1. Find the radius and the interval of convergence of the power series
k (x − 1)k
2. Find the Taylor polynomial of f(x) = √
x + 1 of degree 4 about x = 0.
3. Find the Taylor series of f(x) = 1
about x = 1 and write it using summation notation.
4. Find the Taylor series of the following functions about x = 0 and write them using
summation notation. Determine the values of x for which the series representation is valid.
a) f(x) = x
b) f(x) = x
· tan−1 x
5. Use Taylor's Theorem to find degree n of the Taylor polynomial of the cosine function
cos x about x = 0 that is needed to guarantee 8 decimal digits accuracy in calculating
cos 1. Verify your result by finding the error En(1) = cos 1 − Pn(1) using a calculator.
6. Find values of ω which x(t) = cos(ωt) satisfies the dierential equation
+ 9x = 0
7. Use Euler method with step size ∆t = 0.1 to find approximating points (x1, y1),(x2, y2),(x3, y3)
of the solution curve of the initial value problem dy
dt = y
2 + t, y(0) = 2.
8. Solve the initial value problem
dx = y
2x + x, y(0) = 1 >
In the fifth century B. C., the Greek philosopher Zeno proposed the following paradox
of Achilles (along with other three paradoxes, the Dichotomy, Arrow, and the Stade, as
retold by Aristotle). The Achilles paradox states that, given head start, tortoise will never
be overtaken by the legendary runner Achilles because, Achilles must first reach the point
P at which the tortoise started, and by the time Achilles arrives at P1. the tortoise will
have advanced to point P2, which Achilles must reach before overtaking the tortoise. By
the time Achilles has reached P1, the tortoise will have moved ahead to point Pa, and so
on, ad innitum. Thus Achilles will never be able to overtake the tortoise.
Obviously this contradicts our common sense, but where is the fiaw in this paradox?
Aristotle called this and other paradooses fallacies but was never be able to refute them.
Bertrand Russell said that the paradoxes are "immeasurably subtle and profound".
It took more than 2000 years for mathematicians to create a framework (infinite series) in
which one could undertale the infinite process of calculus free from such logical difficulties.
To quantify the Achilles paradox, suppose Achilles is running at the constant speed of
200 km/hr, and the tortoise at the constant speed km/hr with 40 km head start. Point
out where the fiaw in Zeno's paradox is and show, by using infinite series, that Achilles
will indeed overtake the tortoise (in fact in short amount of time).
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