Facts to Remember
1. Properites of Inequalities:
(a) If x < y and y < z then x < z.
(b) If x < y then x +z < y + z.
(c) If x < y and z is positive, then x z < y z.
However, if x < y and z is negative, then x z > y z.
(d) |x| < y if and only if −y < x < y.
(e) |x| > y if and only if either x > y or x < −y.
2. Properties of Exponents: Be aware that there are some natural assumptions a
x0 = 1 and
(d) These are obtained by combining the above rules:
3. The determinant of a 2 by 2 matrix:
4. Cramer’s Rule: To solve the system of equations
If Δ ≠ 0 then the system has a unique solution
If Δ = 0 and one of Δx, Δy is non
zero, then the system is inconsistent i.e. has no
If Δ = Δx = Δy = 0, then the equations are essentially the same and have infinitely
many solutions, provided at least one term with the variables is present.
5. The distance between two points A and B on the real line is d (A,B) = |A − B|.
6. The distance between two points and in the xy-plane: is
7. For two points and
, the midpoint is
This evaluates to:
8. For the line containing two points
and , a
parametric two point
9. For the line containing two points
and , a
two point form is
10. For the line containing two points and , the two point form is
11. For the line containing two points and , the slope is
Further, the slope intercept form of the line is
y = m x + c,
where m is the slope and c is the y-intercept given by
12. If p is the x-intercept and q is the
y-intercept of a line, then the intercept form
of the line is
13. The equation of a circle with center at (a, b) and of radius r is
14. A parametric form of a circle centered at the origin and of radius r:
where m is the parameter.
15. The quadratic formula for solutions to ax2 + bx + c = 0, when a ≠ 0, is
16. Trigonometric Functions:
17. Trigonometric Identites:
That is, Sine is odd.
That is, Cosine is even.
Addition rule for sine.
Addition rule for Cosine.
Double angle formula for Sine.
Double angle formula for Cosine.
18. Derivative Formulas:
(a) If f(x) = p, where p is a constant, then f' (x) = 0.
(b) Power rule If , then
(c) If c is a constant and g (x) = c f (x) then g' (x) = c f' (x).
(d) Sum rule
(e) Product rule If then
(f) Chain rule then
19. Linear Approximation:
Linear approximation to f (x) at x = a:
20. Newton’s Method for finding a root x = a of f
Start with a guess x = a and improve it to
Guess: x = a Improve to
21. Combinations of n objects taken r at a time are given by:
22. The Binomial Theorem:
Thus, the coefficient of is simply