Note: Recall that Sin(X) = X-X3/3!+X5/5!-X7/7!…..

Part 7: Evaluate Expressions Graphically

(a) Algebraically, evaluate the expression: X⁵+1/X⁵, when X²+1/X² = 7.

(b) Evaluate the same expression graphically.

Part 8: Finding Limits Graphically

(a) Algebraically, find the limit of A(X) = (X-2)/(X-2) as X approaches to 2. Explore this limit graphically on your calculator. Discuss connections between the graphical and algebraic representations.

(b) Do the same for B(X) = (X²-X-2)/(X-2) as X approaches 2.

(c) Do the same for C(X) = 1/(X-1) as X approaches 1.

(d) Do the same for E(X) = (1+1/X)^X as X increases without bound.

(e) Do the same for F(X) = sin(1/X) as X approaches 0.

Part 9: Using Graphs to Solve Max/Min Problems I

Consider the following problem (from Frank Pullano, Winthrop University):

The Mama Mia Sauce Company wants to package their famous sauce in 980 cc cans. They would like to minimize the amount of material used to make the cans. Your job as production manager for Mama Mia is to determine what can dimensions, radius and height, will produce a can which will hold the desired amount of sauce yet use the least amount of material.

(a) Solve this problem using algebraic methods.

(b) Solve this problem by generating a graph of the relationship between the surface area of the can and the radius of the can.

(c) Compare both methods.

 

Part 10: Using Graphs to Solve Max/Min Problems II

Consider the following situation (from Frank Pullano, Winthrop University):

The marketing director at Mama Mia has recently been given research that leads her to believe that at a price of $2.75 per can of sauce, sales will be 3200 cans per week. The research also indicates that for every $.15 increase in price, sales will drop by 100 cans per week.

(a) Derive a function relating revenue (price per can times the number of cans sold) to the number of 15-cent increments, predict the shape of the graph of this function, and then graph the function.

(b) Use this function to determine: (i) the maximum revenue Mama Mia can expect, (ii) the price per can that yields maximum revenue, and (iii) the number of cans of sauce Mama Mia should expect to sell at this price to achieve maximum revenue.

(c) Discuss the graph, make connections to the algebraic representation, and interpret various regions and points of interest.

Part 11: Using Graphs to Solve Max/Min Problems II

Consider the following situation (from Frank Pullano, Winthrop University):

Assume that Mama Mia produces only what it can sell and that the cost of production (C) to Mama Mia is $1.35 per can of sauce, regardless of how many cans are produced.

(a) Use the sales forecast in the problem above to derive and graph a function relating total costs to the number of cans of sauce sold in terms of the number of 15-cent increments. Using your functions for revenue and total cost, write a function for profit. How many jars of sauce must Mama Mia sell, and at what price, to maximize profits? What do you notice about the price per jar at maximum revenue compared to the price per jar at maximum profit?

(b) Discuss the graph, make connections to the algebraic representation, and interpret various regions and points of interest.

Part 12: Solving a Numberical Comparison Problem Graphically

(a) Algebraically, without calculating these values, determine which is greater e^π or π^e.

(b) Solve this problem graphically.

(c) Compare the two approaches.

Part 13: Simulating Exponential Growth

Perform the following demonstration:

Take a sheet on paper, rip it in half, and then pile one half on top of the other. We will now have a pile with height equal to the thickness of two sheets of paper. Take that pile, rip it in half, and pile one half on top of the other. The resulting pile would have height equal to 4 sheets of paper. Continue this process three more times, and show the resulting pile to the class.

(a) Without performing any calculations, estimate the answer the following question: What would be the height of the pile if I continued this process for a total of 50 rips?

(b) Calculate the height of this resulting pile.

Hint:  Use parametric graphing, with X(t)=1 and Y(t)=(.004/(12*5280))*2t, with t ranging from 1 to 50. For emphasis, display the word SUN on your screen at a distance of approximately 93,000,000 miles.

Part 15: Graphically Analyzing the World Series

Consider the following problem (from Larry Cavanaugh, Louisa High School, VA):

The World Series is a best-of seven competition. That is, two teams play each other until one of them has won a total of 4 games. Hence, a World Series can be won in as little as four games and in as many as seven games. Suppose that two teams are playing in a World Series and that the probability of Team A beating Team B in any game is p.

(a) Calculate the probability that Team A wins the World Series in 4 games.

(b) Predict the shape of the graph of the relationship between the probability of Team A winning the Series in 4 games and the probability of Team A winning any single game.

(c) Graph this relationship, and analyze the graph.

(d) Discuss your observations.

(e) Suppose that the teams are evenly matched; that is, the probability of each team winning a single game is 50-50. Calculate the probability of that Team A will win in 4 games.

(f) Repeat tasks (a)-(e) for 5 games, 6 games, and 7 games.

(g) Use the results of (e) to show that the overall probability of either winning the Series is also 50-50.

(h) For the above situation, discuss the relationship between the probability of evenly matched teams winning in 6 games or 7 games. 

(i) Overall, regardless of the probability of any team winning a game, what is the expected number of games that need to be played to determine the winner?