Section 1.3 Rates of Change and Tangents
A.
Average Rate of Change
The average rate of change gives a measure of how much one quantity changes with respect to another
Familiar calculations: miles per gallon, miles per hour, cost per kilowatt
Compute the a
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1 1 .7 St ra t eg y f o r Test ing Series
Module Goal:The student will review the application of certain tests to determine the convergence
properties of series.
M
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1 1 .8 Po wer Series
Module Goal:The student will obtain the ability to analyze power series and determine the radius and
interval of convergence.
Module Outline:
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Sect io n 6 .1 Inv erse Funct io ns
Problem: Throughout high school physics class, you learned how to find the temperature in degrees
Celsius from a given temperat
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Sect io n 6 .2 Ex po nent ia l Funct io ns a nd t heir Deriv a t iv es
Problem: Instead of taking the derivative of the function
, how about the derivative of
? Fu
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Sect io n 6 .3 Lo g a rit hmic Funct io ns
Problem: All functions have an inverse function. What about
? Such inverse functions are
called logarithmic functions.
M
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Sect io n 6 .4 Deriv a t iv es o f Lo g a rit hmic Funct io ns
Module Goal:The student will obtain the ability to work with derivatives and integrals of logarithmi
Name
Math 1920-R5X
Date
Outline for Section 7.1, Inverse Functions
I. Inverse Functions
A. Representation of functions
1. Table
2. Graph
3. Mathematical expression
B. Definition: A function has an inverse over its domain if it is one-to-one,
or equally wr
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6 .5 Ex po nent ia l G ro wt h a nd Deca y
Module Goal: The student will obtain the ability to solve simple first order differential equations involving
exponentia
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Sect io n 6 .6 Inv erse Trig no met ric Funct io ns
Module Goal: The student will obtain the ability to work with inverse trigonometric functions, their domain
of
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1 1 .6 Abso lut e Co nv erg ence a nd t he Ra t io a nd Ro o t Test s
Module Goal:The student will obtain the ability to apply the Ratio test and the Root test (wi
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1 1 .5 Alt erna t ing Series
Module Goal:The student will obtain the ability to apply the Alternating series test to determine the
convergence properties of series
Section 2.1 Tangents and Derivatives at a Point
A. Slope of Tangent Line
Previously, we approximated the slope of the tangent line by taking the limit of the slopes of the secant lines
More formally, the slope of the line tangent to the curve
at the poi
Section 2.2 Derivative as a Function
A. Derivative at a specific point
In the previous section we computed the slope of the tangent line and the derivative of
as follows:
at the point
The value that we obtain from this calculation represents the derivati
Section 1.6 One-Sided Limits
A. Limit Notation
Previously, we discussed the meaning and notation for the limit of a function.
()
This notation is read as the limit of ( ) as approaches is equal to
This means that as gets CLOSE to the value , on both the
Section 1.7 Continuity
A. Understanding Continuity
Intuitively, we think of something as being continuous if it keeps going without any breaks or
interruptions
Similarly, we can think of a continuous function as one whose graph we can trace from end-to-
Section 1.4 Limit of a Function and Limit Laws
A. Limit Notation
The idea of a limit was introduced in the previous section to approximate the slope of the tangent line
Denote the limit as follows:
lim ( ) =
This notation is read as the limit of ( ) as
Section 1.8 Limits Involving Infinity
A. Infinite Limits
We say that a limit is infinite if () as . That is, the values of ( ) become larger and
larger as gets closer and closer to the value .
Example: Consider the function ( ) = 1/ 2 . Use the graph and
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1 1 .2 Series
Module Goal:The student will obtain the ability to analyze series to include establishing the value of the
limit if convergent.
Module Outline:If we
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1 1 .3 The Int eg ra l Test a nd Est ima t es o f Sums
Module Goal:The student will obtain the ability to apply the Integral test (also used to estimate the limit
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1 1 .4 The Co mpa riso n Test
Module Goal:The student will obtain the ability to apply the Comparison test to determine the convergence
properties of series.
Modul
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Sect io n 6 .7 H y perbo lic Funct io ns
Module Goal: The student will obtain the ability to work with hyperbolic functions and be able to perform
known calculus o
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Sect io n 6 .8 Indet ermina t e Fo rms a nd L'H o spit a l's Rule
Module Goal: The student will obtain the ability to evaluate limits of expressions that are class
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Cha pt er 7 Int ro duct io n
The last chapter discussed the properties of inverse functions. An inverse function maps elements from a
function's range `back" to it
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Cha pt ers 9 a nd 1 0 Int ro duct io n
Differential equations are very important and useful when modelling real world phenomena. Often it is not
possible to calcul
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9 .1 Mo deling wit h Dif f erent ia l Equa t io ns
Module Goal:The student will obtain the ability to solve simple first order differential equations.
Module Outli
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1 0 .1 Curv es Def ined by Pa ra met ric Equa t io ns
Module Goal:The student will obtain the ability to use techniques of parametric equations to transform
curves
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1 0 .2 Ca lculus wit h Pa ra met ric Curv es
Module Goal:The student will obtain the ability to apply calculus to parametric and polar representations of
curves to
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1 0 .3 Po la r Co o rdina t es
Module Goal:The student will obtain the ability to use techniques of polar coordinates to transform curves
(that do not satisfy the
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1 0 .4 Area s a nd Leng t hs in Po la r Co o rdina t es
Module Goal:The student will obtain the ability to apply calculus to polar representations of curves to
det