a preview of calculus lesson 1.1. what is calculus it is the mathematics of change it is the...
TRANSCRIPT
![Page 1: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/1.jpg)
A Preview of Calculus
Lesson 1.1
![Page 2: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/2.jpg)
What Is Calculus
• It is the mathematics of change
• It is the mathematics of – tangent lines– slopes– areas– volumes
• It enables us to model real life situations
• It is dynamic– In contrast to algebra/precalc which is static
![Page 3: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/3.jpg)
What Is Calculus
• One answer is to say it is a "limit machine"
• Involves three stages1. Precalculus/algebra mathematics process
• Building blocks to produce calculus techniques
2. Limit process• The stepping stone to calculus
3. Calculus• Derivatives, integrals
![Page 4: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/4.jpg)
Contrasting Algebra & Calculus
• Use f(x) to find the height of the curve at x = c
• Find the limit of f(x) as x approaches c
![Page 5: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/5.jpg)
Contrasting Algebra & Calculus
• Find the average rate of change between t = a and t = b
• Find the instantaneous rate of change at t = c
![Page 6: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/6.jpg)
Contrasting Algebra & Calculus
• Area of a rectangle • Area between two curves
![Page 7: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/7.jpg)
A Preview of Calculus (cont’d)
![Page 8: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/8.jpg)
How do we get the area under the curve?
![Page 9: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/9.jpg)
TANGENT LINE
![Page 10: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/10.jpg)
Tangent Line Problem
• Approximate slope of tangent to a line– Start with slope of secant line
![Page 11: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/11.jpg)
The Tangent Line Problem
![Page 12: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/12.jpg)
Tangent Line Problem
• Now allow the Δx to get smaller
tangent0
( ) ( )limx
f c x f cm
x
tangent
0
( ) ( )limx
f c x f cm
x
![Page 13: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/13.jpg)
The Area Problem
![Page 14: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/14.jpg)
![Page 15: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/15.jpg)
The Area Problem
• We seek the area under a curve, the graph f(x)
• We approximatethat area witha number ofrectangles
• Sum = 31.9• Actual = 33.33
![Page 16: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/16.jpg)
The Area Problem
• The approximation is improved by increasing the number of rectangles
• Number ofrectangles = 10
• Sum = 32.92
• Actual = 33.33
![Page 17: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/17.jpg)
The Area Problem
• The approximation is improved by increasing the number of rectangles
• Number ofrectangles = 25
• Sum = 33.19
• Actual = 33.33
![Page 18: A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It](https://reader036.vdocuments.mx/reader036/viewer/2022081520/5697c0121a28abf838ccc46c/html5/thumbnails/18.jpg)
• In other words!!!! As we increase the number of rectangles, we get closer and
closer to the actual area of under the curve!!
• Or we could say “as the limit of the number of rectangles approaches
infinity”!!!! “we get closer and closer to the actual area under the curve!!!!