Inverse Trig Functions & Derivatives of Trig Functions

Transcription

1 Inverse Trig Functions & Derivatives of Trig Functions Cole Zmurchok Math 102 Section 106 November 28, 2016

2 Today More Trig Function Derivatives 2. Inverse Trig Functions 3. Related Rates with Trig Functions Course Evaluations: do these for all of your courses! Let me know if you need the website. WW 12 due Thursday Final Exam: Dec 12, 8:30 am

3 Warm-up Q1. What is the period, T, of h(t) = 8 6 sin(4t)? A. T = 4 B. T = 1/4 C. T = 1 D. T = 2π E. T = π/2

4 Warm-up Q2. What is the amplitude of h(t) = 8 6 sin(4t)? A. 14 B. 8 C. 12 D. 6 E. -6

5 Warm-up Q3. What is the derivative of tan(e x2 )? A. sec 2 (e x2 ) B. sec 2 (e x2 )e x2 C. sec 2 (e x2 )e x2 x 2 D. sec 2 (e x2 )e x2 2x E. sec 2 (e x2 )e x2 x 2 2x

6 Derivatives of Trigonometric Functions We already know that d dx d dx sin x = cos x cos x = sin x We used the quotient rule to find d dx tan x = d ( ) sin x = sec 2 x dx cos x

7 Derivatives of Trigonometric Functions What is the derivative of cot(x)? d dx cot x = d dx = ( cos x ) sin x sin x sin x cos x cos x sin 2 x = 1 sin 2 x = csc 2 x

8 Inverse Trigonometric Functions f(x) = sin x f 1 (x) = arcsin(x) the angle whose sine is x Inverse Trigonometric functions 1 x θ 1 x 2 Figure The This above triangle triangle has been is constructed so sothat that θ is an angle wh =x. Thismeansthatθ =arcsin(x) sin θ = x This means that θ = arcsin x

9 Inverse Trigonometric Functions Q4. Simplify the expression y = tan(arcsin(x)) A. 1 x 2 B. x 1 x 2 C. 1 x D. x E. 1 x 2 x

10 Inverse Trigonometric Functions The functions sin(x) and arcsin(x), reverse(or invert )eachother seffect, Inverse Trigonometric functions y = tan(arcsin(x)) 1 x θ 1 x 2 θ = arcsin(x) Figure This triangle has been constructed so that θ is an angle wh =x. Thismeansthatθ =arcsin(x) y = tan θ = x 1 x 2

11 Inverse functions

12 Inverse functions on restricted domains

13 On restricted domains f(g(x))=x and g(f(x))=x

14 Inverse Trigonometric Functions f(x) = sin x f 1 (x) = arcsin(x) the angle whose sine is x Because sin x is periodic, we need to restrict the domain to define an inverse function. zx6c0htith

15 (2) Domains The functions sin(x) and arcsin(x) are inverse functions on the following domains: (A) π x π and -1 x 1 (B) π/2 x π/2 and -1 x 1 (C) -1 x 1 and π/2 x π/2 (D) -1 x 1 and π x π (E) π/2 x π/2 and π/2 x π/2

16 Derivatives of Inverse Trigonometric Functions Q6. If y = arcsin x, with 1 x 1, what is dy dx? A. arccos x B. arctan x C. 1 x 2 D. 1 1 x 2 E. 1 1+x 2

17 Derivatives of Inverse Trigonometric Functions Q6. If y = arcsin x, with 1 x 1, what is dy dx? A. arccos x B. arctan x C. 1 x 2 D. 1 1 x 2 E. 1 1+x 2

18 Derivatives of Inverse Trigonometric Functions 1. y = arcsin x 2. Inverse: x = sin y 3. Implicit Differentiation: 1 = cos y dy dx 4. Use identity or triangle to write cos y = 1 sin 2 y = 1 x 2 5. Solve for dy dx : dy dx = 1 1 x 2

19 Extra Practice 1. What are the domains of f(x) = tan x and f 1 (x) = arctan x? 2. Show that the derivative of arctan x is 1 1+x 2

20 Derivatives of Inverse Trig Functions

21 Subtle points! Inverse trigonometric functions are only inverses of the trigonometric functions on their restricted domains! Note that ( arcsin sin ( )) 5π = π 6 6

22 Sin(5π/6)=1/2

23 Arcsin(1/2)=π/6

24 arcsin(sin(x)) x In this example, we found that: arcsin(sin(5π/6))= π/6 5π/6 However, on the restricted domains, π/2 x π/2 and -1 x 1 arcsin(sin(x))=x and sin(arcsin(x))=x.

25 Trig and related rates 1. If the height of an isosceles triangle with base 2m changes at a rate of 3 m/s, how quickly is the angle opposite the base changing when the height is 3 m? 2. A ladder of length L is leaning against a wall so that its point of contact with the ground is a distance x from the wall, and its point of contact with the wall is at height y. The ladder slips away at a constant rate C. How fast does the angle between the ladder and wall change?

26 Summary Using differentiation rules, we can find the derivatives of other trigonometric functions Inverse trigonometric functions also have derivatives Triangles and identities are useful when simplifying inverse trigonometric relationships Caution required: when thinking about the domains of inverse trig functions Trigonometric functions are useful in related rates problems

27 Answers 1. E 2. D 3. D 4. B 5. B

28 Related Exam Problems 1. Use Newton s method to find the smallest positive critical point of the function g(x) = e x sin(10x)