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EXAMPLE 5
Connecting the derivative of a sinusoidal function to the slope
of a tangent
Determine the equation of the tangent to the graph of y x cos 2x at x p 2 .
Solution
When x p y p cos p p 2 , 2 2 .
NEL
The point of tangency is Q p 2 , p 2 R.
The slope of the tangent at any point on the graph is given by
dy
dx dx
d1cos 2x2
cos 2x x
dx dx
At
1121cos 2x2 x1sin 2x2122
cos 2x 2x sin 2x
x p dy
cos p p1sin p2
2 , dx
1
The equation of the tangent is
(Product and chain rules)
(Simplify)
(Evaluate)
y p 2 a x p 2 b
or y x.
EXAMPLE 6
Connecting the derivative of a sinusoidal function to its extreme values
Determine the maximum and minimum values of the function f 1x2 cos 2 x on
the interval x30, 2p4.
Solution
By the algorithm for finding extreme values, the maximum and minimum values
occur at points on the graph where f ¿ 1x2 0 or at endpoints of the interval.
The derivative of f 1x2 is
f ¿1x2 2 1cos x21sin x2
2 sin x cos x
sin 2x
Solving f ¿1x2 0,
sin 2x 0
sin 2x 0
2x 0, p, 2p, 3p, or 4p
so x 0, p or 2p
2 , p, 3p
2 ,
(Chain rule)
(Using the double angle identity)
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5.4 THE DERIVATIVES OF y sin x AND y cos x