Math 165 M2
Your Name Below: (MUST BE WRITTEN CLEARLY)
September 23, 2008
HW 1
(You must show your work fully completely to gain the 3 extra credits.)
Due on Sep 23, No late HW will be accepted!
1. (Extra points: 2 )
Prove that the curves y=√2sin xand y=√2 cos xintersect at right angles at a certain point
with 0 < x < π/2.
Proof: It is sufficient to show that the tangent lines of the two functions at intersections per-
pendicular to each other. First, we isolate the intersection by
√2 sin x=√2 cos x⇒tan x= 1 ⇒x=kπ +π
4
where k= 0,±1,±2,···. Then, the slopes of tangent lines of √2 sin xat x=kπ +π
4is
√2 cos(kπ +π
4) = (−1)|k|1,
while the slopes of tangent lines of √2 cos xat x=kπ +π
4is
−√2 sin(kπ +π
4) = (−1)|k|+11.
Thus, for each k, the product of the slopes of two tangent lines is
(−1)|k|1·(−1)|k|+11 = (−1)
which implies the tangent lines are perpendicular to each other. We thus show that the two curves
intersect at right angles at all of the intersections. Q.E.D.
2. (Extra points: 1 )
Find ALL points on the graph of y=f(x) = x−sin xwhere the tangent line is horizontal.
Solution: If the tangent line is horizontal, then the slope must be zero at that point. Hence,
we have
mt=f0(x) = 1 −cos x= 0 ⇒cos x= 1 ⇒x= 2kπ
where k∈Z. Hence, all of the points, at which the tangent line of f(x) is horizontal, are
(2kπ, f (2kπ)) = (2k π, 2kπ),where k∈Z.
1