MATH 3C COURSE TOPICS: January 28, 2013
**…**: Not on Exams
Exam 1 Material
1/7/13
1. Formula for Area of Rectangle
2. Understanding the Formulas for Area of a Triangle, Parallelogram,
Trapezoid
1/8/13
1. Rope experiment: triangles, unlike squares, are rigid.
2. The Egyptians (4000 years ago) used a 3-4-5 triangle made of
rope to create a right triangles to be used in construction. For
them, knowing “why” was useless when it came to building things.
3. Derivation of the Pythagorean Theorem: The Greeks (2500 years
ago) asked WHY
4. Application: aspect ratio of television
1/9/13
1. Dividing strings into common subunits has musical applications
and matches the Greeks’ desire for order everywhere.
2. Strings that can be divided into common subunits correspond to
rational numbers.
3. Consider the length of the diagonal of a square with side length 1.
By the Pythagorean Theorem, it must have length √2 .
4. A calculator gives √2 = 1.4142 … this isn’t magic.
5. “Greek ladder” method for approximating √2
6. **Explanation why √2 cannot be rational and is therefore
irrational**
7. Knowing which square roots are rational and which are irrational.
8. Rational vs. irrational numbers in the decimal system.
9. Converting repeating decimals to fractions.
1/11/13
1. Real numbers = rational numbers + irrational numbers.
2. We represent real numbers by points on the “real line.”
3. Intervals on the real line.
4. Unions and intersections of intervals.
5. Introduction to the coordinate plane.
6. Distance formula between two points in the plane.
7. How can we define a line by an equation in the coordinate plane?
8. Simple cases: vertical and horizontal lines
9. Slope between two points: the percentage change in y over change
in x
1/14/13
1. Recall from geometry: What are similar triangles?
2. The ratio between two corresponding sides in similar triangles
must always be the same.
3. Definition of a line: the slope is the same between any two points
(uses similar triangles).
4. Point-slope form of a line.
5. Slope-intercept form of a line.
6. Application of lines to real phenomena: converting units of
temperature and money
1/15/13
1. Intersection of two lines and applications
2. Explanation that parallel lines have the same slope.
3. Explanation that perpendicular lines have slopes whose product =
-1.
4. Midpoint formula.
5. Equation for unit circle.
1/16/13
1. Derivation of standard form for circle centered at (h, k) with
radius r
2. Formulas for area and perimeter of a circle of radius r.
3. Completing the square
4. Using completing the square to change circle into standard form.
5. Archimedes’ approximation of pi by using the method of
exhaustion (IDEA only)
1/18/13
1. Definition of ellipse
2. **Derivation of ellipse from foci and distance constant**
3. Standard form for ellipse having foci on the x-axis and y-axis
4. Finding x- and y-intercepts of an ellipse
5. Converting formula of ellipse to standard form.
6. A circle as a specific form of an ellipse.
7. Practical purpose of an ellipse’s foci and applications.
8. (From homework): Finding foci of ellipse given equation of
ellipse.
1/22/13
1. How a number can have more than one decimal representation.
2. Definition of parabola.
3. The vertex of a parabola.
4. Shifting and scaling transformations of parabolas and their effect
on the vertex.
5. Standard Form of Parabola
6. Practical purpose of a parabola’s focus and applications
1/23/13
1. Parabolas and projectile motion
2. Polynomial form of a parabola
3. Converting a given parabola in polynomial form to standard form
using completing the square.
4. **Converting a general parabola in polynomial form to standard
form**
5. Finding x-intercepts of a parabola in standard form
6. **Derivation of quadratic formula for finding x-intercepts of a
general parabola in polynomial form**
7. Applying the quadratic formula to find x-intercepts of parabolas in
polynomial form
Exam 2 Material
1/25/13
1. Archimedes’ Death Ray and Fermat’s Last Theorem
2. Golden rectangles and the golden ratio
3. Shifting graphs up/down/left/right with applications to
circles/ellipses
4. When are three points collinear?
1/29/13
1. Properties of integer exponents
2. Graphing monomials and shifts
3. Definition of polynomial
4. Rational zero theorem for polynomials with integer coefficients
1/30/13
1. Factor theorem
2. Applying factor theorem to long division
3. Two simple factoring rules
4. Multiplicities of zeroes
2/1/13
1. End behavior of polynomials
2. The need for complex numbers
3. Definition of complex numbers
4. Fundamental theorem of algebra
2/4/13
1. Graphing polynomial from zeroes and end behavior
2. Complex addition/subtraction/multiplication
3. Definition of complex conjugate; a complex number times its
conjugate is real
4. Using conjugates for complex division
5. Conjugate zeroes theorem
6. Definition of function
7. Function as table
2/5/13
1. Function as equation and why not all equations are functions
2. Function as graph
3. Vertical line test
4. Not all functions can be represented as clean equations
5. Graphing y =1/xk
6. Piecewise functions and absolute value
7. Domain and range of functions
8. Definition of rational functions
2/6/13
1. Finding zeroes of rational functions
2. Domain/range of rational functions
3. “Holes”/vertical asymptotes of rational functions
4. End-behavior of rational functions and horizontal asymptotes
2/8/13
1. Long division of polynomials
2. Simple interest and why it doesn’t make sense
3. Compound interest compounded annually
4. Summing payments that grow according to compound interest
(e.g., tuition, credit card payments); **derivation of this formula is
not on exams**
5. A closed formula for adding payments that grow by compound
interest
2/11/13
1. Properties of exponents
2. Defining a0 and a-m (when a is not 0)
3. The cases of rational and irrational exponents
4. Compounding interest every period and translating to annual
interest
5. The derivation of e
6. Continuously compounded interest
2/12/13
1. Translating compound interest to annual interest
2. Definition of exponential function and two cases for graph
(growth vs. decay)
3. Exponential function through two points
4. y-intercept of exponential function is its initial value; no xintercepts
5. Using exponential functions to model doubling and half-life
6. “Continuous growth and decay”  use P(t) = (er)t
2/13/13
1. Why we need logarithms: solving for x in exponential functions
2. Definition of logarithm
3. Properties of logarithms
4. Applying logarithms to exponential functions
2/15/13
1. More on radioactive decay and a general formula involving halflife
2. Equations involving logarithms and exponential functions
3. Shifts, scales, and flips of functions
Monday, February 20: President’s Day Holiday
2/16/13
1. Combining shifts, scales, flips of functions together
2. Sum/Different/Product/Quotient of two functions
3. Function composition with tables, equations, and graphs
2/18/13
1. Idea of inverse
2. Conditions for when an inverse of a given function will exist in
table/equation/graph notation
3. Finding an inverse of a given function in table/equation/graph
notation
4. Increasing and decreasing functions
2/22/13: Catchup and Exam Review
2/25/13: Midterm 2
Post-Exam 2 Material
2/26/13:
1. Positive and negative angles around a circle
2. Finding the length of a circular arc given an angle and vice-versa
3. Eratosthenes’s method for finding the circumference of the earth
2/27/13:
1. Definition of sine/cosine/tangent in right triangles
2. Calculation of sine/cosine/tangent of 30 degrees and 60 degrees
3. Finding the height of a tall tree using tangent
3/1/13:
1. Two vital identities:
a. tan(θ) = sin(θ)/cos(θ)
b. [cos(θ)]2 + [sin(θ)]2 = 1
2. General formula for area of triangle with sine
3. Using area formula to see that sin(180° - θ) = sin(θ)
4. Finding area of regular polygons
3/4/13:
1. Approximation of pi with large regular polygon
2. Law of Sines (including derivation)
3. Applying Law of Sines to finding missing sides of triangles
(including problem of finding distance to faraway object in space)
3/5/13:
1. Extending sine/cosine/tangent to angles outside the interval
0° < θ < 90° via the unit circle
2. Bounds on sine and cosine (but not tangent)
3/6/13:
1. Introduction to radians
2. Adding radians to the unit circle
3. Graph of y = sin(x), y = cos(x) and y = tan(x) along with domain
and range
3/8/13:
1. Period and amplitude
2. Horizontal scaling of functions f(cx)
3. Transformations (shifts/scales/flips) of trigonometric functions
and how they affect period and amplitude
3/11/13:
1. Inverse trigonometric functions
2. **Derivation of law of cosines**
3. Three forms of law of cosines
4. Applying inverse trigonometric functions to law of sines/cosines
3/12/13:
1. Introduction to systems of equations
2. Graphing systems of equations
3. The substitution method for solving systems of equations
3/12/13:
1. Introduction to systems of linear equations
2. Systems of linear equations can have 0, 1, or infinitely many
solutions
3. Gaussian elimination for finding solutions to systems of linear
equations
Friday, March 15: Review Session (Optional)