More Of The Same

6. Find the arc length for the spiral whose polar equation is $r = e^\theta$ over the interval $\theta \in [0, 1]$.

The “key example”: the easiest problem of its type. Should this be on a final exam? Heck no, it should be assigned early on and then referred to often. Here it is now. The old “unchanged by differentiation” trick…

5. Find a formula for the slope of the tangent to the “epicycloid” whose parametric equations are $x = 5cos(t) - cos(5t)$ and $y=5sin(t)-sin(5t)$.

Routine; I passed over it in near-silence on the review day. Also… owen by the way… played ‘em ILMB and For John Henry.

4.Recall that the Taylor Series for the exponential function is $e^x = \sum_{n=0}^\infty {{x^n}\over{n!}}$. Use the $3^{rd}$ Taylor polynomial $P_3$ to obtain a (rational number) estimate for $e^2$ (write out the first four terms of the series; put $x=2$; simplify). Find a reasonable upper bound on the “error term” $R_3$ for this estimate.

The class that had this final didn’t “get” the control-the-error parts of the course; this one I didn’t even really try. You should… whoever you are… but if you’re my student, I’ll go ahead and admit there won’t be an “error term” problem on this quarter’s exam. Be able to write out a Taylor Series and use it in estimating a number; the rest, for us, is gravy.

More still to come I imagine.

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One Response to “More Of The Same”

1. Kavin Says:

Maths too hard for me when i was in school and college i used to get low marks in maths the formulas it have o my god.