Assignment Homework #4 - MATH 1014 Section A
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Assignment Homework #4 - MATH 1014 Section A
1. Consider the curve C with parametric equations x(t) = cos(2t),y(t) = sin(t), where − ≤ t ≤ .
a) Find a Cartesian equation for C . Then make a rough sketch of the curve.
b) The curvature κ of a curve C at a given point is a measure of how quickly the curve changes direction at that point. For example, a straight line has curvature κ = 0 at every point. At any point, the curvature can be calculated by
' d2 y '
( 1 + ( )2 ) 2 .
Show that the curvature of the curve C is:
κ(t) = 4
2. Consider the polar curves C1 : r = 2sinθ and C2 : r = 2cosθ, where 0 ≤ θ ≤ 2π .
a) Find the Cartesian equations of C1 and C2 and make a neat sketch of these curves.
b) Using polar equations of C1 and C2 to find their intersection points (Do not use their Cartesian equations!).
c) Using polar equations of C1 and C2 to show that they intersect each other at right angles (Do not use their Cartesian equations!).
3. Flower power! Find the area of the region outside r = , but inside r = cos(2aθ), where a is a positive integer. As part of your solution, you should include a sketch of the area for a = 2 and a = 3. (Hint: you may want to plot this using computer graphing software before you start in order to plan out your solution!).
4. Show that the following series are convergent and find their sums:
a) + + ... + + ... b) 习
5. The Cauchy-Schwarz inequality is considered one of the most important and widely used inequalities in mathematics. This inequality states that
pi qi ≤ pq
Now, assume that the series 对 an with nonnegative terms an ≥ 0 is a convergent series. Use
the Cauchy-Schwarz inequality and show that the series 对 is convergent.
2022-07-20