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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

In particular,
x0 = 1 and
.
(c)

(d) These are obtained by combining the above rules:

and

3. The determinant of a 2 by 2 matrix:

4. Cramer’s Rule: To solve the system of equations

first calculate:

and

If Δ ≠ 0 then the system has a unique solution

and

If Δ = 0 and one of Δx, Δy is non zero, then the system is inconsistent i.e. has no
solution!

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
form
is

9. For the line containing two points and , a compact parametric
two point form
is

or

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:

Fundamental Identity.
Variant 1.
Variant 2.

Basic Identity.

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.

Variant 1.

Variant 2.

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 (x):
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