Here is the given information: The owner of a commercial fishing operation decides to upgrade the hydraulic pumps, hydraulic lines, and the drum drive to a new system that is supposed to save costs over time. The projected savings rate for the new equipment is given by S′(x)=225−x2S′(x)=225−x2 where xx is the number of years the machinery will be used. The rate of additional costs to run and maintain the new equipment is expected to be C′(x)=x2+25x+150C′(x)=x2+25x+150 Both functions have units of thousands of dollars per year. Here is a procedure for finding an approximation of the value of the net savings accumulated from x=ax=a to x=bx=b years after the new hydraulic system has been installed: Let f(x)=f(x)= net savings [in thousands of dollars] after xx years Step 1: Find the value of f(0)f(0) . This is the _____________ at the beginning of the 1st year. Step 2: Break the interval [a,b][a,b] up in to nn subintervals, each of equal length, dxdx . Example 1: If a=0a=0 , b=10b=10 , and n=10n=10 , find the length of each of the subintervals. Example 2: If a=0a=0 , b=10b=10 , and n=50n=50 , find the length of each of the subintervals. Example 3: If a=0a=0 , b=xb=x , and n=100n=100 , find the length of each of the subintervals. FYI: In this last case, the length of the subinterval will be a function of xx. Step 3: Find the midpoint from each of the intervals in Examples 1 - 3 in Step 2. For each example, you should have a collection of nn midpoints. Use xixi to denote the iith midpoint: For example, x3x3 represents the midpoint from the 3rd subinterval (when counted left to right). You should form a table as your answer. Here's the table for Example 1: iith subinterval xixi is the iith midpoint [0,1][0,1] x1=0.5x1=0.5 [1,2][1,2] x2=1.5x2=1.5 [2,3][2,3] x3=2.5x3=2.5 ⋮⋮ ⋮⋮ [9,10][9,10] x10=9.5x10=9.5 The n=10n=10 midpoints for Example 1 are x1=0.5,x2=1.5,x3=1.5,…,x10=9.5x1=0.5,x2=1.5,x3=1.5,…,x10=9.5 Make your own [similar] tables for Examples 2 and 3. Be careful and take note of the values that bb and nn are equal to in each of these examples. You are welcome to use "dots" if the pattern is apparent. Give a formula that finds the iith midpoint in Example 2. Step 4: Find the net savings associated to each of the subintervals in Example 1. Your answer will again be in table form. Read on to see how to do this: For Example 1, the rate of net savings over the iith subinterval is given by f′(xi)=S′(xi)−C′(xi)f′(xi)=S′(xi)−C′(xi) Note that the rate of net savings has the following units: net savingslength of timenet savingslength of time The length of time is the same for each subinterval: dx=b−andx=b−an Remember how rate×time=distancerate×time=distance ? In this case, we instead have rate×time=net savingsrate×time=net savings . This means f′(xi)dxf′(xi)dx is the approximate net savings over the iith subinterval. Here is the table of approximations of net savings (over each subinterval) for Example 1: iith subinterval xixi : iith midpoint f′(xi)f′(xi) : iith rate of net savings f′(xi)dxf′(xi)dx : iith net savings amount (approx) [0,1][0,1] x1=0.5x1=0.5 f′(0.5)=?f′(0.5)=? f′(0.5)⋅dx=?f′(0.5)⋅dx=? [1,2][1,2] x2=1.5x2=1.5 f′(1.5)=?f′(1.5)=? f′(1.5)⋅dx=?f′(1.5)⋅dx=? [2,3][2,3] x3=2.5x3=2.5 f′(2.5)=?f′(2.5)=? f′(2.5)⋅dx=?f′(2.5)⋅dx=? ⋮⋮ ⋮⋮ ⋮⋮ ⋮⋮ [9,10][9,10] x10=9.5x10=9.5 f′(9.5)=?f′(9.5)=? f′(9.5)⋅dx=?f′(9.5)⋅dx=? Please fill out each entry with a "?" in this table. Please watch the June 9th presentation for a different example that shows how to put this table together. This means that f(10)≈f(0)+sum of all the f′(xi)dxf(10)≈f(0)+sum of all the f′(xi)dx Use Desmos to compute the sum (see the June 9th presentation) Find the [similar] table that is associated to Example 2 using Desmos.
Here is the given information:
The owner of a commercial fishing operation decides to upgrade the hydraulic pumps, hydraulic lines, and the drum drive to a new system that is supposed to save costs over time. The projected savings rate for the new equipment is given by
S′(x)=225−x2S′(x)=225−x2
where xx is the number of years the machinery will be used.
The rate of additional costs to run and maintain the new equipment is expected to be
C′(x)=x2+25x+150C′(x)=x2+25x+150
Both functions have units of thousands of dollars per year.
Here is a procedure for finding an approximation of the value of the net savings accumulated from x=ax=a to x=bx=b years after the new hydraulic system has been installed:
Let f(x)=f(x)= net savings [in thousands of dollars] after xx years
Step 1: Find the value of f(0)f(0) . This is the _____________ at the beginning of the 1st year.
Step 2: Break the interval [a,b][a,b] up in to nn subintervals, each of equal length, dxdx .
Example 1: If a=0a=0 , b=10b=10 , and n=10n=10 , find the length of each of the subintervals.
Example 2: If a=0a=0 , b=10b=10 , and n=50n=50 , find the length of each of the subintervals.
Example 3: If a=0a=0 , b=xb=x , and n=100n=100 , find the length of each of the subintervals.
FYI: In this last case, the length of the subinterval will be a function of xx.
Step 3: Find the midpoint from each of the intervals in Examples 1 - 3 in Step 2.
For each example, you should have a collection of nn midpoints.
Use xixi to denote the iith midpoint: For example, x3x3 represents the midpoint from the 3rd subinterval (when counted left to right).
You should form a table as your answer.
Here's the table for Example 1:
| iith subinterval | xixi is the iith midpoint |
| [0,1][0,1] | x1=0.5x1=0.5 |
| [1,2][1,2] | x2=1.5x2=1.5 |
| [2,3][2,3] | x3=2.5x3=2.5 |
| ⋮⋮ | ⋮⋮ |
| [9,10][9,10] | x10=9.5x10=9.5 |
The n=10n=10 midpoints for Example 1 are
x1=0.5,x2=1.5,x3=1.5,…,x10=9.5x1=0.5,x2=1.5,x3=1.5,…,x10=9.5
Make your own [similar] tables for Examples 2 and 3. Be careful and take note of the values that bb and nn are equal to in each of these examples. You are welcome to use "dots" if the pattern is apparent.
Give a formula that finds the iith midpoint in Example 2.
Step 4: Find the net savings associated to each of the subintervals in Example 1.
Your answer will again be in table form. Read on to see how to do this:
For Example 1, the rate of net savings over the iith subinterval is given by
f′(xi)=S′(xi)−C′(xi)f′(xi)=S′(xi)−C′(xi)
Note that the rate of net savings has the following units: net savingslength of timenet savingslength of time
The length of time is the same for each subinterval: dx=b−andx=b−an
Remember how rate×time=distancerate×time=distance ? In this case, we instead have
rate×time=net savingsrate×time=net savings .
This means
f′(xi)dxf′(xi)dx is the approximate net savings over the iith subinterval.
Here is the table of approximations of net savings (over each subinterval) for Example 1:
| iith subinterval | xixi : iith midpoint | f′(xi)f′(xi) : iith rate of net savings | f′(xi)dxf′(xi)dx : iith net savings amount (approx) |
| [0,1][0,1] | x1=0.5x1=0.5 | f′(0.5)=?f′(0.5)=? | f′(0.5)⋅dx=?f′(0.5)⋅dx=? |
| [1,2][1,2] | x2=1.5x2=1.5 | f′(1.5)=?f′(1.5)=? | f′(1.5)⋅dx=?f′(1.5)⋅dx=? |
| [2,3][2,3] | x3=2.5x3=2.5 | f′(2.5)=?f′(2.5)=? | f′(2.5)⋅dx=?f′(2.5)⋅dx=? |
| ⋮⋮ | ⋮⋮ | ⋮⋮ | ⋮⋮ |
| [9,10][9,10] | x10=9.5x10=9.5 | f′(9.5)=?f′(9.5)=? | f′(9.5)⋅dx=?f′(9.5)⋅dx=? |
Please fill out each entry with a "?" in this table.
Please watch the June 9th presentation for a different example that shows how to put this table together.
This means that
f(10)≈f(0)+sum of all the f′(xi)dxf(10)≈f(0)+sum of all the f′(xi)dx
Use Desmos to compute the sum (see the June 9th presentation)
Find the [similar] table that is associated to Example 2 using Desmos.
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