Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.
-
Upload
milton-blankenship -
Category
Documents
-
view
238 -
download
0
description
Transcript of Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.
![Page 1: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/1.jpg)
![Page 2: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/2.jpg)
Chapter 5:Applications of the
Derivative
Chapter 4:Derivatives
Chapter 5:Applications
![Page 3: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/3.jpg)
Objectives: To be able to use the derivative to analyze function Draw the graph of the function based on the analysis Apply the principles learned to problem situations
![Page 4: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/4.jpg)
Example 1: Find the equation of the tangent
line to the parabola, y = x1/2 at a. (0,0) b. (1,1).
![Page 5: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/5.jpg)
Example 2: Find the point on the parabola
y = x2 – 2x + 1 where the tangent line is horizontal.
![Page 6: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/6.jpg)
Example 3: Locate the point where the
tangent line is a. horizontal b. Vertical.
x3 + y3 = 6xy
![Page 7: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/7.jpg)
Example 4: Find the equation of the tangent
line to the ellipse at the end of the latus rectum found in the first quadrant. Equation of ellipse is x2/16 + y2/25 = 1
![Page 8: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/8.jpg)
Example 5: Find the equation of the tangent
line at t = 0. x = t2 + t y = t2 - t
![Page 9: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/9.jpg)
Example 6: Find the equation of the tangent
line to y = 1 + x – 2x3 at x = 1.
![Page 10: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/10.jpg)
Example 7: Find the equation of the tangent
line to y = x / (2x – 1) at x = 1.
![Page 11: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/11.jpg)
Example 8: Find the equation of the tangent
line to y = 2 /(3-x)1/2 at x = - 1.
![Page 12: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/12.jpg)
Example 9: Find the equation of the tangent
line to y = x / (x2-3) at x = 2.
![Page 13: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/13.jpg)
Example 10: Find the equation of the
horizontal tangent line of x = t(t2 – 3) y = 3(t2 – 3)
![Page 14: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/14.jpg)
Example 11: 2(x2+y2)2 = 25(x2 –y2) is an
equation of a curve called the lemniscate.
(a) find the equation of its tangent line at (3, 1).
(b) Locate the points where the tangent line is horizontal.
![Page 15: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/15.jpg)
Example 12: Find the equation of the tangent
line at the given point. x2/16 - y2/9 = 1 at ( -
5 , 9/4 )
![Page 16: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/16.jpg)
Example 13: Find the equation of the tangent
line at the given point.y2 = 5x4 – x2 at ( 1 , 2
)
![Page 17: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/17.jpg)
Example 14: Find the equation of both lines
through ( 2, - 3) that are both tangent to the parabola y = x2 + x.
![Page 18: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/18.jpg)
Example 15: Where does the normal line to the
parabola y = x – x2 at ( 1 , 0 ) intersect the curve a second time?
![Page 19: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/19.jpg)
Example 16: Find the cubic function y = ax3 +
bx2 + cx + d where its graph has horizontal lines ( -2 , 6) and ( 2, 0).
![Page 20: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/20.jpg)
Example 17: The vertex of a parabola is the
point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of y2 = -2x + 8
![Page 21: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/21.jpg)
Example 18: The vertex of a parabola is the
point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of y = x2 + 8x – 5.
![Page 22: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/22.jpg)
Example 19: The vertex of a parabola is the
point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of x2
– 4x + y = 0.
![Page 23: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/23.jpg)
Example 20: Locate the points where the
tangent line is horizontal. y = x3 – x2 – x + 1
![Page 24: Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications.](https://reader035.fdocuments.us/reader035/viewer/2022081421/5a4d1b6f7f8b9ab0599b4bc6/html5/thumbnails/24.jpg)
Example 20: Locate the points where the
tangent line is horizontal. y = 2x3 – 3x2 – 6x + 37