5128_Ch03_pp098-184

Section 3.5 Derivatives of Trigonometric Functions 145
d
tan x sec d x
2 x,
d
cot x csc d x
2 x,
d
sec x sec x tan x
d x
d
csc x csc x cot x
d x
EXAMPLE 4
Finding Tangent and Normal Lines
Find equations for the lines that are tangent and normal to the graph of
f x tan x
x
at x 2. Support graphically.
SOLUTION
Solve Numerically Since we will be using a calculator approximation for f 2 anyway,
this is a good place to use NDER.
We compute tan 22 on the calculator and store it as k. The slope of the tangent line
at 2, k is
NDER ( tan x
,
x
2) ,
y 1 = tan (x)/ x
y 2 = 3.43x – 7.96
y 3 = –0.291x – 0.51
which we compute and store as m. The equation of the tangent line is
y k mx 2, or
y mx k 2m.
Only after we have found m and k 2m do we round the coefficients, giving the
tangent line as
y 3.43x 7.96.
The equation of the normal line is
1
y k x 2, or
m
y m
1 x k m
2 .
X=2
Y=–1.09252
[–3/2, 3/2] by [–3, 3]
Figure 3.40 Graphical support for
Example 4.
Again we wait until the end to round the coefficients, giving the normal line as
y 0.291x 0.51.
Support Graphically Figure 3.40, showing the original function and the two lines,
supports our computations. Now try Exercise 23.
EXAMPLE 5
Find y if y sec x.
SOLUTION
A Trigonometric Second Derivative
y sec x
ysec x tan x
d
ysec x tan x
d x
d
d
sec x tan x tan x sec x
d x
d x
sec x sec 2 x tan x sec x tan x
sec 3 x sec x tan 2 x Now try Exercise 36.