Chapter 6: The Denite Integral
Lesson 2: Slope Fields and Differential Equations
Slope Fields:
Slope fields provide an excellent way to visualize a family of solutions of
differential equations. Some differential equations can be solved algebraically,
and
Chagter 6: The Denite Integral
Lesson 6: The Second Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus iPart ill:
anaemic)
This says that everyr continuous function f is the derivative of some other function.
It says that every continuo
Chapter 5: Antiderivatives
Lesson 2: The Antiderivatives of Trigonometric Functions
Pythagorean Identities:
sin 2 cos 2 1
1 cot 2 csc2
tan 2 1 sec 2
Reciprocal and Quotient Identities:
sec
1
cos
csc
tan
1
sin
sin
cos
cot
cot
1
tan
cos
sin
Doub
Chagter 6: The Definite Integral
Lesson 4: The Trapezoidal Rule for Approximating Definite integrals
Last day we approximated the a definite integral (area under the curve) by using
Left, Right, and Midpoint Riemann Sums using RECTANGLES.
Todayr we are
Chagter 6: The Denite Integral
Lesson 1: Introduction to Differential Equations
Basic Differential Eguations:
Any equation involving a derivative is called a Differential Equation. Differentiai
equations arise in chemistry, rihirsicsr mathematics and all
Chapter 4: Applications of Derivatives
Lesson 7: Related Rates Part |
Rate of Change:
The rate at which a variable changes with respect to time (E '1' t
Jr. d! "
.).
Related Rate Eguation:
Any equation involving two or more variables that are diff
Chapter 5: Antiderivatives
Lesson 3: Integrating Using the Chain Rule
Example 1: Differentiate the following. What pattern do you notice about
integration with the chain rule?
a)
d
20
( 5 x +3) =
dx
b)
20
d 2
( x +3) =
dx
c)
d
20
( sin x +3) =
dx
Note: To
Cha ter 4: A lications of Derivatives
Lesson 6: Li nearization
Linear Aggroximation: Using the tangent line to approximate values of the function
close to the point of tangency.
Examgle 1: Let f{x)=x3. xv
a) Find the line tangent to the graph of f(x
Chapter 4: Applications of Derivatives
Lesson 4: Connecting f(x) and f(x) with the Graph of f(x)
Example 1: Draw a possible graph, given each set of conditions
a) f(2)=4, f(2)<O, f(2)<O
7
deccci
b) f(2)=4, f(2)>O, f(2)>O
t
c) f(2)=4, f(2)=O, f(2)<O
1
d
f(
Chapter 5: Antiderivatives
Lesson 4: More Integration Formulas
Nifty Little Trick for Substitution Problems:
Example 1: Use substitution to integrate the following.
Example 2: Integrate the following:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
Chapter 4: Applications of Derivatives
Lesson 8: Related Rates Part II
Sometimes more than two quantities are changing within a certain relationship. At
this point, you must consider the product rule the quotient rule, and the chain rule
when differentiat
Chapter 4: Applications of Derivatives
Lesson 5: Modeling and Optimization
Method for Solving Extreme Value Problems:
Step 1.
Draw an appropriate figure and label the quantities relevant to the
problem.
Step 2.
Find a formula for the quantity to be maximi
Chapter 5: Antiderivatives
Lesson 1: The Antiderivative
Example 1: Find f ( x ) given f ' ( x ) (Find the antiderivative):
a) f ' ( x ) =2 x
2
b) f ' ( x ) =3 x
8
c) f ' ( x ) =9 x
d) f ' ( x ) =3
e) f ' ( x ) = x
2
f) f ' ( x ) = x
4
g) f ' ( x ) =7 x
h)
Chagter 2: Limits and Continuity
Lesson 2: Limits of Functions (Algebraic-ally]
The Progerties of Limits
if L, M. c, and k are real numbers and
mnjlx)=£landling[x)=4,then
rN' 1'
. Sum Rule: lim(j{.r)+g[x]) : L+ M
' lim (ab 4-
1 Difference Rule: g[.r] LM
Cheater 2: Limits and Continuig
Lesson 1: Limits
Formal Denition of a limit:
Let c and L be real numbers. The mention 1' has iimit i. as it approaches a if,
given any positive number a, there is a positive number 6 such that for ail x
U<lx-c|-:§=>|f[x)L
Chapter 2: Limits and Continuity
Lesson 3: Limits Involving In nity
Examgle 1: Graph y=l then determine the following limits:
x
Finding Horizontal nematotes:
Note: i-.lo.i'izontaii asymptotes and limits at innity go hand in hand. Determining
the HM? a
Chagter 2: Limits and Continuity
Lesson 4: Continuity
Continuous Functions:
A function is said to be continuous if the function has no gaps over its entire domain.
A continuous function can be drawn without lifting your pencil off the paper,
f
.1
ii]
I
No
Chapter 7: Agglications of Integration
Lesson 3: Rotating about lines Parallel to the axes
Examgle 1: The region bounded by the curves y =e , y=e, _1'=C- is rotated
about the xaxis to generate a solid. Find the volume of the solid Y
_ -X
R: Y4
x1 7'
="' (
Chagter 3: Derivatives
Lesson 7: Implicit Differentiation
Aii the differentiation prohiems we ve iooired at so far are functions iike
y = x: + 5x or _'i-' : sin .1.- . in such cases, y is written expiicirty as a function of X.
it is not aiways convenien