MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid - - PowerPoint PPT Presentation

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Aug 01, 2023 170 likes •271 views

MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid November 5th, 2020 sin( x ) . Warm-up question : Consider the function f ( x ) = arcsin 1 What is the domain of f ? 2 Where is f differentiable? 3 What is f ?

SLIDE 1

MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Muhammad Mohid November 5th, 2020 Warm-up question: Consider the function f (x) = arcsin

sin(x) .

1 What is the domain of f ? 2 Where is f differentiable? 3 What is f ′?

SLIDE 2

Definition of arctan

1 Sketch the graph of tan.

SLIDE 3

Definition of arctan

1 Sketch the graph of tan. 2 Prove that tan is not one-to-one.

SLIDE 4

Definition of arctan

1 Sketch the graph of tan. 2 Prove that tan is not one-to-one. 3 Select the largest interval containing 0 such that the restriction

f tan to it is (defined and) one-to-one.

SLIDE 5

Definition of arctan

1 Sketch the graph of tan. 2 Prove that tan is not one-to-one. 3 Select the largest interval containing 0 such that the restriction

f tan to it is (defined and) one-to-one. We define arctan as the

inverse of this restriction. Let x, y ∈ R arctan y = x ⇐ ⇒ ???

SLIDE 6

Definition of arctan

1 Sketch the graph of tan. 2 Prove that tan is not one-to-one. 3 Select the largest interval containing 0 such that the restriction

f tan to it is (defined and) one-to-one. We define arctan as the

inverse of this restriction. Let x, y ∈ R arctan y = x ⇐ ⇒ ???

4 What is the domain of arctan? What is the range of arctan?

SLIDE 7

Definition of arctan

1 Sketch the graph of tan. 2 Prove that tan is not one-to-one. 3 Select the largest interval containing 0 such that the restriction

f tan to it is (defined and) one-to-one. We define arctan as the

inverse of this restriction. Let x, y ∈ R arctan y = x ⇐ ⇒ ???

4 What is the domain of arctan? What is the range of arctan?

Sketch the graph of arctan.

5 Compute 1 arctan

tan (1)
2 arctan
tan (3)
3 arctan
tan

π 2

4 arctan
tan
−6)
5 tan
arctan (0)
6 tan
arctan (10)

SLIDE 8

Derivative of arctan

Obtain (and prove) a formula for the derivative of arctan. Hint: Call f (t) = arctan t and differentiate ∀t ∈ . . . tan(f (t)) = t

SLIDE 9

Computations - Inverse trig functions

Compute the derivatives of these functions, and simplify them as much as possible:

1 f (x) = arcsin

x3/2

2 f (x) = 2x2 arctan(x2) − ln(x4 + 1)

SLIDE 10

Before next class...

Watch videos 5.2, 5.3, and 5.4.
Download the next class’s slides (no need to look at them!)