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MATH 105 Lecture Notes - Quadratic Equation, List Of Trigonometric Identities, Differential Calculus

Course Code
MATH 105
Julia Kam

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In taking this class, you are expected to have proficiency in basic pre-calculus and differential calculus. Having
this basic knowledge will make your time in the course a lot more enjoyable; weakness in basic skills is the
biggest stumbling block to math, not the new material itself. It is your responsibility to review this material
and any unfamiliar terminology below. This is essential to understanding the material and in succeeding on
exams. This is not all-inclusive, but it is a good start.
- Solving a quadratic equation (ax2 + bx + c = 0) and quadratic formula
- How to complete the square e.g. -x2 + 6x + 2 = -(x-3)2 + 11
- How to factor a general polynomial (with integer or rational roots) e.g. x3 + x2 8x 12 = (x+2)2(x-3)
- Logic in equalities e.g. (x2 + y2) = x + y, 1/(x+y) = 1/x + 1/y are not true
- How to solve inequalities e.g. x2 > 4 means x < -2 or x > 2.
- Solving 2 by 2 or 3 by 3 systems of linear equations e.g. x+3y=7, 4x-11y = 18.
Trigonometry and Geometry
- Special angles and special triangles (sin, cos, tan for 0, π/6, π/4, π/3, π/2, π)
- Basic trigonometric identities e.g. sin(a+b) = sin a cos b + cos b sin a
- Similar triangles
- Domain/range
- Recognizing if a function is even/odd
- Notation for composition of functions
- Difference between a function and its value at a single point (f(x) vs f(4))
- Rules for exponents and logarithms e.g. ln (ab) = ln a + ln b, (xa)b = xab
- Plots of simple functions like exp(x), ln(x), sin(x), cos(x), tan(x), parabolas, lines, root functions,
circles, y=1/x
- Characteristic values and points-of-interest for simple functions e.g. e0 = 1, ln 1 = 0, -1 sin x 1
- Meaning of fractional and negative exponents
- How to find intersection points between two functions (lines, quadratics, polynomials)
- Knowing ballpark numerical values to π, e
Differential Calculus:
- Definition of a derivative
- Derivatives of polynomial, trigonometric, logarithmic, exponential functions, functions with
rational/irrational powers, inverse trig functions
- Sum, product, and chain rules for derivatives
- Asymptotes (vertical, horizontal) for functions
- Limits at infinity and physical meaning
- Interpretation of derivatives: increasing/decreasing, concavity
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