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lecture15

# lecture15 - tan x sec x Find f x and each x-value for which...

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Lecture 15: Derivatives of Trigonometric Functions (Chapter 3, Section 15) ex. Find the following limits: 1) lim ± ! 0 sin ± = 2) lim ± ! 0 cos ± = To diﬁerentiate the trig functions we need to ﬂnd the important limit: lim ± ! 0 sin ± ± .

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2 Recall (from lecture 2) the area of the sector of a circle with central angle ± and radius r is given by A = 1 2 r 2 ± . To evaluate lim ± ! 0 sin ± ± , consider the following sector of a circle with radius 1, and the related areas:
3 We can then evaluate: lim ± ! 0 cos ± ¡ 1 ± ex. Evaluate the limit: lim x ! 0 tan x x .

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4 ex. Let f ( x ) = 8 < : sin x x < 0 1 ¡ cos x x 0 : Is f ( x ) continuous at x = 0? Find f 0 (0) if possible.
5 Evaluate the following limits: ex. lim x ! 0 sin 8 x x ex. lim x ! 0 sin 2 ( …x ) 3 x 2

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6 Evaluate the limits: ex. lim x ! 0 cos x ¡ 1 sec x ¡ 1 = ex. lim x ! 0 ¡ sin x 1 ¡ cos x
7 Evaluate the following limits using substitution: ex. lim x ! 2 sin(cos x ) cos x ex. lim x ! sin x x ¡

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8 Derivatives of Trigonometric Functions 1) d dx (sin x ) = 2) d dx (cos x ) =
9 3) d dx (tan x ) = 4) d dx (cot x ) =

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10 5) d dx (sec x ) = 6) d dx (csc x ) = ex. Evaluate: d dx ( x cot x )
11 ex. Find the slope of the tangent line to f ( x ) = cos x 4 + tan x at x = 4 .

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12 ex. Let f ( x ) = 1 ¡

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Unformatted text preview: tan x sec x . Find f ( x ) and each x-value for which the graph of f has a horizontal tangent line. 13 ex. A 60 foot ±agpole casts a shadow that changes with the sun’s elevation. If y is the length of the shadow and ± is the angle cast by the sun, ﬂnd the rate at which the length of the shadow is changing with respect to ± when ± = … 4 . 6-¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ± 14 Try these!! 1. Evaluate the limits: (a) lim x ! sin 3 x tan 3 x (b) lim x ! 1 ¡ cos x x 2 (c) lim x ! 2 sin( x ¡ 2) x 2 + x ¡ 6 (d) lim x ! … 2 cos x x ¡ … 2 (Use a substitution, trig identity) 2. Find the equation of the tangent line to f ( x ) = e x sec x at x = 0. 3. Find each point at which f ( x ) = cos x 2 + sin x has a horizontal tangent line on the interval [0 ; 2 … ]....
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lecture15 - tan x sec x Find f x and each x-value for which...

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