Multiplying Fractions and Mixed Numbers
When we ask, "What is 4/5 of 55?" or "What is 1/6 of 18/5?", we are really
asking, "What is 4/5 times 55?" and "What is 1/6 times 18/5?". When dealing with
fractions, the word "of" indicates multiplication. So how d
Least Common Denominator (LCD)
A common denominator of two numbers is a number that can be divided by the
denominators of both numbers. For example, 1/6 and 4/9 have common
denominators of 18, 36, 54, 72, etc. The least common denominator, or LCD, is
the
Fractions
A fraction describes a part of a whole. The number on the bottom of the fraction
is called the denominator, and it denotes how many equal parts the whole is
divided into. The number on the top of the fraction is called the numerator, and it
deno
Expressing Fractions as Decimals
Sometimes we will want to work with decimals instead of fractions. To convert
fractions to decimals, simply divide the numerator by the denominator, either
using a calculator or by hand using the usual method of long divis
Expressing Decimals as Fractions in Lowest Terms
Sometimes it is easier to work with fractions than to work with decimals. It is
therefore important to learn how to change decimals into fractions. The secret to
this task lies in understanding the meaning
Adding and Subtracting Fractions
We can only add or subtract fractions when they have the same denominator.
Therefore, the first step in adding or subtracting fractions is writing them as
fractions with the same denominator (see Reducing Fractions and the
Equivalent Fractions
Two fractions are equivalent if they express the same part of a whole. For
example, 2/3 and 4/6 express the same part of a whole. 12/9 and 4/3 are also
equivalent.
Two fractions are equivalent if there is a number by which both the nu
AP Calculus: Limits
Names_
A. Sketch a graph that meets the following criteria:
1.
2.
lim f (x) 2
lim f (x) 4
x2
3.
lim f (x) 2
x4
4.
lim f (x) 1
x
x1
lim f (x) 4
x1
5.
lim f (x) 3
x2
lim f (x) 2
x2
f (2) 2
6.
lim f (x) 2
x
lim f (x) 2
x
lim f (x)
x1
lim
AP Calculus AB
Chapter 3
1.
Name_
2.
3(x h)2 3x2
lim
h0
h
3.
f (x)
7 x h 2 x h 7x 2x
4
lim
5.
h
4
6.
2
11 x h 3 x h 4 11x 3x 4
3
lim
h
d
11x4 7x3 8x2
dx
8.
3
h0
9x4 9a4
xa
x a
lim
y x 8x 5x 3x 4
dy
dx
5
7.
9.
4.
4
h0
f (x) 12x4 8x3 7x 4
10.
12x6 3x12a6
Chapter 2 Review
1. lim ( x x 1)
x
2. f ( x) 2
x 16
x3 2 x
3. lim 2
x 3 x 5
x2 4
4. f ( x)
x2
3
2
x 2
5. lim(e x x)
x
(find VA, HA and holes)
(find VA, HA and Holes)
x 3 9 x 14
x
2 x 3
6. lim
6. f ( x) 2 x 2 5 x
a) Write the equation for the tangent li
Calculus Chapter 2 Limits (Day 3)
HOMEWORK
Name_
Find the (a) lim f ( x), (b) lim f ( x), and (c) all asymptotes (horizontal and vertical) and holes in the
x
graphs.
1.
f ( x)
2. f ( x)
3.
4
x 1
x
2
x2
x 5x 6
f ( x)
4.
f ( x)
5.
f ( x)
2
2
2( x 9)
2
LHpitals Rule
Indeterminate Forms 0,-, 00, 1,0
Examples:
1.
=
2 x
lim x e
x
2.
1
lim x sin
x
x
3.
=
1
1
ln x x 1
lim
x 1
4.
1
lim 1
x
x
5.
lim x x
x 0
x
=
=
Homework:
1.
=
1 1
lim
x 0 x
x
2.
1
2
x
x
=
lim
x 0
3.
lim cos x cos x
x
4.
=
2
lim 1
AP Calculus
Chapter 2
1.
a.
b.
c.
d.
e.
Name_Pd_
Given f(x) = x3 5x2.
Find the average rate of change over [-2, 5]
Find the instantaneous rate of change at x = 5
Give the slope of the curve at x = -1
Give the equation for the line tangent to f(x) at x = 1
AP Calculus
Name_
Given the graph of f(x), graph f (x).
1.
2.
y
y
x
x
AP Calculus
Given the graph of f(x), graph f (x).
Name_
1.
2.
y
y
x
x
Given the graph of f (x), graph f(x)
3. f(1)=2
4. f(-3)= 1
y
y
x
x
Given the graph of f (x), graph f(x)
3.
Advanced Calculus
Improper Integrals Quiz Review
Using the direct comparison or limit comparison test, determine if the following
integrals diverge or converge.
1.
2.
1
ln x
1 cos 2 x
dx
x
3 dx
x
e
0
Evaluate the following integrals. If they converge, g
Advanced Calculus
Name_
Improper Integrals
Evaluate the following integrals or state that it diverges. If you use the direct
comparison test or the limit comparison test to determine the integral diverges,
show your work. If it converges, you must find th
Limit Review
Limit
x 2 16
lim
x 4 x 4
2 x3
lim
x x 5
lim x3 2 x 2 1
x 2
2 x2 3
x 5 x 2 7
lim
ex
x x 3
lim
lim
x 0
7x
x 5
5x2
lim 2
x 2 x 4
lim
x 0
ln x
x
2 x 3, x 0
lim
x 0
2
x 3, x 0
x2
lim 2
x 1 x 2 x 1
e2 x 1
lim
x 0
x
lim
x 0
1
x2
Reason
Answer
LHpitals Rule
The Indeterminate Forms 0/0 and
LHpitals Rule: If f(a)=g(a)=0, f and g are differentiable on an open interval I containing a, and
f ( x)
f ( x)
f (a )
lim
or
that g(x) 0 on I if x a, then lim
x a g ( x )
x a g ( x )
g (a )
Remember: Take t
4670 - 1 - Page 1
AP CALCULUS
EXACT VALUES TRIG REVIEW
Name _
_ 1)
Find the exact value of sin
.
_ 8)
Find the exact value of tan
.
_ 2)
Find the exact value of sin
.
_ 9)
Find the exact value of tan
.
_ 3)
Find the exact value of sin
_ 10)
Find the exact
Calculus I
Worksheet #8
Review for Test 2 Limits and Continuity
1
sin5x
lim
x 0 cos4 x
2
sin2 3x
lim 2
x 0 x cos x
3
sin5x
lim 1
x 0
sin x
3
4
x
lim
x 0 tan x
5
lim
x
1 + sin x
1 cos x
6
cos2x
x
x2
lim
2
6x 5x
2
3
x x + 4x
8
4a 2 x 2
li
Calculus 1 Worksheet 7
3 Part Definition of Continuity
Show (THREE STEPS) that each of the following functions is either continuous or
discontinuous at the given value of x.
1. f (x) = x + 5 at x = 1
x 2 16
at x = 4
x4
5. f (x) = [x] at x = 2
3. f (x) =
7
Calculus 1
Worksheet #10
Power Rule
_
f ( x ) = a ( ) f ' ( x ) = a ( )
Learn: The Power Rule:
(
1)
_
_
Example: f ( x) =
1 8
1
x f '( x) = 8 x81 = 4 x 7
2
2
_
Calculus 1 Worksheet #7
Limits approaching points from left or right: lim f ( x) or lim f ( x)
xa
1. Sketch a graph of the function f(x)
xa +
1
x < 1
x2 ,
1 x < 1
2,
3,
x =1
f(x) =
x + 1, 1 < x 2
1 ,
x>2
( x 2 )2
2. Using your graph from problem 1
Calculus 1 Worksheet #6
Limits approaching points from left or right: lim f ( x) or lim f ( x)
xa
1. Sketch a graph of the function f(x)
xa
1
x 1
x2 ,
1 x 1
2,
3,
x 1
f(x)
x 1, 1 x 2
1 ,
x2
x 2 2
2. Using your graph from problem 1, determine the
Calculus 1 Worksheet #5
Limits involving approaching infinity: lim f ( x)
x
TO INFINITY AND BEYOND !
1
lim = 0
Important theorem: x x
Limits Involving Infinity
(Principle of Dominance)
xa
, if a < b. Then, limit = 0. (Look for the highest degrees/powers
Calculus 1 Worksheet 6
3 Part Definition of Continuity
Show (THREE STEPS) that each of the following functions is either continuous or
discontinuous at the given value of x.
1. f (x) = x + 5 at x = 1
x 2 16
at x = 4
x4
5. f (x) = [x] at x = 2
3. f (x) =
7
Calculus 1 Worksheet #4
Limits involving trigonometric functions: lim
sin( )
x0
KNOW THE FOLLOWING THREE THEOREMS:
A. lim
sin
x0
=1
B. lim
x0
sin
=1
C. lim
1 cos
x0
=0
Examples:
sin 3x
sin 3 x 3
sin 3x
lim
lim 3
= 3
x 0
x 0
x 0
x
x
3
3x
1 cos 7 x
Calculus I
Worksheet 18
dy
for #1 12.
dx
2 x
1. y
3x 1
Find
3. y
2. y ln
ex
ex 1
4. y cos 2 x
eln x
x
5. y x ln x
6. y x(ln 3 x)
7. x t sin t and y 1 cos t
8. y 32 x .
9. y x 2 ln x.
x 3
11. y
2 5x
10. y 5 2 x
13. If y x 2 1 , then find the derivative