding of the six trigonometric functions relative to a right triangle and to the unit circle.
- recall and apply the fundamental trigonometric identities, the co-function formulas and the formulas for negatives.
- sketch the graphs of the six basic trigonometric functions and recognise which functions are odd and which functions are even.
- find the exact values of the remaining trigonometric functions given the values of two trigonometric functions or the value of one trigonometric function and the quadrant.
- find the exact values of the trigonometric functions for an angle in standard position given a point on the terminal side.
- find the reference angle of any angle in degrees and/or radians.
- express any trigonometric function as a function of a given trigonometric function.
- recall the exact values of the trigonometric functions for reference angles of 30o, 45o, and 60o and the axis angles.
- use a calculator to approximate the value of the trigonometric function of any real number.
- use a calculator to approximate the reference angle given the value of the trigonometric function.
- determine the amplitude, period and the phase shift of any trigonometric function and sketch its graph showing all intercepts and turning points.
- demonstrate an understanding of the terms 'angle of depression' and 'angle of elevation' and solve 'word problems' involving right triangles.
ANALYTIC TRIGONOMETRY AND APPLICATIONS
- recall or derive and demonstrate an understanding of the addition and subtraction formulas, the double angle formulas and the half-angle identities for sine, cosine and tangent.
- demonstrate an understanding of the product-to-sum and sum-to-product formulas when given the formulas.
- combine a sine function and a cosine function of the same period into a single cosine function when given the formula.
- verify trigonometric identities.
- find all the solutions of trigonometric equations and find solutions on a restricted interval.
- sketch graphs of the six inverse trigonometric functions and state the domain and range of each function.
- sketch the graph of simple inverse trigonometric functions.
- find the exact value of inverse trigonometric expressions.
- simplify given composites of trigonometric and inverse trigonometric functions.
- solve 'word problems' that require the use of the inverse trigonometric functions.
- verify inverse trigonometric identities.
- solve 'word problems' that require the use of the Law of Sines and/or Law of Cosines.
PARABOLAS, ELLIPSES AND HYPERBOLAS
- find the vertex, focus and directrix of a parabola and sketch its graph.
- find the vertices and foci of an ellipse and sketch its graph.
- find the vertices and equations of the asymptotes of a hyperbola and sketch its graph.
- find an equation of a parabola or ellipse that satisfies given conditions.
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:
- Stewart, Redlin, Watson, Precalculus: Mathematics for Calculus., Current Edition, Brooks Cole.
- graphing calculator