Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate $f^{'} (c)$ . Illustrate your work with an appropriate sequence of graphs of secant lines.
role="math" localid="1648705352169" $f (x) = e^{x}, c = 0$

We have approximated the slope by using the concept of the secant line.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

We have to use a sequence of approximations to estimate $f^{'} (c)$

$f (x) = e^{x}, c = 0$

Step 2. Use sequence of approximation

Let,

$h = 1, 0.5, 0.1, 0.01$

Consider the expressions,

$\begin{aligned} \frac{f (1) - f (0)}{1 - 0} = \frac{[e] - 1}{1} \\ = 1.718 \\ \frac{f (0.5) - f (0)}{0.5 - 0} = \frac{[e^{0.5}] - 1}{0.5} \\ = 1.297 \\ \frac{f (0.1) - f (0)}{1.1 - 0} = \frac{[e^{0.1}] - [1]}{0.1} \\ = 1.051 \\ \frac{f (0.01) - f (0)}{0.01 - 0} = \frac{[e^{0.01}] - [1]}{0.01} \\ = 1.005 \end{aligned}$

The slope of tangent will be :

$f^{'} (0) = 1$

The graph is ;

Step 3. First secant graph

Take c=0 , c+h=1, then the corresponding values are:

$f (0) = 1, f (1) = 2.718$

The secant line can be drawn as:

Step 4. Second secant graph

Take c=0 and c+h = 0.1 then the corresponding values are :

$f (0) = 1, f (0.1) = 1.1051$

The secant graph is :

Step 5. Third secant graph

Take c=0 and c+h=0.01, then the corresponding values are :

$f (0) = 1, f (0.01) = 1.0101$

The secant graph is :

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Calculus

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Short Answer

For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate $f^{'} (c)$ . Illustrate your work with an appropriate sequence of graphs of secant lines.
role="math" localid="1648705352169" $f (x) = e^{x}, c = 0$

Step by Step Solution

Step 1. Given information.

Step 2. Use sequence of approximation

Step 3. First secant graph

Step 4. Second secant graph

Step 5. Third secant graph

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Calculus

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For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate f'(c). Illustrate your work with an appropriate sequence of graphs of secant lines. role="math" localid="1648705352169" f(x)=ex,c=0

Step by Step Solution

Step 1. Given information.

Step 2. Use sequence of approximation

Step 3. First secant graph

Step 4. Second secant graph

Step 5. Third secant graph

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Pure Maths

For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate $f^{'} (c)$ . Illustrate your work with an appropriate sequence of graphs of secant lines.
role="math" localid="1648705352169" $f (x) = e^{x}, c = 0$