10.43
esc (3* - 5).
10.44
10.45
For what value of A does 3 sin Ax have a period of 2?
10.46
Find the angle of intersection of the lines 3!,: y = x - 3 and 3! 2 : y = -5x + 4.
10.47
Find the angle of intersection of the tangent lines to the curves xy = 1 and y = x
3
at the common point
(1,1).
68
CHAPTER 10
10.42
cot
2 x.
D x [csc (3x - 5)] = [-esc (3x - 5) cot (3* - 5)] • D,(3x - 5) = -esc (3x - 5) cot (3* - 5) • (3)
= -3 csc (3*-5) cot (3*-5).
Evaluate
Remember that
and use the definition of the derivative.
The angle 0, that .$?, makes with the Jt-axis has a tangent that is equal to the slope of the line. The angle 0 2 that
.S?, makes with the *-axis has a tangent equal to the slope of & 2 . Thus tan 0, = 1 and tan 0 2 = -5. The
angle 0 between ^ and <£ 2 is 0 2 - 0 } . So, tan 6 = tan (0 2 - 0^ =
Reference to a table of tangents reveals that 0 = 56°.
Fig. 10-7
Let 0 l be the angle between the horizontal and the tangent line to y = x
3 , and let 0 2 be the angle between
the horizontal and the tangent line to xy = 1. Now, tan 0, is the slope of the tangent line to y = jc
3
, which is
the derivative of x
3 evaluated at (1,1), that is, 3x
2 evaluated at x = 1 or 3. So, tan 0, = 3. Likewise, since
the derivative of 1/JC is —(1/Jt
2 ), which, when evaluated at x = 1, is —1, we have tan 0 2 = —1. Hence,
A table of tangents yields 0 2 — 0, » 63°.
10.48
Evaluate
Thus, the desired limit is f.
D^cot
2 x) = 2 cot x • D, (cot x) = 2 cot x (-esc
2 *) = -2 cot x esc
2 *.
= [D,(cos x)](ir/3) = -sin (w/3) = -V5/2.
The period p = 2ir/A. Thus, 2 = 2ir/X, 2>l = 27r, yl = IT.
But,
and