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Warmup:

Warmup:. 1). 2). 3). 5.1 Estimating with Finite Sums. Greenfield Village, Michigan. A car travels using their cruise control at 65 mph from 4 pm to 6 pm What is the total distance traveled?. Distance = rate x time. Let’s look at a graph of it:.

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Warmup:

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Presentation Transcript

  1. Warmup: 1) 2) 3)

  2. 5.1 Estimating with Finite Sums Greenfield Village, Michigan

  3. A car travels using their cruise control at 65 mph from 4 pm to 6 pm What is the total distance traveled? Distance = rate x time Let’s look at a graph of it: Look at the area of the resulting rectangle, (2)(65) = 130 . The distance traveled is equal to the area under the line.

  4. At 4 pm, a car slowly decreases their speed from 65 mph all the way to 0 mph by 6pm at a constant rate. What is the total distance traveled? Use area of a triangle: 65 miles

  5. SO we can use DIFFERENT geometric shapes to find our area under the curve. Remember this for later! Find the area under this line on this interval

  6. If the velocity is not constant, the distance traveled is still equal to the area under the curve. (The units work out.) Example: (left hand) We could estimate the area under the curve by drawing rectangles touching at their left corners. v(0)= 1 v(1)=1.125 v(2)=1.5 v(3)=2.125 0-1 1-2 2-3 3-4 This is called the Left-hand Rectangular Approximation Method (LRAM).

  7. Approximate Area: Rectangles 1 , 2, 3, 4 Area 1= length x width = 1 x 1 = 1 sq unit Area 2 = length x width = 1.125 x 1=1.125 Area 3 = length x width = 1.5 x 1= 1.5 v(0)= 1 v(1)=1.125 v(2)=1.5 v(3)=2.125 0-1 1-2 2-3 3-4 Area 4 = length x width = 2.125 x 1= 2.125 Total Area is the sum of these 1 + 1.125 + 1.5 + 2.125 = 5.125

  8. Use the right side of the interval Approximate area: v(1)=1.125 v(2)=1.5 v(3)=2.125 v(4) = 3 0-1 1-2 2-3 3-4 We could also use a Right-hand Rectangular Approximation Method (RRAM).

  9. Approximate area: Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). In this example there are four subintervals. .

  10. Approximate area: The exact answer for this problem is . With 8 subintervals: width of subinterval

  11. Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve:

  12. We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation.

  13. Group Problem: Rain is falling outside during a massive storm. Data is collected on the rate of change of the rainfall for the first hour of the storm. Find the total amount of rainfall that has fallen after 1 hour using a left and right handed method using 6 intervals of equal width

  14. The End Use worksheet 5.1a

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