<p>1. For the math prob,</p>
<ul>
<li><p>Yes, you're right, the function can not be a line</p></li>
<li><p>We see that the first three points of the table (10,6); (20,12.5); 'n (30,19) lie on the line (d): y = 0.65x - 0.5</p></li>
</ul>
<p>If the relationship between x and y represents by the function y = f(x)
=> f(x) intersects line (d) at at least 3 points
=> the equation: f(x) - (0.65x-0.5) = 0 has at least 3 roots. (*)</p>
<p>Here, we have two cases:
a) If f(x) has polynomial form: (<strong>)
Because of (*) and (</strong>), the degree of f(x) >= 3 </p>
<p>Furthermore, there are only four given points of f(x) => the problem just can be solved if the degree of f(x) is 3. (1)</p>
<p>=> f(x) = A3.X^3 + A2.X^2 + A1.X^1 + A0 (***) </p>
<p>Substitute values of (x,y) into equation (***), we have</p>
<p>6 = A3.10^3 + A2.10^2 + A1.10^1 + A0
12.5 = A3.20^3 + A2.20^2 + A1.20^1 + A0
19 = A3.30^3 + A2.30^2 + A1.30^1 + A0
25 = A3.40^3 + A2.40^2 + A1.40^1 + A0</p>
<p>I myself solved with this system of equations and it has no root at all.</p>
<p><a href="Note:%20I%20wanna%20clarify%20more%20about%20(1).%20If%20the%20degree%20of%20f(x)%20%3E=%204,%20then%20we'll%20have%20%3E=%205%20variables%20in%20the%20system%20of%20equations%20(2).%20In%20other%20words,%20we%20have%204%20equations%20with%20%3E=%205%20variables%20-%20that%20system%20of%20equations%20can%20not%20be%20solved">i</a> *</p>
<p>b) If f(x) doesn't have polynomial form
This case is too complex to examine because f(x) can be anything else (fraction, square root or combination of them, etc.)</p>
<p>Conclusion I strongly believe that there are some errors in the subject. For instance, just change the last point (40,25) into (40,25.5), then we have nothing to worry about this prob.</p>
<p>Meanwhile, the problem asks about the slope of the graph. If f(x) is not a line, then the idea "the slope of the graph" is nonsense. We only have "The derivative of the function at a point is the **slope of the line tangent* to the curve at the point"*</p>
<p>==> All of my attempt to solve this is useless, hehe. Thus, Redwood, you don't need to think about this prob anymore. It doesn't make any sense. </p>
<p>2. I do not have a official definition of tolerance in engineering. Yet, here is what I think:
No matter what type of a measuring instrument, it can not measure exactly 100%. There must be slight difference between the real number and the nominal measure (nominal measure is the number you can read from a measuring instrument). So, tolerance tells us about that difference. </p>
<p>It can look like this: 10±0.1 -- which means if the nominal measure you read is 10, the real number is between 0.9 and 10.1 </p>
<p>or 10+0.2/−0.1 -- which means that the real number is between 0.9 and 10.2</p>
<p>Hope you find my writing is readable. :)</p>