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Lab 4: Calculating the length of a curve

1 Introduction

A function f of one variable x describes a curve in the xy-plane in the following way:

The curve consists of all points (x, y) whose coordinates satisfy the equation

y = f(x).

The purpose of this lab is to compute the length of this curve using the arclengthformula.You should read the course notes, especially chapter 5, to refresh your memory on themeaning of arclength and how it is calculated.

Recall that the length of the curve y = f(x) for a x b is given by the arclengthformula

(x) =

ba

d

dxdx

where

d

dx=

1 +

dy

dx

2

.

We sometimes refer to d as an element of arclength, and to ddx

as the rate of change ofarclength. Here we will investigate how the function f, the rate of change of arclength,and the total accumulated length depend on x in the given interval.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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2 Problems

2.1 Problem Set 1

You are told that the derivative of a certain function f is

f(x) = 1.7x sin(x2)

and that the point (0, 0) lies on the curve described by that function. Use the spread-sheet to create one page that contains the following graphs:

The graph y = f(x) of the function f (whose derivative is given to you). Thisshould be plotted over the interval from 0 to 2.5.

The graph of the derivative of the arclength function

d

dx(x) =

1 + f(x)2,

showing how it varies across the same interval.

The graph of the approximation to the arclength (x) of the curve from 0 to x, for

all values x between 0 and 2.5.You should use a very small step size to get good approximations to the exact solutions.If you see any sharp corners on your graph, then you must use a smaller step size. Makesure that the graph doesnt change considerably when you decrease the step size.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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2.1.1 Solution to Problem Set 1

Here is how we could organize the spreadsheet: We tabulate x values (in incrementsof x = 0.001 in column a, and the values of the given f at these x in column b. Wethen compute f(x) in column c by the linear approximation formula

f(x + x) = f(x) + f(x)x.

Thus, we start with f(0) = 0 in cell c0 and compute the other, approximate, values off along column c as we did in the last two labs. In column d we calculate

d

dx(x) =

1 + f(x)2

and we add up these values in column e to get the total length (x) as it accumulates

along the curve.The resulting plot is shown in the figure below.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5

Figure 1 The results for lab 4 problem set 1. The graph y = f(x) isshown in red, the derivative of the arc length in green and the total arclength in blue

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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2.2 Problem Set 2

You are told that the derivative of a certain function f is

f(x) = 2.1x sin(x2)

and that the point (0, 0) lies on the curve described by that function. Use the spread-

sheet to create one page that contains the following graphs:

The graph y = f(x) of the function f (whose derivative is given to you). Thisshould be plotted over the interval from 0 to 2.5.

The graph of the derivative of the arclength function

d

dx

(x) = 1 + f(x)2,showing how it varies across the same interval.

The graph of the approximation to the arclength (x) of the curve from 0 to x, forall values x between 0 and 2.5.

You should use a very small step size to get good approximations to the exact solutions.

If you see any sharp corners on your graph, then you must use a smaller step size. Makesure that the graph doesnt change considerably when you decrease the step size.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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Next Page

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2.2.1 Solution to Problem Set 2

Here is how we could organize the spreadsheet: We tabulate x values (in incrementsof x = 0.001 in column a, and the values of the given f at these x in column b. Wethen compute f(x) in column c by the linear approximation formula

f(x + x) = f(x) + f(x)x.

Thus, we start with f(0) = 0 in cell c0 and compute the other, approximate, values off along column c as we did in the last two labs. In column d we calculate

d

dx(x) =

1 + f(x)2

and we add up these values in column e to get the total length (x) as it accumulates

along the curve.The resulting plot is shown in the figure below.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

Previous Page

Next Page

Exit ]

http://close/http://close/

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0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5

Figure 2 The results for lab 4 problem set 2. The graph y = f(x) isshown in red, the derivative of the arc length in green and the total arclength in blue

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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2.3 Problem Set 3

You are told that the derivative of a certain function f is

f(x) = 1.3x sin(x2)

and that the point (0, 0) lies on the curve described by that function. Use the spread-

sheet to create one page that contains the following graphs:

The graph y = f(x) of the function f (whose derivative is given to you). Thisshould be plotted over the interval from 0 to 3.5.

The graph of the derivative of the arclength function

d

dx

(x) = 1 + f(x)2,showing how it varies across the same interval.

The graph of the approximation to the arclength (x) of the curve from 0 to x, forall values x between 0 and 3.5.

You should use a very small step size to get good approximations to the exact solutions.If you see any sharp corners on your graph, then you must use a smaller step size. Makesure that the graph doesnt change considerably when you decrease the step size.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

Previous Page

Next Page

Exit ]

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2.3.1 Solution to Problem Set 3

Here is how we could organize the spreadsheet: We tabulate x values (in incrementsof x = 0.001 in column a, and the values of the given f at these x in column b. Wethen compute f(x) in column c by the linear approximation formula

f(x + x) = f(x) + f(x)x.

Thus, we start with f(0) = 0 in cell c0 and compute the other, approximate, values off along column c as we did in the last two labs. In column d we calculate

d

dx(x) =

1 + f(x)2

and we add up these values in column e to get the total length (x) as it accumulates

along the curve.The resulting plot is shown in the figure below.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

Previous Page

Next Page

Exit ]

http://close/http://close/

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0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5

Figure 3 The results for lab 4 problem set 3. The graph y = f(x) isshown in red, the derivative of the arc length in green and the total arclength in blue

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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2.4 Problem Set 4

You are told that the derivative of a certain function f is

f(x) = 1.8x cos(x2)

and that the point (0, 0) lies on the curve described by that function. Use the spread-

sheet to create one page that contains the following graphs:

The graph y = f(x) of the function f (whose derivative is given to you). Thisshould be plotted over the interval from 0 to 2.

The graph of the derivative of the arclength function

d

dx

(x) = 1 + f(x)2,showing how it varies across the same interval.

The graph of the approximation to the arclength (x) of the curve from 0 to x, forall values x between 0 and 2.

You should use a very small step size to get good approximations to the exact solutions.If you see any sharp corners on your graph, then you must use a smaller step size. Makesure that the graph doesnt change considerably when you decrease the step size.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

Previous Page

Next Page

Exit ]

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2.4.1 Solution to Problem Set 4

Here is how we could organize the spreadsheet: We tabulate x values (in incrementsof x = 0.001 in column a, and the values of the given f at these x in column b. Wethen compute f(x) in column c by the linear approximation formula

f(x + x) = f(x) + f(x)x.

Thus, we start with f(0) = 0 in cell c0 and compute the other, approximate, values off along column c as we did in the last two labs. In column d we calculate

d

dx(x) =

1 + f(x)2

and we add up these values in column e to get the total length (x) as it accumulates

along the curve.The resulting plot is shown in the figure below.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

Previous Page

Next Page

Exit ]

http://close/http://close/

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-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2

Figure 4 The results for lab 4 problem set 4. The graph y = f(x) isshown in red, the derivative of the arc length in green and the total arclength in blue

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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2.5 Problem Set 5

You are told that the derivative of a certain function f is

f(x) = 1.9x cos(x2)

and that the point (0, 0) lies on the curve described by that function. Use the spread-

sheet to create one page that contains the following graphs:

The graph y = f(x) of the function f (whose derivative is given to you). Thisshould be plotted over the interval from 0 to 3.

The graph of the derivative of the arclength function

d

dx

(x) = 1 + f(x)2,showing how it varies across the same interval.

The graph of the approximation to the arclength (x) of the curve from 0 to x, forall values x between 0 and 3.

You should use a very small step size to get good approximations to the exact solutions.If you see any sharp corners on your graph, then you must use a smaller step size. Makesure that the graph doesnt change considerably when you decrease the step size.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

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Next Page

Exit ]

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2.5.1 Solution to Problem Set 5

Here is how we could organize the spreadsheet: We tabulate x values (in incrementsof x = 0.001 in column a, and the values of the given f at these x in column b. Wethen compute f(x) in column c by the linear approximation formula

f(x + x) = f(x) + f(x)x.

Thus, we start with f(0) = 0 in cell c0 and compute the other, approximate, values off along column c as we did in the last two labs. In column d we calculate

d

dx(x) =

1 + f(x)2

and we add up these values in column e to get the total length (x) as it accumulates

along the curve.The resulting plot is shown in the figure below.

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

Previous Page

Next Page

Exit ]

7 I t d ti

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-1

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3

Figure 5 The results for lab 4 problem set 5. The graph y = f(x) isshown in red, the derivative of the arc length in green and the total arclength in blue

[

Introduction

ProblemSet 1

ProblemSet 2

ProblemSet 3

ProblemSet 4

ProblemSet 5

Previous Page

Next Page

Exit ]

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