I. Entering Functions.
II. Increasing and decreasing functions.
III. Functions and Slopes of Secant Lines.

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• Tools for Winplot used in this lab:

• Equation menu
• User Function
• name  box
• name (x) =  box  
  
• enter
• Point    (x,y)
• Segment  (x,y)
• One Function Menu
• Extremes...
• Mark point
  
• Slider...
• x =  .....
• Secant check box and slope.
• Anim Menu
•  individual
• A...
  

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I.Entering Functions. Demonstration.

• Equation menu
• User Function
• name   g
  
• name (x) =  2x^3 - 5x^2 - x + 4 
• enter
  
• Explicit
• y =      g(x)
  

• Point  (x,y) 
  
• x=0 y=g(0) ; 
  
• x=a  y = g(a)
• Use animation slider for A...
  
• Segment  (x,y)
• x1 = 0 y1 = g(0)
• x2 = a  y2 = g(a)
• Use animation slider for A...

II. Increasing and decreasing functions.*
Example:
Consider the function f
  where  f(x) = 2x³ - 5x² - x + 4 with x in the domain [-5, 5].

• Using Winplot find the local extreme points and values for the function.
• ANS: 
• local min: (x,y) = (1.76129406047167,-2.34445731940242)
• local max::(x,y) = (-0.09462739380500,4.04816102310612)
  
• Determine approximately the interval(s) where the function is increasing  .....decreasing.
• ANS:
• The function is increasing for the approximate intervals [-5,-0.0946] and  [1.7613, 5].
• The function is decreasing for the approximate interval [-0.09463,  1.76129]

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Record your answers for the next work and submit them on Moodle by Wednesday, Sept.19

For each of the following functions determine approximately the interval(s) where the function is increasing on the domain [-5, 5].

• f(x) = -x³ + 3x² - 4
• f(x) = 2x³ - 3x² + 4
  

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III. Functions and Slopes of Secant Lines.

Basic Terminology:
If  a line passes through a curve at two distinct points,
                    the line is described as a secant (cutting) line.

We will investigate secant lines determined by two points on the graph of functions. In particular, your task will be to find the slopes of these lines, using Winplot.

Example: Consider the function f where f(x) =-5x² - 3x + 4.

• Use Winplot's slider to look for the values of f(1) and f(3).

• Use these values to find the slope of the secant line passing through the function's graph at the points (1, f (1)) and (3,f(3)).  
• Verify this result on Winplot using the secants feature of the slider as follows:
• Position your slider with "x =" 1. <enter>
• Check of the box labelled "secants at  base point."
• In the "x = " box enter 3. <enter>
• Read the result on the slider from "slope:       " 
• Use the slider  to find the slopes of secant lines passing through the point (1, f (1)) and each of the following points:
• (2,f(2)); (1.5,f(1.5)); 
• (-1,f(-1)); (0,f(0)); (0.5,f(0.5))

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Record your answers for the next work and submit them on Moodle by Wednesday, Sept.19

For                     f(x) = -x³ + 3x² - 4  
find the slopes of secant lines passing through the point (1, f (1)) and each of the following points:

• (1, f (1)) and (2,f(2)); 
• (1, f (1)) and (1.1,f(1.1));
• (1, f (1)) and (0,f(0));
• (1, f (1)) and (0.9,f(0.9))

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