Mathematical Notation and Mean Value Theorem Problems

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In this assignment you need to explain clearly and in detail how you solved
the problems. Use correct mathematical notation and complete sentences. 1. Calculate the arc length of two familiar curves.
a.Calculate the circumference of a circle of radius r.
b. Find the exact length of the spiral defined by r(t) =< t cos(t), tsin(t), t >
on the interval [0, 2π]. 2. We can adapt the arc length formula to curves in 2-space that define
y as a function of x: Let y = f(x) define a smooth curve in 2-space.
Parameterize this curve and use the definition of arc length to show
that the length of the curve defined by f on an interval [a, b] is
Z b
a
p
1 + [f
0
(x)]2 dx 3. Find the arc length function s(t) for the helix r(t) =< 3 cos(t), 3 sin(t), 4t > ,
t ≥ 0. Then, use the relationship between the arc length and the parameter t to find an arc length parameterization of r(t) in terms of
s.

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In this assignment you need to explain clearly and in detail how you solved
the problems. Use correct mathematical notation and complete sentences.
1. Calculate the arc length of two familiar curves.
a.Calculate the circumference of a circle of radius r.
b. Find the exact length of the spiral dened by r(t) =< t cos(t), t sin(t), t >
on the interval [0, 2π].
2. We can adapt the arc length formula to curves in 2-space that dene
y as a function of x: Let y = f (x) dene a smooth curve in 2-space.
Parameterize this curve and use the denition of arc length to show
that the length of the curve dened by f on an interval [a, b] is
Z bp
1 + [f 0 (x)]2 dx
a
3. Find the arc length function s(t) for the helix r(t) =< 3 cos(t), 3 sin(t), 4t > ,
t ≥ 0. Then, use the relationship between the arc length and the parameter t to nd an arc length parameterization of r(t) in terms of
s.
1

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square root

mean value theorem

numerator and denominator

integral changes

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