Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 3.5, Problem 50E

To determine

To graph: The function $f(h)=sin(h)h$ and estimate $limh→0f(h)$ and then to compare the estimate with $π180$ and then to state if there is a reason to believe that limit should be $π180$ .

Expert Solution

Answer to Problem 50E

Estimate is equal to $π180$ .

Explanation of Solution

Given Information: $limh→0sin(h)h$ is to be estimated.

Graph:

Calculus: Graphical, Numerical, Algebraic, Chapter 3.5, Problem 50E , additional homework tip 1

Observing the graph, it can inferred that the limit is near about $0.0175$ , comparing with $π180$ , it is found that the estimate is equal to $π180$ . It is so because $π180$ is the conversion factor for converting degrees to radians.

To determine

To estimate: $limh→0cos(h)−1h$ using graphing utility in degree mode.

Expert Solution

Answer to Problem 50E

$limh→0cos(h)−1h=0$

Explanation of Solution

Given Information: $limh→0cos(h)−1h$ is to be estimated.

Graph:

Calculus: Graphical, Numerical, Algebraic, Chapter 3.5, Problem 50E , additional homework tip 2

Observing the graph it can be easily estimated that $limh→0cos(h)−1h=0$ .

To determine

The formula obtained for the derivative of $sinx$ .

Expert Solution

Answer to Problem 50E

The formula is $ddx(sinx)=π180(cosx)$ .

Explanation of Solution

Given Information: Degree mode limits are to be used.

Calculation:

$ddx(sinx)=limh→0sin(x+h)−sinxh =limh→0sinxcos h+cos xsin h−sin xh =limh→0sinx(cos h−1)+cosxsin hh =(limh→0sinx)(limh→0cos h−1h)+(limh→0cos x)(limh→0sin hh) =(sinx)(0)+(cosx)(π180) =π180(cosx)$

Thus, the formula is $ddx(sinx)=π180(cosx)$ .

To determine

The formula obtained for the derivative of $cosx$ .

Expert Solution

Answer to Problem 50E

The formula is $ddx(cosx)=−π180(sinx)$ .

Explanation of Solution

Given Information: Degree mode limits are to be used.

Calculation:

$ddx(cosx)=limh→0cos(x+h)−cosxh =limh→0cosxcos h−sin xsin h−cos xh =limh→0cosx(cos h−1)−sinxsin hh =(limh→0cosx)(limh→0cos h−1h)−(limh→0sin x)(limh→0sin hh) =(cosx)(0)−(sinx)(π180) =−π180(sinx)$

Thus, the formula is $ddx(cosx)=−π180(sinx)$ .

To determine

To calculate: The second and third degree-mode derivatives of $sinx$ and $cosx$ .

Expert Solution

Answer to Problem 50E

The derivatives are $d2dx2sinx=−( π180)2sinx, d3dx3sinx =− π31803cosx$ and $d2dx2cosx=−( π180)2cosx, d3dx3cosx=π31803sinx$ .

Explanation of Solution

Given Information: Degree mode derivatives are to be calculated.

Calculation:

$d2dx2sinx=ddx(π180cosx) =π180(−π180sinx) =−( π180)2sinxd3dx3sinx=ddx[−( π180)2sinx] =− π21802(π180cosx) =− π31803cosx$

$d2dx2cosx=ddx(−π180sinx) =−π180(π180cosx) =−( π180)2cosx$

$d3dx3cosx=ddx[(− π180)2cosx] =− π21802(−π180sinx) =π31803sinx$

Thus, the derivatives are $d2dx2sinx=−( π180)2sinx, d3dx3sinx =− π31803cosx$ and $d2dx2cosx=−( π180)2cosx, d3dx3cosx=π31803sinx$ .

Chapter 3.5, Problem 49E

Chapter 3.5, Problem 51E

Chapter 3 Solutions

Calculus: Graphical, Numerical, Algebraic

Ch. 3.1 - Prob. 1QR Ch. 3.1 - Prob. 2QR Ch. 3.1 - Prob. 3QR Ch. 3.1 - Prob. 4QR Ch. 3.1 - Prob. 5QR Ch. 3.1 - Prob. 6QR Ch. 3.1 - Prob. 7QR Ch. 3.1 - Prob. 8QR Ch. 3.1 - Prob. 9QR Ch. 3.1 - Prob. 10QR

Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Prob. 20E Ch. 3.1 - Prob. 21E Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Prob. 24E Ch. 3.1 - Prob. 25E Ch. 3.1 - Prob. 26E Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Prob. 36E Ch. 3.1 - Prob. 37E Ch. 3.1 - Prob. 38E Ch. 3.1 - Prob. 39E Ch. 3.1 - Prob. 40E Ch. 3.1 - Prob. 41E Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Prob. 44E Ch. 3.1 - Prob. 45E Ch. 3.2 - Prob. 1QR Ch. 3.2 - Prob. 2QR Ch. 3.2 - Prob. 3QR Ch. 3.2 - Prob. 4QR Ch. 3.2 - Prob. 5QR Ch. 3.2 - 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Prob. 4RE Ch. 3 - Prob. 5RE Ch. 3 - Prob. 6RE Ch. 3 - Prob. 7RE Ch. 3 - Prob. 8RE Ch. 3 - Prob. 9RE Ch. 3 - Prob. 10RE Ch. 3 - Prob. 11RE Ch. 3 - Prob. 12RE Ch. 3 - Prob. 13RE Ch. 3 - Prob. 14RE Ch. 3 - Prob. 15RE Ch. 3 - Prob. 16RE Ch. 3 - Prob. 17RE Ch. 3 - Prob. 18RE Ch. 3 - Prob. 19RE Ch. 3 - Prob. 20RE Ch. 3 - Prob. 21RE Ch. 3 - Prob. 22RE Ch. 3 - Prob. 23RE Ch. 3 - Prob. 24RE Ch. 3 - Prob. 25RE Ch. 3 - Prob. 26RE Ch. 3 - Prob. 27RE Ch. 3 - Prob. 28RE Ch. 3 - Prob. 29RE Ch. 3 - Prob. 30RE Ch. 3 - Prob. 31RE Ch. 3 - Prob. 32RE Ch. 3 - Prob. 33RE Ch. 3 - Prob. 34RE Ch. 3 - Prob. 35RE Ch. 3 - Prob. 36RE Ch. 3 - Prob. 37RE Ch. 3 - Prob. 38RE Ch. 3 - Prob. 39RE Ch. 3 - Prob. 40RE Ch. 3 - Prob. 41RE Ch. 3 - Prob. 42RE Ch. 3 - Prob. 43RE Ch. 3 - Prob. 44RE Ch. 3 - Prob. 45RE Ch. 3 - Prob. 46RE Ch. 3 - Prob. 47RE Ch. 3 - Prob. 48RE Ch. 3 - Prob. 49RE Ch. 3 - Prob. 50RE Ch. 3 - Prob. 51RE Ch. 3 - Prob. 52RE Ch. 3 - Prob. 53RE Ch. 3 - Prob. 54RE Ch. 3 - Prob. 55RE Ch. 3 - Prob. 56RE Ch. 3 - 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Knowledge Booster

To graph: The function f ( h ) = sin ( h ) h and estimate lim h → 0 f ( h ) and then to compare the estimate with π 180 and then to state if there is a reason to believe that limit should be π 180 .

Answer to Problem 50E

Explanation of Solution

Answer to Problem 50E

Explanation of Solution

The formula obtained for the derivative of sinx<m:math><m:mrow><m:mi>sin</m:mi><m:mi>x</m:mi></m:mrow></m:math> .

Answer to Problem 50E

Explanation of Solution

The formula obtained for the derivative of cosx<m:math><m:mrow><m:mi>cos</m:mi><m:mi>x</m:mi></m:mrow></m:math> .

Answer to Problem 50E

Explanation of Solution

To calculate: The second and third degree-mode derivatives of sinx<m:math><m:mrow><m:mi>sin</m:mi><m:mi>x</m:mi></m:mrow></m:math> and cosx<m:math><m:mrow><m:mi>cos</m:mi><m:mi>x</m:mi></m:mrow></m:math> .

Answer to Problem 50E

Explanation of Solution

Chapter 3 Solutions

Additional Math Textbook Solutions

The formula obtained for the derivative of $sinx$ .

The formula obtained for the derivative of $cosx$ .

To calculate: The second and third degree-mode derivatives of $sinx$ and $cosx$ .