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Warm Up: pg 164 #59-61, 63, 64 and pg 156 QQ #4 61) A.

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Presentation on theme: "Warm Up: pg 164 #59-61, 63, 64 and pg 156 QQ #4 61) A."β€” Presentation transcript:

1 Warm Up: pg 164 #59-61, 63, 64 and pg 156 QQ #4 61) A

2 Derivatives of Inverse Trig Functions
Ch 3.8 Derivatives of Inverse Trig Functions

3 Review from PreCalc: sin(ΞΈ) = π‘œπ‘π‘ β„Žπ‘¦π‘
a ratio an angle 60Β° 2 1 EX: sin(30Β°) = 1 2 30Β° 3 But inverse trig asks the opposite question EX: Arcsin( 1 2 ) = find the angle whose ratio is 1 2 =30Β°

4 The 3 main Inverse Trig Functions

5 Remember the restricted ranges …
Sine and Tangent are in Quadrant I and IV Cosine is in Quadrant I and II

6 The Reciprocal Function is NOT the inverse function
Inverse notation: Reciprocal Notation: arcsin(ratio) or 𝑠𝑖𝑛 βˆ’1 (π‘Ÿπ‘Žπ‘‘π‘–π‘œ) 1 sin⁑(30Β°) = csc(30Β°) This does not mean reciprocal

7 What is the derivative of the arcsine?
** YOUR FRQ on the next TEST will include deriving one of the 3 main trig functions We find the derivative of y = arcsin(x) as follows: Start with y = sin(x) Inverse: x = sin(y) Take d/dx = cos(y)y’ Isolate y’ cos⁑(𝑦) =𝑦′ Replace cos(y) With info from βˆ’ π‘₯ 2 =𝑦′ Triangle picture 1 x y 1βˆ’ π‘₯ 2 We know y = arcsin(x) only exists between - Ο€ 2 ≀ y ≀ Ο€ 2 and -1 < x < 1

8 What is the derivative of the arctan?
** YOUR FRQ on the next TEST will include deriving one of the 3 main trig functions Start with y = tan(x) Inverse: x = tan(y) Take d/dx = secΒ²(y)y’ Isolate y’ secΒ²(𝑦) =𝑦′ Replace cos(y) With info from π‘₯Β² =𝑦′ Triangle picture 1βˆ’ π‘₯ 2 x 1

9 You Try! y = arccos(x) Start with y = cos(x) Inverse: x = cos(y)
Take d/dx = -sin(y)y’ Isolate y’ sin(𝑦) =𝑦′ Replace cos(y) With info from βˆ’ π‘₯ 2 =𝑦′ Triangle picture 1 1βˆ’ π‘₯ 2 x

10 What are the formal definitions of the inverse trig functions”
If u is a differentiable function of x with |u| < 1, we apply the chain rule to get: 𝑑 𝑑π‘₯ 𝑠𝑖𝑛 βˆ’1 𝑒= 1 1βˆ’π‘’Β² 𝑑𝑒 𝑑π‘₯ 𝑑 𝑑π‘₯ π‘π‘œπ‘  βˆ’1 𝑒=βˆ’ 1 1βˆ’π‘’Β² 𝑑𝑒 𝑑π‘₯ 𝑑 𝑑π‘₯ π‘‘π‘Žπ‘› βˆ’1 𝑒= 1 1+𝑒² 𝑑𝑒 𝑑π‘₯ 𝑑 𝑑π‘₯ π‘π‘œπ‘‘ βˆ’1 𝑒=βˆ’ 1 1+𝑒² 𝑑𝑒 𝑑π‘₯ 𝑑 𝑑π‘₯ 𝑠𝑒𝑐 βˆ’1 𝑒= 1 |𝑒| 𝑒 2 βˆ’1 𝑑𝑒 𝑑π‘₯ 𝑑 𝑑π‘₯ 𝑐𝑠𝑐 βˆ’1 𝑒=βˆ’ 1 |𝑒| 𝑒 2 βˆ’1 𝑑𝑒 𝑑π‘₯

11 Find Full Assignment Online

12 EX 1: Applying the Arcsine formula
a) Find 𝑑 𝑑π‘₯ ( 𝑠𝑖𝑛 βˆ’1 ( 3 π‘₯Β² )) b) Find 𝑑 𝑑π‘₯ ( 1 𝑠𝑖𝑛 βˆ’1 (2π‘₯) )

13 EX 2: Applying the Arctan formula
A particle moves along the x-axis so its position at any time t β‰₯ 0 is x(t) = π‘‘π‘Žπ‘› βˆ’1 ( 𝑑 ). What is the velocity of the particle when t = 16?

14 You try! 𝑑 𝑑π‘₯ ( 𝑠𝑖𝑛 βˆ’1 (1βˆ’π‘₯)) b) Find 𝑑 𝑑π‘₯ ( 𝑐𝑠𝑐 βˆ’1 ( π‘₯ 2 )) c) 𝑑 𝑑π‘₯ ( 𝑠𝑒𝑐 βˆ’1 (5 π‘₯ 4 )) d) A particle moves along the x-axis so its position at any time t β‰₯ is x(t) = 𝑠𝑖𝑛 βˆ’1 ( 𝑑 4 ). What is the velocity of the particle when t = 4?

15 EX 3: Derivatives of inverse functions
a) Find an equation for the line tangent to the graph of y = tan(x) at the point ( πœ‹ 4 , 1) b) Find an equation for the line tangent to the graph of y = π‘‘π‘Žπ‘› βˆ’1 (x) at the point (1, πœ‹ 4 )

16 What did you notice in the last EX?
*The Derivative of a function is the reciprocal of the derivative of its inverse*

17 EX 4: Derivatives of inverse functions
Let f(x) = π‘₯ 5 +2 π‘₯ 3 +π‘₯βˆ’1. Find f(1) b) Find f’(1) c) Find 𝑓 βˆ’1 (3) d) Find 𝑓 βˆ’1 (3)β€² So for #28: F(1) = 3 f’(1) =12 (3,1) f’(3) = 1/12

18 You Try! Let f(x) = cos π‘₯ +3π‘₯ Find f(0) b) Find f’(0)
c) Find 𝑓 βˆ’1 (1) d) Find 𝑓 βˆ’1 (1)β€²

19 3.8: pg 170 QR #1-3, 6- 10, EX: #1- 19odd, 24, 26, 29 DIDN’T GET TO 3.8B


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