009A Sample Final A, Problem 9 - Revision history

MathAdmin at 03:41, 28 May 2015

2015-05-28T03:41:19Z

MathAdmin at 03:41, 28 May 2015

2015-05-28T03:41:02Z

MathAdmin at 13:44, 13 April 2015

2015-04-13T13:44:23Z

MathAdmin at 17:37, 29 March 2015

2015-03-29T17:37:34Z

MathAdmin at 17:36, 29 March 2015

2015-03-29T17:36:55Z

MathAdmin at 02:51, 28 March 2015

2015-03-28T02:51:47Z

MathAdmin at 02:50, 28 March 2015

2015-03-28T02:50:45Z

MathAdmin at 02:49, 28 March 2015

2015-03-28T02:49:06Z

MathAdmin at 02:46, 28 March 2015

2015-03-28T02:46:24Z

MathAdmin at 05:00, 27 March 2015

2015-03-27T05:00:22Z

@@ Line 34: / Line 34: @@
 !Step 3: &nbsp;
 |-
-|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: 0%;">30</math>, so we should really write <math style="vertical-align: -23%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
+|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: 0%;">30</math>, so we should really write <math style="vertical-align: -22%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
 |-
 |&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{dz}{dt} = \frac {3}{5}\cdot(-30) = -18.</math>

@@ Line 34: / Line 34: @@
 !Step 3: &nbsp;
 |-
-|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: 0%;">30</math>, so we should really write <math style="vertical-align: -21%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
+|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: 0%;">30</math>, so we should really write <math style="vertical-align: -23%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
 |-
 |&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{dz}{dt} = \frac {3}{5}\cdot(-30) = -18.</math>

@@ Line 16: / Line 16: @@
 !Step 1: &nbsp;
 |-
-|'''Write the Basic Equation:''' From the picture, we can see there is a triangle involving both the bug and the point <math style="vertical-align: -21%;">(3,4)</math>.  From this, we can see that  <math style="vertical-align: -8%;">z^2=x^2+4^2</math>.
+|'''Write the Basic Equation:''' From the picture, we can see there is a right triangle involving both the bug and the point <math style="vertical-align: -21%;">(3,4)</math>.  From this, we can see that  <math style="vertical-align: -8%;">z^2=x^2+4^2</math>.
 |}

@@ Line 34: / Line 34: @@
 !Step 3: &nbsp;
 |-
-|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: -3%;">30</math>, so we should really write <math style="vertical-align: -21%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
+|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: 0%;">30</math>, so we should really write <math style="vertical-align: -21%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
 |-
 |&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{dz}{dt} = \frac {3}{5}\cdot(-30) = -18.</math>

@@ Line 34: / Line 34: @@
 !Step 3: &nbsp;
 |-
-|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: -1%;">30</math>, so we should really write <math style="vertical-align: -21%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
+|'''Evaluate and Solve:''' When the bug is at the origin, we have <math style="vertical-align: 1%;">x=3</math>.  By the Pythagorean Theorem, <math style="vertical-align: 0%;">z=5</math>.  Based on our drawing, our variable <math style="vertical-align: 0%;">x</math> is actually <u>''decreasing''</u> at a rate of <math style="vertical-align: -3%;">30</math>, so we should really write <math style="vertical-align: -21%;">dx/dt=-30</math>. We now simply plug in to the result of our implicit differentiation to find
 |-
 |&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{dz}{dt} = \frac {3}{5}\cdot(-30) = -18.</math>

@@ Line 9: / Line 9: @@
 ! Foundations: &nbsp;
 |-
-|Like most geometric word problems, you should start with a picture.  This will help you declare variables and write meaningful equation(s). In this case, we will have to use implicit differentiation to arrive at our related rate. In particular, we need to choose variables to describe the distance between the bug and the point <math style="vertical-align: -23%;">(3,4)</math>, which we can call <math style="vertical-align: 0%;">z</math>.  By the given information, we can consider the position on the <math style="vertical-align: 0%;">x</math>-axis simply as <math style="vertical-align: 0%;">x</math>.
+|Like most geometric word problems, you should start with a picture.  This will help you declare variables and write meaningful equation(s). In this case, we will have to use implicit differentiation to arrive at our related rate. In particular, we need to choose variables to describe the distance between the bug and the point <math style="vertical-align: -21%;">(3,4)</math>, which we can call <math style="vertical-align: 0%;">z</math>.  By the given information, we can consider the position on the <math style="vertical-align: 0%;">x</math>-axis simply as <math style="vertical-align: 0%;">x</math>.
 |}
 &nbsp;'''Solution:'''
@@ Line 16: / Line 16: @@
 !Step 1: &nbsp;
 |-
-|'''Write the Basic Equation:''' From the picture, we can see there is a triangle involving both the bug and the point <math style="vertical-align: -23%;">(3,4)</math>.  From this, we can see that  <math style="vertical-align: -8%;">z^2=x^2+4^2</math>.
+|'''Write the Basic Equation:''' From the picture, we can see there is a triangle involving both the bug and the point <math style="vertical-align: -21%;">(3,4)</math>.  From this, we can see that  <math style="vertical-align: -8%;">z^2=x^2+4^2</math>.
 |}