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547=4 · 2020. 11. 29. · Example 7: Find the 51st derivative of f(x)=sinx. Specifically, find...
Transcript of 547=4 · 2020. 11. 29. · Example 7: Find the 51st derivative of f(x)=sinx. Specifically, find...
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LECTURE: 3-3 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Recall last time we found ddx (sinx) = cosx and d
dx (cosx).
Example 1: Using the derivative of sinx and cosx find derivatives of:
(a) y = cotx (b) y = cscx
Derivatives of Trigonometric Functions:
• ddx (sinx) =
• ddx (cosx) =
• ddx (tanx) =
• ddx (cscx) =
• ddx (secx) =
• ddx (cotx) =
Example 2: Find the second derivatives of the following functions:
(a) g(t) = 4 sec t+ tan t. (b) y = x2 sinx.
Example 3: Find an equation of the tangent line to the curve y =1
sinx+ cosxat the point (0, 1).
UAF Calculus I 1 3-3 Derivatives of Trigonometric Functinos
= COSI =I
six six
y'
I sinxccosx )'
- co , xc six )'
y)
=six ¥-1
( sin
Tsin Tsin
= - sin2x-cos= -s.IS#x=-eotxcsaxJSin2x
= =-c
cos X - cotxcscx- sinx secxtonx
seek - csix
first•
547=4sect Emttsec 't y
'= 2xsinx + x7osx
=sect(4tuttsec =x(2sin*txcos
yr =Cinxtcosx )C0 ) - I ( cosy - six )
-
-( sinxtcosx )2
=
sinx-cosx-x-oi.jcd-syjfo.IO?-o,s=-Is=-I(sinxtcosxP
y - I = - I ( x - O )
y-
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Example 4: For what values of x does the graph of f(x) = x+ 2 sinx have a horizontal tangent?
Example 5: Differentiate f(x) =secx
1� tanxand determine where the tangent line is horizontal.
Generalized Product Rule: How does the product rule genearlize to more than two functions? For example, whatis the derivative of y = f(x)g(x)h(x)?
Example 6: Differentiate h(✓) = ✓2 tan ✓ sec ✓.
UAF Calculus I 2 3-3 Derivatives of Trigonometric Functinos
f'
C D= I +2 cos x = O
con .⇒ . "¥x:÷÷÷ ⇒ .
f'
C ⇒ = ( I - tax ) Csecxtonx ) - secx ( - Seis )-
Cl - tan x )-
cats ? I
=secx ( tax - tank t seok ) I + tar ? see
'
-
( I - tax )2
= Secx ( tax + D see -
- O X - Tues- ten x -11=0
( I - tan xpfan ×=
- I
¥374x=3tf C x ) . ( s hlx ))
powerrule within power rule !
D= f G) ( gcx ) .li Ix ) -15634×7 ) -1 f' Cx ) ( glx )hLx7 )
( )
h' ④ = of ( tmo seco
't
20-tmo-se-O-fftano.seco-tuo-tsefo-seco-3-2.frtonoseco
÷:÷::::÷÷:÷:
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Example 7: Find the 51st derivative of f(x) = sinx. Specifically, find the first four or five derivatives and look fora pattern.
Example 8: A mass on a spring vibrates horizontally on a smooth level surface. Its equation of motion is x(t) =8 sin t, where t is in seconds and x is in centimeters.
(a) Find the velocity at time t.
(b) Find the position and velocity of the mass at time t = 2⇡/3. In what direction is it moving at this time?
Example 9: A ladder 12 feet long rests against a vertical wall. Let ✓ be the angle between the top of the ladder andthe wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides awayfrom the wall, how fast does x change with respect to ✓ when ✓ = ⇡
6 .
UAF Calculus I 3 3-3 Derivatives of Trigonometric Functinos
f'
L x ) -_ cos Cx )
f-"
( × ) = - sin X hmm . 51 is odd .Either
Cos x or - cos x
f" '
( x ) = - cost52 is divisible by
,so is
fu ) ( ⇒ = sire
y
so fC5' KD =f' "
C x ) = -c
positive x ,to the right
x' G) = 8 cost
--
X (2 "G ) x
'( 273 )
position : x ( F) = 8 sin (2*5)=8 Fsz =453cLvelocity : x'
(F) = 8 cos = 8C -
I
s ) = -4cnLeft
- - de
sin ⑤ = ¥do
- ⇒ x = 12 sinox
diff -- If C 12 sin A
= 12 cos ⑦
x' ( %) = 12 cos €1213=6r3