Chapter 2: Q. 37 (page 166)

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.
$f (x) = x + x^{3}, c = 0$

Short Answer

Expert verified

The value is $f' (0) = 1$ .

Step by step solution

Step 1. Given Information

We are given a function $f (x) = x + x^{3}, c = 0$ and using a sequence of approximations we have to estimate $f' (c)$ .

Step 2. Using a sequence of approximations

$h = 1, \frac{f (1) - f (0)}{1 - 0} = \frac{(1 + {(1)}^{3}) - (0 + {(0)}^{3})}{1} = \frac{2 - 0}{1} = 2 h = 0.5, \frac{f (0.5) - f (0)}{0.5 - 0} = \frac{(0.5 + {(0.5)}^{3}) - (0 + {(0)}^{3})}{0.5} = \frac{0.625 - 0}{0.5} = 1.25 h = 0.1, \frac{f (0.1) - f (0)}{0.1 - 0} = \frac{(0.1 + {(0.1)}^{3}) - (0 + {(0)}^{3})}{0.1} = \frac{0.101 - 0}{0.1} = 1.01 h = 0.01, \frac{f (0.01) - f (0)}{0.01 - 0} = \frac{(0.01 + {(0.01)}^{3}) - (0 + {(0)}^{3})}{0.01} = \frac{0.010001}{0.01} = 1.0001$

We might guess that the slope of the tangent line is $f' (0) = 1$ .

The graph of the function is

Step 3. Graphs of secant lines

When $c = 0, c + h = 1$ with $h = 1$ we have $f (0) = 0, f (1) = 2$ the secant line can be drawn as follows:

When $c = 0, c + h = 0.5$ with $h = 0.5$ we have $f (0) = 0, f (0.5) = 0.625$ the secant line can be drawn as follows:

When $c = 0, c + h = 0.1$ with $h = 0.1$ we have $f (0) = 0, f (0.1) = 0.101$ the secant line can be drawn as follows:

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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.
$f (x) = x + x^{3}, c = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using a sequence of approximations

Step 3. Graphs of secant lines

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

For each function f and value x=cin Exercises 35–44, use a sequence of approximations to estimate f'(c). Illustrate your work with an appropriate sequence of graphs of secant lines.f(x)=x+x3,c=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using a sequence of approximations

Step 3. Graphs of secant lines

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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.
$f (x) = x + x^{3}, c = 0$