chapter 3: applications of the derivative and curve...
TRANSCRIPT
![Page 1: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/1.jpg)
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How have you sketched graphs in the past?
Polynomials: Zeros (xints) & yint; Relative/Absolute Extrema (min/max)
Rationals: Asymptotes, intercepts, holes.
Chapter 3: Applications of the Derivative and Curve Sketching
3.1 Increasing and Decreasing Functions
![Page 2: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/2.jpg)
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You will still be responsible for the following (We will discuss min/max by hand)
Zeros: set y=0; Factor & solve or Calculator
yint: set x=0 (evaluate)
Vert Asym: Denominator = 0 and Numerator ≠ 0
Horiz. Asym.
Holes: Numerator and Den equal Zero (typically).
![Page 3: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/3.jpg)
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The first thing we will add to this list is describing how the curve is behaving over certain intervals (a, b); We will describe whether the curve is increasing, decreasing, or neither.
![Page 4: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/4.jpg)
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Increasing Interval:
A function is said to be increasing on the interval (a, b) if the y value of any point is higher than that of the point to the left of it. In calc...
A function is increasing iff...
f(x1) < f(x2) for all x1 < x2
on the interval.
From this we can get a similarly worded definition of decreasing intervals.
How will we determine if a graph is increasing or decreasing? Will we plot a plethora of points and examine yvalues? Not accurate enough!
What describes how a function moves?
Derivatives!!!
![Page 5: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/5.jpg)
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Derivative: Describes slope Slope is positive → Increasing Slope is negative → Decreasing Slope is zero → Constant (Horiz)
If we have the derivative of a function, we can describe how the function moves/changes.
How do I find where we change from inc to dec?
How do we find the various intervals?
,
Our intervals are formed by places where the derivative either equals zero, or is undefined. These places are called critical values.
Once we find critical values, we test the derivative in each interval (positive or negative) to determine if the original function is increasing or decreasing.
This is the FirstDerivativeTest
![Page 6: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/6.jpg)
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IF...
1. x=c is a critical value from the first derivative
2. f(c) is defined (it exists. no asymptote, no hole.)
3. FirstDerTest yields sign change over x=c
THEN...
x=c represents a minimum or a maximum of f(x)
![Page 7: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/7.jpg)
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Describe the inc/dec behavior of the following functions.
Describe / classify critical values.
Use chart.
chart:
When is a critical value a minimum or a maximum?
![Page 8: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/8.jpg)
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chart:
![Page 9: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/9.jpg)
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There are times when an undefined derivative does NOT necessarily imply an ASYM.
Ex]
Domain:
Zeros:
yint:
We find the locations of asymptotes from the original function!!
If a function is undefined at c, the derivative will be undefined at c.
If a function is defined at c, the derivative can be defined or undefined.
![Page 10: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/10.jpg)
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Determine if y = (x+4)3 is increasing everywhere.
Domain:
Zeros:
yint:
![Page 11: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/11.jpg)
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Pg 184 Ex 3
Find & classify critical points. Describe Increasing/Decreasing intervals.
Domain:
Zeros:
yint:
![Page 12: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/12.jpg)
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Ex]
Domain:
Zeros:
yint:
![Page 13: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/13.jpg)
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Pg 175: 533 EOO
![Page 14: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/14.jpg)
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![Page 15: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/15.jpg)
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![Page 16: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/16.jpg)
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Finding Absolute Extrema on Closed Intervals [a,b]
1. Find critical values on [a, b]
2. Evaluate critical values, a, and b, in function.
3. Find highest / lowest values.
3.2 (Part of it) Extrema and the FirstDerivative Test
![Page 17: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/17.jpg)
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Ex] Find absolute extrema on [3,5] for
y = t 2t
![Page 18: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/18.jpg)
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![Page 19: Chapter 3: Applications of the Derivative and Curve Sketchingdc-pope.weebly.com/uploads/3/7/7/5/37756673/3.1... · February 13, 2017 How have you sketched graphs in the past? Polynomials:](https://reader035.vdocuments.us/reader035/viewer/2022062602/5edc7a3dad6a402d66672688/html5/thumbnails/19.jpg)
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Pg 175: 533 EOO
Pg 184: 19, 21, 23