05&06-Applications of Integrals



Assessment Standards
6.2 Can compute the derivatives and integrals involving exponential functions.
6.4  Can compute the derivative and integrals involving log functions.

Applications of Integration
5.1 Can compute the area between curves
5.2 Can compute volumes of revolution using disks
5.5 Can compute the average value of a function

 

We're working through the un-starred units in chapter 6.

Chapter 6 - Logarithmic and Exponential Functions

 

Inverse Functions

·    The use of multiple representations (verbal, numeric, visual, algebraic) to understand inverse functions, always returning to the central idea of reversing inputs and outputs.

·    Use of implicit differentiation to find the derivative of an inverse function.

Functions relate inputs to outputs. What if we’re given the output and we need to find the input? Most functions have inverse functions, and we’ll learn how inverse functions are treated in calculus.

Inverse Functions

Non-Invertible Functions

Exponential Functions and Their Derivatives

·    Comparing relative magnitudes of functions, their rates of change (e.g. exponential vs. polynomial vs. logarithmic).

·    Algebraic and geometric properties of exponential functions.

·    Translation and reflection of exponential functions, from both symbolic and geometric perspectives.

·    Derivatives of exponential functions. The definition of e.

Exponential functions are everywhere in nature, from biology, to nuclear physics. We need to understand their behavior, and how to study the rate of change of exponential functions.

Exponential Functions

The number e

The Natural Exponential Function

Calculus of General Exponentials Functions

Logarithmic Functions

·    Comparing magnitudes of functions, their rates of change.

·    Logarithmic functions and their properties, including their geometric properties as inverses of exponentials.

·    Graphs of logarithmic functions, including asymptotic behavior.

The logarithm is the inverse of the exponential. What are logarithmic functions, and why are they useful?

Logarithmic Functions

Solving Exponential and Logarithmic Functions

Changing the Base of a Logarithmic Function

Derivatives of Logarithmic Functions

·    Derivatives and integrals of logarithmic functions.

We will discover the huge importance of the derivative of logarithm functions.

Natural Logarithmic Functions and their Derivatives

Calculus of General Logarithmic Functions

Chapter 5 - Applications of Integration

 

5.1 More About Areas

·    Appropriate integrals are used in a variety of applications to model physical, social, or economic situations.

·    Specific application should include finding the area of a region.

The area under a curve and between two curves can be interpreted in many powerful ways. We’ll learn how to apply this simple area concept to very different real world situations.

The Area Between Curves

KHW: Area Between Curves

5.2 Volumes via Cross-sections and Rings

·    Finding the volume of a solid with known cross-sections.

·    Volumes of solids of revolution.

We’ll learn how to find the volumes of complex shapes using calculus. These shapes can often times be hard to pictures, so we’ll build real-world models, and 3D computer models to better understand them.

Volumes of Solids of Revolution: Disks and Rings (12 Videos)

KHW: Disks and Rings

Volume of a Solid with Known Cross-Section

KHW: Cross Sections

5.3 Volumes via Cylindrical Shells

·    The method of cylindrical shells to compute volumes of revolution.

·    Comparisons between the shell and washer methods.

We’ll use the natural symmetries of rotation to find the volumes of even more solids.

Volumes of Solids of Revolution: Cylindrical Shells (7 videos)

KHW: Shells

5.5 The Average Value of a Function

·    Finding the average value of a function and the Mean Value Theorem.

·    A geometric interpretation of the MVT for integrals.

We know how to find the average rate of change of a function. What does it mean to find the average value of a function?

The Mean Value Theorem for Integrals (4 Videos)

KHW: Mean Value of Functions



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HomeworkProblems
5A  [u5.1: 1, 3, 5, 9, 11, 13, 17, 19, 27, 31, 35, 37, 46][u5.2: 1, 5, 7, 9, 11, 13, 15, 25, 29, 49] 
5B [u5.3: 1, 5, 7, 9, 13, 15, 17, 21, 37, 39, 45, 47] 
5C [u5.4: 1, 3, 7, 9, 11, 13, 15, 19, 21] [u5.5: 1, 7, 9, 11, 13, 17, 19] 
6A - Inverse Functions and Exponentials [u6.1: 1, 3, 5, 9, 13, 17, 19, 21, 23, 25, 31, 35, 39, 41] [u6.2: 1, 5, 11, 15, 25, 29, 33, 37, 43, 51, 67, 69, 79, 83, 87] 
6B - Logarithms and Derivatives of Logs [u6.3: 1, 5, 7, 11, 13, 15, 21, 25, 29, 35, 37, 39, 47, 49, 57, 59, 61, 65] [u6.4: 7, 9, 15, 19, 27, 33, 35, 47, 73, 75, 81] 
6C - Inverse Trig Functions [u6.6: 1, 3, 5, 7, 13, 15, 23, 29, 31, 37, 43, 45, 59, 61, 63] 
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Integrals Test Reflection.docx
(23k)
Nikhil Joshi,
Mar 31, 2015, 2:10 PM
v.4