# 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,

x^{0} = 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 Δ_{x}, Δ_{y} is non
zero, then the system is** inconsistent **i.e. has no

solution!

If Δ = Δ_{x} = Δ_{y} = 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 ax^{2}
+ 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