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Starbucks Coffee Company Total Product Sales An exploration of the nations coffee obsession through Integrals and Riemann Sums

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Basic Info n n Starbucks has done some of the legwork for you! Lucky you! n n They know the total thousands of pounds of base consumed nationally for each month of the past year and have used a graphing calculator to find the function that expresses this information. n n The function is: f(x)=-.568x x Month Frappuccino Base (Thousands of Pounds)

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Starbucks Needs Your Help! n n Its the end of the year, and Starbucks is getting ready to order Frappuccino Base from its suppliers for the upcoming year. n n Therefore they need to know the average pounds of Frappuccino Base they process and sell over the course of the year. n n They can always order more or less base as the year progresses, but they need an approximation. n n As the Calculus Guru you are, Starbucks Coffee Company has turned to you and asked you to help them order more Frappuccino Base.

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Graph n n The data is presented in a graph like the one on the right. Knowing the graphical representation of this information will help your understanding of the Integral and Riemann Sum process n n As you can see, the consumption of Frappuccino Base reaches its peak in the summer months, maybe because its hot??

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What is an Integral? n n An integral takes a function, such as f(x) and looks at it as if it were the function f(x). Then by taking the integral we find the original function that f(x) is the derivative of. n n For example, we have the function f(x)=x 2 n n Say we take the integral of f(x) from 1 to 12, so we write it as 1 12 (x 2 )dx n n Then we undo this to find the function that f(x) is the derivative of, which we write as (1/3)x 3 ] 1 12 n n Then we plug in f(12) and subtract f(1) from it, like so: (1/3)(12) 3 – (1/3)(1) 3 n n This gives us , so 1 12 (x 2 )dx=

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So How Does That Relate to This Problem? n n Integrals are a way to find the total area under a curve or the total change over time. n n In this situation, an integral would express the total thousands of pounds of Frappuccino Base consumed nationally. n n Now, there are a few ways to take an integral – you can find the integral, or take Left-hand, Right-hand, Midpoint, or Trapezoid Riemann sums

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Method 1: Taking an Integral n n f(x)= -.568x x n n 1 12 (-.568x x )dx n n -.189x x x] 1 12 n n [-.189(12) (12) (12)] – [-.189(1) (1) (1)] n n n n thousands of pounds of Frappuccino Base consumed n Original function n The integral of f(x) from 1 to 12 n Take the integral of f(x) n Plug in x=12 and x=1 n Subtract f(12)-f(1) n Find the answer! (this is only an approximation because rounding may influence the actual answer)

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Method 2: Left-Hand Riemann Sums n n Now, for this we are going to take the Left-Hand Riemann Sum from 1 to 12, splitting the graph into 6 segments n n Notice that the red lines create 6 rectangles whose left- hand corner intersects with points on the graph. n n You will be taking the area of each of these rectangles and adding them together.

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n The first rectangles width (along the x- axis) is 2, and its height or length is 2 (refer to the chart for exact heights) n Each rectangle has a width of 2 and a length that is specified by the chart, except for the last rectangle, whos height is 1 Month Frappuccino Base (Thousands of Pounds)

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n n Now, using the area formula (A=W * L), we find the area of each rectangle. Remember, the width is 2 for each rectangle except the last one, and the heights are highlighted in red in the chart. 2( )+ 10=154 Month Frappuccino Base (Thousands of Pounds)

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Important Riemann Sum Secrets n n The area under the curve using the Left- Hand Riemann Sum is 154 n n This means Starbucks consumed 154 millions of pounds of Frappuccino base in this year. n n Now, because the rectangles do not exactly fill the graph completely, the Left-Hand Riemann Sum is an underestimate of the actual area. n n The Left-Hand, Right-Hand, and Midpoint Riemann sums follow the same method. All of these are either over or underestimates.

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Guess what?! Calculators can do all of this for you!

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Integrals n n Now that you know how to do this the hard way… calculators can help you do this much easier! n n First, put the function f(x)= -.568x x into Y 1 n n To find the integral of f(x) from 1 to 12, return to the home screen, press math, 9, vars, go to y-vars, then press 1 twice, then include x,1,12) and press enter. n n The answer, , is the integral of f(x) from 1 to 12.

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n n To find a Riemann Sum with a calculator, youll have to find and enter a program called ALLSUMS, a Google search should find it for you. n n Once the program is loaded, press PRGM, go to ALLSUMS, press enter twice, enter 1 for A,12 for B, then for N put in the number of segments you need to find n n The answers on the right of the screen correspond with the letters on the top of the screen n n L (Left-Hand) goes with the first number, R (Right-Hand) goes with the second number, and so on. n n T is for Trapezoids and M is for Midpoint Riemann Sums

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You did it!! n n You now know how to find the integral of a function using the integral method and Riemann Sums, and you know the shortcuts too! n n Oh and you found that Starbucks consumes thousands of pounds of Frappuccino base. Theyd probably like to know that so they can order more.* *This was a fictional situation. Please dont call Starbucks.

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