# Calculus 1.0

## Introduction

### Quick Start

You can download the software online from The Calculus 1.0 Website. Unzip the software and run Calculus. The features in the main menu are demonstrated by over 200 examples. For any item in the menu, simply select an appropriate example to see how to input the parameters. The examples have been carefully designed to demonstrate all aspects of the software. For details of specific features, read on.
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## Expressions

### Symbols, Operations and Rules of Precedence

All expressions are written in the usual infix notation, following the usual (BEDMAS) rules of precedence, with the following symbols and operations:

 Variable x Variable (polar and parametric) u Left Parenthesis ( Right Parenthesis ) Exponentiation ^ Division / Multiplication * Addition + Subtraction and Negation -

The compiler and interpreter for Calculus 1.0 automatically converts expressions to reverse polish notation for internal evaluation. Incorrectly written expressions are interpreted as zero.
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### Notation for Standard Functions

The Calculus 1.0 compiler and interpreter recognizes the following standard functions:

 Natural Logarithm log(x) Exponential exp(x) Sine sin(x) Cosine cos(x) Tangent tan(x) Inverse Sine arcsin(x) Inverse Cosine arccos(x) Inverse Tangent arctan(x) Hyperbolic Sine sinh(x) Hyperbolic Cosine cosh(x) Hyperbolic Tangent tanh(x) Absolute Value abs(x) Ceiling ceil(x) Floor floor(x) Square Root sqrt(x)

Expressions for all other functions are constructed from the standard functions, symbols, operators and decimal numbers.
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## Functions

### Graph Function

From the menu select Functions, Graph Function.
Enter the function y=f(x) following the usual infix notation using decimal numbers, parentheses (, ), variable symbol x, operations division /, multiplication *, addition +, subtraction -, exponentiation ^ and elementary functions log, exp, sin, cos, tan, arcsin, arccos, arctan, sinh, cosh, tanh, abs, ceil, floor and sqrt. The function is automatically parsed using standard precedence rules. If the function does not parse correctly, it is interpreted as the zero function by default.
For example, try the polynomial 5*x^3-2*x^2+x-4.
For example, try the function exp(abs(x-1)).
For example, try the function -sin(x^2+1)/(x^2+1).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Press Graph Function f(x). The graph ____ is displayed in the graph window. You can move your cursor over the graph to read the x,y coordinates.
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### Evaluate Function and Limit

From the menu select Functions, Evaluate Function and Limit.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter c. (To estimate limits, you must select appropriate values c near the limiting value at which the function is defined).
Press Evaluate Function f(c). The graph of the function f(x) ____ and the point (c, f(c)) are displayed in the graph window. The function value f(c) is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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### Standard Functions

From the menu select Functions, Standard Functions. The graphs ____ of all the standard functions are available.
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## Derivatives

### First Derivative

From the menu select Derivatives, First Derivative.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Press Graph First Derivative f '(x). The graphs of the original function f(x) ____ and its derivative f '(x) ____ are displayed in the graph window. You can move your cursor over the graphs to read the x,y coordinates.
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### Second Derivative

From the menu select Derivatives, Second Derivative.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Press Graph Second Derivative f ''(x). The graphs of the original function f(x) ____ and its second derivative f ''(x) ____ are displayed in the graph window. You can move your cursor over the graphs to read the x,y coordinates.
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### Tangent Line

From the menu select Derivatives, Tangent Line.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter c.
Press Draw Tangent Line. The graph of the function f(x) ____ and the tangent line at x=c ____ are displayed in the graph window. The derivative value f '(c) (slope of the tangent line) is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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## Integrals

### Indefinite Integral

From the menu select Integrals, Indefinite Integral.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Press Graph Indefinite Integral F(x). The graphs of the function f(x) ____ and the integral F(x) ____ are displayed in the graph window.You can move your cursor over the graph to read the x,y coordinates.
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### Definite Integral and Area

From the menu select Integrals, Definite Integral and Area.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter a (lower limit of integration).
Enter b (upper limit of integration).
Enter Partition N. This fixes the partition size of Simpson's Rule for integration.
Press Compute Definite Integral and Area. The graph of the function f(x) ____ and the corresponding area ||||| under the curve from x=a to x=b are displayed in the graph window. The definite integral value F(b)-F(a) (area under the curve) is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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### Arc Length

From the menu select Integrals, Arc Length.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter a (lower limit of integration).
Enter b (upper limit of integration).
Enter Partition N. This fixes the partition size of Simpson's Rule for integration.
Press Compute Arc Length. The graph of the function f(x) ____ and the corresponding arc length ____ on the curve from x=a to x=b are displayed in the graph window. The value of the arc length is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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## Roots

### Roots of f(x)=0

From the menu select Roots, Roots of f(x)=0.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter Initial Value x. This fixes the initial approximation for a root.
Enter Number of Iterations. This fixes the number of iterations performed by Newton's method.
Press Find Root f(x)=0. The graph of the function f(x) ____ and the corresponding root are displayed in the graph window. The value of the root is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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## Max / Min

### Maxima and Minima

From the menu select Max / Min, Maxima and Minima.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter Initial Value x. This fixes the initial approximation for a local max / min.
Enter Number of Iterations. This fixes the number of iterations performed by Newton's method.
Press Find Max / Min. The graph of the function f(x) ____ and the corresponding local max / min are displayed in the graph window. The value of the local max / min is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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## Inflection

### Points of Inflection

From the menu select Inflection, Points of Inflection.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter Initial Value x. This fixes the initial approximation for a point of inflection.
Enter Number of Iterations. This fixes the number of iterations performed by Newton's method.
Press Find point of Inflection. The graph of the function f(x) ____ and the corresponding point of inflection are displayed in the graph window. The value of the point of inflection is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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## Series

### Taylor Series

From the menu select Series, Taylor Series.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter Center x=a. This defines the center x=a for the Taylor series approximation.
Enter Degree n. This defines the degree of the polynomial that approximates the Taylor series.
Enter Step Size dx. This defines the step size for Lagerange interpolation during calculation of derivatives.
Press Compute Taylor Series. The graph of the function f(x) ____ and the corresponding graph of the Taylor series approximation ____ are displayed in the graph window. The Taylor series approximation is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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### Fourier Series

From the menu select Series, Fourier Series.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter Period p. This defines the period for the Fourier series approximation.
Enter Degree n. This defines the degree of the polynomial that approximates the Fourier series.
Enter Partition Size N. This defines the partition size for Simpson's rule during calculation of integrals.
Press Compute Fourier Series. The graph of the function f(x) ____ and the corresponding graph of the Fourier series approximation ____ are displayed in the graph window. The Fourier series approximation is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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## Interpolation

### Lagrange Interpolation

From the menu select Interpolation, Lagrange Interpolation.
Enter the x, y coordinates of a point and press Insert (x,y). This inserts the point (x,y) into the list of points. Repeat this process to insert any number of points into the list of points.
Press Lagrange Interpolation. The graph of the Lagrange Interpolation Polynomial ____ and the list of points are displayed in the graph window. The Lagrange Interpolation Polynomial is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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### Method of Least Squares

From the menu select Interpolation, Method of Least Squares.
Enter the x, y coordinates of a point and press Insert (x,y). This inserts the point (x,y) into the list of points. Repeat this process to insert any number of points into the list of points.
Press Method of Least Squares. The graph of the Least Squares Line of Best Fit ____ and the list of points are displayed in the graph window. The equation of the Least Squares Line of Best Fit is displayed in its box.You can move your cursor over the graph to read the x,y coordinates.
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## Curves

### Parametric Equations

From the menu select Curves, Parametric Equations.
Enter the Parametric Equations x(u) and y(u) (note that the variable here must be u).
Enter u-min, u-max (parameter range) for the graph.
Press Draw Parametric Curve. The curve ____ is displayed in the graph window. You can move your cursor over the curve to read the x,y coordinates.
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### Polar Equation

From the menu select Curves, Polar Equation.
Enter the Polar Equation r=f(u) (note that the variable here must be u and x=r*cos(u), y=r*sin(u)).
Enter u-min, u-max (parameter range) for the graph.
Press Draw Polar Curve. The curve ____ is displayed in the graph window. You can move your cursor over the curve to read the x,y coordinates.
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## Surfaces

### Surface of Revolution / Area and Volume

From the menu select Surfaces, Surface of Revolution / Area and Volume.
Enter the function y=f(x).
Enter x-min, x-max (domain) and y-min, y-max (range) of the graph window.
Enter a (lower limit of integration).
Enter b (upper limit of integration).
Enter Partition N. This fixes the partition size of Simpson's Rule for integration.
Select Draw Wire Frame Surface or Draw Smooth Surface.
Press Generate Surface. The surface of revolution of the function f(x) from x=a to x=b about the x-axis is displayed in the graph window. The value of the surface area is displayed in its box. The value of the enclosed volume is displayed in its box.You can click and drag the surface in the graph window to look at it from all angles.
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## Calculator

### Calculate

From the menu select Calculator, Calculate.
Enter an expression involving standard functions, symbols, operations and decimal numbers.
(The examples show how to write long expressions to estimate limits of some Sequences and Series.)
Press Calculate. The answer is displayed in its box.
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## Edit

### Copy and Paste with 5 Memories

From the menu select Edit, Copy, Memory 1. This copies the current contents of Calculus 1.0 to Memory 1. Similarly, different data can be copied to Memories 2,3,4 and 5.
From the menu select Edit, Paste, Memory 1. This pastes the current contents of Memory 1 to Calculus 1.0. Similarly, different data can be pasted from Memories 2,3,4 and 5.
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