Question

We will estimate the length of the curve along g(x)=-1/4(x-5) to the second power +6 from x=11 using the sum of the lengths of line segments.

a.Which collection of line segments do you think would provide the best estimate of the length of y=g(x) from x=-1 to x=11? Explain why this choice is better than the other two. Which would give it the worst estimate?


B.Use the collection of line segments identified in problem 2a, and the distance formula to estimate the length of y=g(x)=-1 to x=11. Round the result to the nearest hundredth.

Answer 1

Explanation:

just do it

Answer 2

a. collection C will give the best estimate

b. A: 21.63; B: 21.89; C: 22.22

Explanation:

You want to know which of three approximations shown to the curve g(x) = -1/4(x -5)² would give the best estimate of the length of the curve. And you want to know the value of each.

a. Approximation

The curve is likely to be best approximated by the estimate that uses the largest number of line segments to match the curve. The numbers of segments used by (A, B, C) are (2, 3, 4), respectively.

Likely, approximation C will be best because it uses the most line segments.

b. Numerical values

The distance formula is based on the Pythagorean theorem. It computes the hypotenuse of a right triangle with legs equal to the difference in x-coordinates and the difference in y-coordinates.

In the attached calculator screen, the function Hypot(x, y) does the same thing, where 'x' and 'y' are the leg lengths (differences of coordinates) in the x- and y-directions.

The last three calculations shown are the sums of segment lengths in the collections A, B, C, respectively.

Estimates:

• A: 21.63
• B: 21.89
• C: 22.22

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Additional comment

The integral that gives the distance along the curve is shown at the top of the calculator display. For the interval of interest, it tells us the "exact" value of the curve length is about 22.61 units.