You may use any of these TI calculators:

TI-83 Plus

TI-84 Plus series

TI-Nspire CX series

TI-89 Titanium

TI-73 Explorer

The first thing we need to do is to reset the Random Access Memory (RAM).

This will clear everything that was initially stored by a previous user.

Also, after each problem, it is recommended that you reset the calculator.

(1.) | (2.) |

(3.) | (4.) |

Let us do some examples.

__NOTE:__ Please begin from the first example. Do not skip.

Concepts: **Graph polynomials**, **Determine local extrema**

(a.) Graph a polynomial within a range of values

(b.) Determine the local minimum of the polynomial

(c.) Determine the local maximum of the polynomial.

(1.) (a.) Use a graphing utility to approximate the local maximum value and local minimum value of the function $$ f(x) = -0.4x^3 - 0.7x^2 + 5x - 4 \\[3ex] for\;\; -6 \lt x \lt 4 \\[3ex] and\;\; -25 \lt y \lt 10 \\[3ex] $$

(b.) The local minimum is*y* ≈ ............ and it occurs at
*x* ≈ ............

(c.) The local maximum is*y* ≈ ............ and it occurs at
*x* ≈ ............

(Round to the nearest hundredth as needed)

(a.) Graph a polynomial within a range of values

(b.) Determine the local minimum of the polynomial

(c.) Determine the local maximum of the polynomial.

(1.) (a.) Use a graphing utility to approximate the local maximum value and local minimum value of the function $$ f(x) = -0.4x^3 - 0.7x^2 + 5x - 4 \\[3ex] for\;\; -6 \lt x \lt 4 \\[3ex] and\;\; -25 \lt y \lt 10 \\[3ex] $$

(b.) The local minimum is

(c.) The local maximum is

(Round to the nearest hundredth as needed)

(a.) Based on the graph from the TI graphing calculator:

The correct option is option

(b.)

The local minimum is

(c.)

The local maximum is

(1.) |
$X_{min}$ is the minimum value of x on the x-axis $X_{max}$ is the maximum value of x on the x-axis $X_{scl}$ is the scale (1 cm to how many units) on the x-axis $Y_{min}$ is the minimum value of y on the y-ayis $Y_{max}$ is the maximum value of y on the y-ayis $Y_{scl}$ is the scale (1 cm to how many units) on the y-axis $X_{res}$ is the resolution For Question (1.): −6 < x < 4 $X_{min} = -5$ and $X_{max} = 3$ −25 < y < 10 $Y_{min} = -24$ and $Y_{max} = 9$ |

(2.) | (3.) |

(4.) | (5.) |

(6.) | (7.) |

(8.) | (9.) |

(10.) | (11.) |

(12.) | (13.) |

(14.) | (15.) |

(16.) | (17.) |

Concept: **Model quadratic function**

(a.) Develop a quadratic model based on a given data

(2.) Use a graphing utility to find the quadratic function of best fit for the data.

*y* = .............................

(Type an expression using x as the variable. Use integers or decimals for any numbers in the expression. Round to three decimal places as needed.)

(a.) Develop a quadratic model based on a given data

(2.) Use a graphing utility to find the quadratic function of best fit for the data.

x |
25 | 35 | 45 | 55 | 65 | 75 |

y |
66 | 107 | 165 | 243 | 340 | 452 |

(Type an expression using x as the variable. Use integers or decimals for any numbers in the expression. Round to three decimal places as needed.)

The quadratic function of best fit for the data is:

$ y = ax^2 + bx + c \\[3ex] y = 0.09125x^2 - 1.390714286x + 43.62946429 \\[3ex] y \approx 0.091x^2 - 1.391x +43.629 $

(1.) | (2.) |

(3.) | (4.) |

(5.) | (6.) |

(7.) | (8.) |

Concept: **Scatter Diagrams**

(a.) Draw scatter diagrams

(b.) Interpret scatter diagrams

(3.) The following data represent the number of major hurricane strikes in a particular country each decade from 1921 to 2000.

(a.) Draw a scatter diagram of the data.

Comment on the type of relation that may exist between the two variables.

Which of the following shows the correct scatter diagram for these data?

(b.) Which relation best describes these data?

(**I.**) linear with positive slope

(**II.**) no relation

(**III.**) cubic

(**IV.**) linear with negative slope

(c.) The cubic function of best fit to these data is

$ H(x) = 0.159x^3 - 2.320x^2 + 9.330x - 0.2143 \\[3ex] $ Use a graphing utility to verify that this is the cubic function of best fit.

Use this function to predict the number of major hurricanes that struck between 1961 – 1970.

............. hurricanes.

(Round to the nearest integer as needed.)

(d.) With a graphing utility, draw a scatter diagram of the data and then graph the cubic function of best fit on the scatter diagram.

(e.) Concern has risen about the increase in the number and intensity of hurricanes, but some scientists believe this is just a natural fluctuation that could last another decade or two.

Use your model to predict the number of major hurricanes that will strike between 2001 and 2010.

(Round to the nearest integer as needed.)

(f.) Does your result appear to agree with what these scientists believe?

(g.) From 2001 – 2005, 6 hurricanes struck.

Does this support or contradict your prediction in part(f)?

(a.) Draw scatter diagrams

(b.) Interpret scatter diagrams

(3.) The following data represent the number of major hurricane strikes in a particular country each decade from 1921 to 2000.

Decade, x |
Major Hurricanes Striking, H |
---|---|

1921 – 1930, 1 | 7 |

1931 – 1940, 2 | 10 |

1941 – 1950, 3 | 12 |

1951 – 1960, 4 | 10 |

1961 – 1970, 5 | 8 |

1971 – 1980, 6 | 6 |

1981 – 1990, 7 | 7 |

1991 – 2000, 8 | 7 |

(a.) Draw a scatter diagram of the data.

Comment on the type of relation that may exist between the two variables.

Which of the following shows the correct scatter diagram for these data?

(b.) Which relation best describes these data?

(

(

(

(

(c.) The cubic function of best fit to these data is

$ H(x) = 0.159x^3 - 2.320x^2 + 9.330x - 0.2143 \\[3ex] $ Use a graphing utility to verify that this is the cubic function of best fit.

Use this function to predict the number of major hurricanes that struck between 1961 – 1970.

............. hurricanes.

(Round to the nearest integer as needed.)

(d.) With a graphing utility, draw a scatter diagram of the data and then graph the cubic function of best fit on the scatter diagram.

(e.) Concern has risen about the increase in the number and intensity of hurricanes, but some scientists believe this is just a natural fluctuation that could last another decade or two.

Use your model to predict the number of major hurricanes that will strike between 2001 and 2010.

(Round to the nearest integer as needed.)

(f.) Does your result appear to agree with what these scientists believe?

(g.) From 2001 – 2005, 6 hurricanes struck.

Does this support or contradict your prediction in part(f)?

(a.) Based on the scatter diagram from the calculator:

The correct option is option

(b.) The scatter diagram shows a cubic relationship.

(c.) The cubic function of best fit to these data is

$ H(x) = 0.159x^3 - 2.320x^2 + 9.330x - 0.2143 \\[3ex] $ Use a graphing utility to verify that this is the cubic function of best fit.

Use this function to predict the number of major hurricanes that struck between 1961 – 1970.

1961 – 1970 ⇒

$ H(x) = 0.159x^3 - 2.320x^2 + 9.330x - 0.2143 \\[3ex] H(5) = 0.159(5)^3 - 2.32(5)^2 + 9.33(5) - 0.2143 \\[3ex] = 0.159(125) - 2.32(25) + 46.65 - 0.2143 \\[3ex] = 19.875 - 58 + 46.65 - 0.2143 \\[3ex] = 8.3107 \\[3ex] \approx 8 \\[3ex] $ The number of major hurricanes that struck between 1961 – 1970 is approximately 8 hurricanes.

(d.) The scatter diagram and the graph of the cubic function is:

(e.) Following the trend:

2001 – 2010 ⇒

$ H(x) = 0.159x^3 - 2.320x^2 + 9.330x - 0.2143 \\[3ex] H(9) = 0.159(9)^3 - 2.32(9)^2 + 9.33(9) - 0.2143 \\[3ex] = 0.159(729) - 2.32(81) + 46.65 - 0.2143 \\[3ex] = 115.911 - 187.92 + 83.97 - 0.2143 \\[3ex] = 11.7467 \\[3ex] \approx 12 \\[3ex] $ The number of major hurricanes that struck between 2001 – 2010 is approximately 12 hurricanes.

(f.) Does your result appear to agree with what these scientists believe?

Yes, the natural fluctuation is supported because the result is comparable to hurricane data from past decades.

There were 12 major hurricanes between 1941 – 1950

(g.) From 2001 – 2005, 6 hurricanes struck.

Does this support or contradict your prediction in part(f)?

Yes, it supports it.

Half of the hurricanes (6) struck in the first half (first 5 years: 2001 – 2005) of the decade.

It is possible that the other half (6) could occur in the second half (second 5 years: 2005 – 2010) of the decade.

(1.) | (2.) |

(3.) | (4.) |

(5.) | (6.) |

(7.) | (8.) |

The cubic function of best fit to these data is

$ H(x) = 0.159x^3 - 2.320x^2 + 9.330x - 0.2143 \\[3ex] $ Use a graphing utility to verify that this is the cubic function of best fit.

(9.) | (10.) |

(11.) | (12.) |

$ y = ax^3 + bx^2 + cx + d \\[3ex] H(x) = 0.1590909091x^3 - 2.32034632x^2 + 9.33008658x - 0.2142857143 \\[3ex] H(x) \approx 0.159x^3 - 2.320x^2 + 9.330x - 0.2143 \\[3ex] $ With a graphing utility, draw a scatter diagram of the data and then graph the cubic function of best fit on the scatter diagram.

(13.) | (14.) |

(15.) | (16.) |

Concept: **Table of Values**

(a.) Given:

(i.) a polynomial

(ii.) several values of*x* within constant increments

(b.) To Do:

(i) Determine values of*y*

(ii) Make a Table of Values for the polynomial

(4.) In Calculus, certain functions can be approximated by polynomial functions.

Explore such function now.

(a.) Using a graphing utility, create a table of values with $Y_1 = f_1(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_1(x) = 1 + x + x^2 + x^3$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

(Type an integer or decimal rounded to five decimal places as needed. Type N if the function is undefined).

(b.) Using a graphing utility, create a table of values with $Y_1 = f_2(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_2(x) = 1 + x + x^2 + x^3 + x^4$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

(Type an integer or decimal rounded to five decimal places as needed. Type N if the function is undefined).

(c.) Using a graphing utility, create a table of values with $Y_1 = f_3(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_3(x) = 1 + x + x^2 + x^3 + x^4 + x^5$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

(Type an integer or decimal rounded to five decimal places as needed. Type N if the function is undefined).

(d.) What do you notice about the values of the function as more terms are added to the polynomial?

Are there some values of*x* for which the approximations are better?

**A.** As more terms are added, the values of the polynomial function get further and further away from the values of *f*.

The approximations near −1 or 1 are better than those near 0.

**B.** As more terms are added, the values of the polynomial function get closer to the values of *f*.

The approximations near 0 are better than those near −1 or 1.

**C.** As more terms are added, the values of the polynomial function get further and further away from the values of *f*.

The approximations near 0 are better than those near −1 or 1.

**D.** As more terms are added, the values of the polynomial function get closer to the values of *f*.

The approximations near −1 or 1 are better than those near 0.

(a.) Given:

(i.) a polynomial

(ii.) several values of

(b.) To Do:

(i) Determine values of

(ii) Make a Table of Values for the polynomial

(4.) In Calculus, certain functions can be approximated by polynomial functions.

Explore such function now.

(a.) Using a graphing utility, create a table of values with $Y_1 = f_1(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_1(x) = 1 + x + x^2 + x^3$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

(Type an integer or decimal rounded to five decimal places as needed. Type N if the function is undefined).

$x$ | $Y_1$ | $Y_2$ |
---|---|---|

−1 | ............ | ............ |

−0.9 | ............ | ............ |

−0.8 | ............ | ............ |

−0.7 | ............ | ............ |

−0.6 | ............ | ............ |

−0.5 | ............ | ............ |

−0.4 | ............ | ............ |

−0.3 | ............ | ............ |

−0.2 | ............ | ............ |

−0.1 | ............ | ............ |

0 | ............ | ............ |

0.1 | ............ | ............ |

0.2 | ............ | ............ |

0.3 | ............ | ............ |

0.4 | ............ | ............ |

0.5 | ............ | ............ |

0.6 | ............ | ............ |

0.7 | ............ | ............ |

0.8 | ............ | ............ |

0.9 | ............ | ............ |

1 | ............ | ............ |

(b.) Using a graphing utility, create a table of values with $Y_1 = f_2(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_2(x) = 1 + x + x^2 + x^3 + x^4$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

(Type an integer or decimal rounded to five decimal places as needed. Type N if the function is undefined).

$x$ | $Y_1$ | $Y_2$ |
---|---|---|

−1 | ............ | ............ |

−0.9 | ............ | ............ |

−0.8 | ............ | ............ |

−0.7 | ............ | ............ |

−0.6 | ............ | ............ |

−0.5 | ............ | ............ |

−0.4 | ............ | ............ |

−0.3 | ............ | ............ |

−0.2 | ............ | ............ |

−0.1 | ............ | ............ |

0 | ............ | ............ |

0.1 | ............ | ............ |

0.2 | ............ | ............ |

0.3 | ............ | ............ |

0.4 | ............ | ............ |

0.5 | ............ | ............ |

0.6 | ............ | ............ |

0.7 | ............ | ............ |

0.8 | ............ | ............ |

0.9 | ............ | ............ |

1 | ............ | ............ |

(c.) Using a graphing utility, create a table of values with $Y_1 = f_3(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_3(x) = 1 + x + x^2 + x^3 + x^4 + x^5$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

(Type an integer or decimal rounded to five decimal places as needed. Type N if the function is undefined).

$x$ | $Y_1$ | $Y_2$ |
---|---|---|

−1 | ............ | ............ |

−0.9 | ............ | ............ |

−0.8 | ............ | ............ |

−0.7 | ............ | ............ |

−0.6 | ............ | ............ |

−0.5 | ............ | ............ |

−0.4 | ............ | ............ |

−0.3 | ............ | ............ |

−0.2 | ............ | ............ |

−0.1 | ............ | ............ |

0 | ............ | ............ |

0.1 | ............ | ............ |

0.2 | ............ | ............ |

0.3 | ............ | ............ |

0.4 | ............ | ............ |

0.5 | ............ | ............ |

0.6 | ............ | ............ |

0.7 | ............ | ............ |

0.8 | ............ | ............ |

0.9 | ............ | ............ |

1 | ............ | ............ |

(d.) What do you notice about the values of the function as more terms are added to the polynomial?

Are there some values of

The approximations near −1 or 1 are better than those near 0.

The approximations near 0 are better than those near −1 or 1.

The approximations near 0 are better than those near −1 or 1.

The approximations near −1 or 1 are better than those near 0.

(a.) Table of Values for $Y_1 = f_1(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_1(x) = 1 + x + x^2 + x^3$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

$x$ | $Y_1$ | $Y_2$ |
---|---|---|

−1 | 0.5 | 0 |

−0.9 | 0.52632 | 0.181 |

−0.8 | 0.55556 | 0.328 |

−0.7 | 0.58824 | 0.447 |

−0.6 | 0.625 | 0.544 |

−0.5 | 0.66667 | 0.625 |

−0.4 | 0.71429 | 0.696 |

−0.3 | 0.76923 | 0.763 |

−0.2 | 0.83333 | 0.832 |

−0.1 | 0.90909 | 0.909 |

0 | 1 | 1 |

0.1 | 1.11111 | 1.111 |

0.2 | 1.25 | 1.248 |

0.3 | 1.42857 | 1.417 |

0.4 | 1.66667 | 1.624 |

0.5 | 2 | 1.875 |

0.6 | 2.5 | 2.176 |

0.7 | 3.33333 | 2.533 |

0.8 | 5 | 2.952 |

0.9 | 10 | 3.439 |

1 | N | 4 |

(b.) Table of Values for $Y_1 = f_2(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_2(x) = 1 + x + x^2 + x^3 + x^4$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

$x$ | $Y_1$ | $Y_2$ |
---|---|---|

−1 | 0.5 | 1 |

−0.9 | 0.52632 | 0.8371 |

−0.8 | 0.55556 | 0.7376 |

−0.7 | 0.58824 | 0.6871 |

−0.6 | 0.625 | 0.6736 |

−0.5 | 0.66667 | 0.6875 |

−0.4 | 0.71429 | 0.7216 |

−0.3 | 0.76923 | 0.7711 |

−0.2 | 0.83333 | 0.8336 |

−0.1 | 0.90909 | 0.9091 |

0 | 1 | 1 |

0.1 | 1.11111 | 1.1111 |

0.2 | 1.25 | 1.2496 |

0.3 | 1.42857 | 1.4251 |

0.4 | 1.66667 | 1.6496 |

0.5 | 2 | 1.9375 |

0.6 | 2.5 | 2.3056 |

0.7 | 3.33333 | 2.7731 |

0.8 | 5 | 3.3616 |

0.9 | 10 | 4.0951 |

1 | N | 5 |

(c.) Table of Values for $Y_1 = f_3(x) = \dfrac{1}{1 - x}$ and $Y_2 = g_3(x) = 1 + x + x^2 + x^3 + x^4 + x^5$ for $-1 \le x \le 1$ with $\Delta Tbl = 0.1$

$x$ | $Y_1$ | $Y_2$ |
---|---|---|

−1 | 0.5 | 0 |

−0.9 | 0.52632 | 0.24661 |

−0.8 | 0.55556 | 0.40992 |

−0.7 | 0.58824 | 0.51903 |

−0.6 | 0.625 | 0.59584 |

−0.5 | 0.66667 | 0.65625 |

−0.4 | 0.71429 | 0.71136 |

−0.3 | 0.76923 | 0.76867 |

−0.2 | 0.83333 | 0.83328 |

−0.1 | 0.90909 | 0.90909 |

0 | 1 | 1 |

0.1 | 1.11111 | 1.11111 |

0.2 | 1.25 | 1.24992 |

0.3 | 1.42857 | 1.42753 |

0.4 | 1.66667 | 1.65984 |

0.5 | 2 | 1.96875 |

0.6 | 2.5 | 2.38336 |

0.7 | 3.33333 | 2.94117 |

0.8 | 5 | 3.68928 |

0.9 | 10 | 4.68559 |

1 | N | 6 |

(d.) Based on the observations of the three tables, we notice that:

As more terms are added, the values of the polynomial function get closer to the values of

The approximations near 0 are better than those near −1 or 1.

(1.) | (2.) |

(3.) | (4.) |

(5.) | (6.) |

(7.) | (8.) |

(9.) | (10.) |

(11.) | (12.) |

(5.)

(6.)