HOW TO PROVE 2 LINES ARE PARALLEL
1. If corresponding angles are =, then lines are parallel.
2. If alternate interior angles are =, then lines are parallel.
3. If alternate exterior angles are =, then lines are parallel.
4. If consecutive interior angles
Parallelogram: Quadrilateral with both pairs of opposite sides parallel.
Properties of Parallelograms:
1.
Both pairs of opposite sides are parallel.
2.
Both pairs of opposite sides are equal.
3.
Opposite angles are =.
4.
Consecutive angles are supplementa
THE RELATIONSHIP THE FIRST AND SECOND DERIVATIVES
HAVE TO THE FUNCTION
FIRST DERIVATIVE RULES
1. If the first derivative is positive, then the function must be increasing.
2. If the first derivative is negative, then the function must be decreasing.
3. If
Area and Integration Rules
1. Area under the curve (area is above the x-axis):
a, b are the x-intercepts
2. Area under the curve (area is under the x-axis):
- or
a, b are the x-intercepts
3. Area between two curves:
a, b are the x values of the points of
LIMIT DEFINITION:
f(x) will have a limit at x = c IFF the limit as x approaches c
from the left = limit as x approaches c from the right.
CONTINUITY DEFINITION
f(x) will be continuous at x = c IFF the limit as x approaches c
= f(c).
Infinity Rules!
Select the terms from the
numerator and denominator that
have the highest power: (axn/bxm):
1. Highest Power is the Same:
(n = m) Answer will be the fraction
a/b.
2. Highest power in the
denominator: ( n < m)
Answer = 0.
3. Highest power i
If a function is discontinuous at a point c,
but has a limit as x approaches c, then
c is a REMOVABLE DISCONTINUITY.
If a function is discontinuous at a point c,
and the limit as x approaches c does not
exist, then c is an ESSENTIAL
DISCONTINUITY.
LIMIT DEFINITION:
f(x) will have a limit at x = c IFF the limit as x approaches c
from the left = limit as x approaches c from the right.
CONTINUITY DEFINITION
f(x) will be continuous at x = c IFF the limit as x approaches c
= f(c).
lim f(x) = f(c)
x c
(I
Rules for ex and ln(x).
1. If y = ln(x), then ey = x.
2. ln(ef(x) = f(x)
3. eln(f(x) = f(x)
4. ln(a) + ln(b) = ln(ab)
5. ln(a) ln(b) = ln(a/b)
6. ln(ba) = aln(b)
7. ln(1/a) = ln(a-1) = -ln(a)
DERIVATIVES:
If y = ef(x) then = f(x)ef(x)
If y = ln(f(x) then
Approximating Definite Integrals
Riemann Sums and Trapezoidal Rule
Riemann Sum: add the areas of rectangles to approximate the definite
integral.
Area = b x h. In a Riemann sum, the base = x and the height = f(x).
1. Left-endpoint sum: Use the left x-valu
Intermediate Value Theorem
Let f be a continuous function on a closed interval [a,b].
If k is a number between f (a) and f (b), then there exists
at least one number c in [a,b] such that f (c) = k.
5
f(B)
4
3
2
1
A
-8
-6
-4
B
-2
2
-1
f(a)
-2
-3
-4
-5
4
6