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

MAT 137 — LEC 0601 Instructor: Alessandro Malusà 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 ′ ?

Definition of arctan 1 Sketch the graph of tan.

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

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

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 of 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 ⇐ ⇒ ???

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 of 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?

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 of 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) 4 arctan tan − 6) � � 2 arctan tan (3) � � 5 tan arctan (0) � �� � π 3 arctan tan � � 2 6 tan arctan (10)

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

Computations - Inverse trig functions Compute the derivatives of these functions, and simplify them as much as possible: � x 3 / 2 � 1 f ( x ) = arcsin 2 f ( x ) = 2 x 2 arctan( x 2 ) − ln( x 4 + 1)

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