Calculus for engineering: read the trace, then the area
Introduzido| English | Significado |
|---|---|
| displacement/dɪˈspleɪsmənt/ | the net change in position, s(b) minus s(a) |
| derivative/dɪˈrɪvətɪv/ | the instantaneous rate of change, slope of the tangent |
| volume of revolution/ˈvɒljuːm ɒv ˌrevəˈluːʃn/ | The volume of a solid generated by rotating a plane region about an axis. |
The trolley trace, three ways
- The rig's trolley runs $s(t) = t^3 - 6t^2 + 9t$ metres. The certification form wants displacement, distance, velocity and acceleration from one trace - and mixing them certifies the wrong thing.
- Derivative 导数: $v = ds/dt$, $a = dv/dt$. The zeroes of $v$ partition the story.
Displacement reads endpoints, distance reads area
- $v = 3t^2 - 12t + 9$ vanishes at $t = 1$ and $t = 3$; $s(0,1,3,4) = 0, 4, 0, 4$.
- Displacement $0 \to 4$ s $= s(4)-s(0) = 4$ m; distance $= \int|v| = 4 + 4 + 4 = 12$ m.
Which number does the form want?
Route each request to the quantity that answers it.
v(t) = 3t^2 - 12t + 9. v(2), and the times it is zero?
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v(2) = 12 - 24 + 9 = -3; zeroes at t = 1 and 3.
Displacement 0 to 4 s: s(t) = t^3 - 6t^2 + 9t?
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s(4) - s(0) = 4 - 0.
Distance over the same interval?
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4 forward + 4 back + 4 forward.
Match each pair of times.
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The graph is a dictionary.
Fillets and the second way
- The mount's blend $y = x(4-x)$ revolved 0 to 2 m: $V = \pi \times 17.067 = 53.62$ m3 - analytic integral and 400,001-point quadrature agree to four decimals.
- The dimensional check: $[t^3] =$ m makes $[3t^2] =$ m/s; a disagreeing term is an error.
pi times the integral of (x(4-x))^2 from 0 to 2 (two decimals)?
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pi x 17.067 = 53.62.
a(t) = 6t - 12 is zero at t = 2. There:
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dv/dt = 0 means v peaks or dips - here v = -3, its minimum.
If a term of an equation carries m/s while the others carry m, that is a style choice.
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It is an error - the dimensional check exists to catch it.
Three calculus slips
- Reporting displacement as distance. Find the zeroes of v first.
- Dropping units mid-derivation. The dimensions audit the algebra.
- Trusting an unevaluated integral on a drawing. Check it a second way.
Every integral worth ordering steel against is checked a second way.
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Analytic then numeric - the fillet agrees to four decimals.
Recap
- $v = 3t^2 - 12t + 9$; zeroes at 1 and 3; displacement 4 m, distance 12 m.
- Volume of revolution: $\pi \int y^2 dx = 53.62$ m3, checked two ways.
- Dimensional check every line; sign check every area.