Calculus 1: Applications of Differentiation
The Student[Calculus1] package contains three routines that can be used to both work with and visualize the concepts of Taylor approximations, charting a function, and using Newton's method. This worksheet demonstrates this functionality.
For further information about any command in the Calculus1 package, see the corresponding help page. For a general overview, see Calculus1.
Getting Started
While any command in the package can be referred to using the long form, for example, Student[Calculus1][FunctionChart], it is easier, and often clearer, to load the package, and then use the short form command names.
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The following sections show how the routines work.

Function Chart


The FunctionChart routine plots a function and shows regions of positive and negative sign, increasing and decreasing, and positive and negative concavity. By default:
1. Roots are marked by circles.
2. Extreme points are marked by diamonds.
3. Inflection points are marked by crosses.
4. Regions of increase and decrease are marked by red and blue lines, respectively.
5. Regions of positive and negative concavity are marked by azure and purple fill, respectively, with arrows pointing in the direction of the concavity.
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You can also perform curve analysis using the CurveAnalysisTutor command.
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Newton's Method


Given a point and an expression , the intercept of the tangent line through (, ) can be used as an approximation to a root of the expression . The equation of the tangent line is:
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 (3.1) 
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 (3.2) 
Solving for zero:
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 (3.3) 
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 (3.4) 
As an example, consider the function and an initial point .
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 (3.5) 
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 (3.6) 
Repeating this another times:
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 (3.7) 
The routine NewtonsMethod performs the same process.
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 (3.8) 
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The root to which a sequence of Newton iterations converges to depends on the initial point. For example,
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In general, when the root is not a double root, Newton's method is very efficient. In the following example with Digits set to , Newton's method converges to the root after only iterations.
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 (3.9) 
You can also learn about Newton's method using the NewtonsMethodTutor command.
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Main: Visualization
Previous: Various Theorems about Derivatives
Next: Integration
