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

MAT 137 — LEC 0601 Instructor: Alessandro Malusà TA: Muhammad Mohid October 22nd, 2020 Warm-up question : think of examples of 1 A function with a vertical asymptote at x = 2; 2 A function with a vertical tangent line at x = 2.

Write a proof for the quotient rule for derivatives Theorem • Let a ∈ R . • Let f and g be functions defined at and near a . Assume g ( x ) � = 0 for x close to a . • We define the function h by h ( x ) = f ( x ) g ( x ). IF f and g are differentiable at a , THEN h is differentiable at a , and h ′ ( a ) = f ′ ( a ) g ( a ) − f ( a ) g ′ ( a ) . g ( a ) 2 Write a proof directly from the definition of derivative. Hint: Imitate the proof of the product rule in Video 3.6.

Check your proof 1 Did you use the definition of derivative? 2 Are there words or only equations? 3 Does every step follow logically? 4 Did you only assume things you could assume? 5 Did you assume at some point that a function was differentiable? If so, did you justify it? 6 Did you assume at some point that a function was continuous? If so, did you justify it? If you answered “no" to Q ?? , you probably missed something important.

Critique this proof f ( x ) g ( x ) − f ( a ) h ( x ) − h ( a ) g ( a ) h ′ ( a ) = lim = lim x − a x − a x → a x → a f ( x ) g ( a ) − f ( a ) g ( x ) = lim g ( x ) g ( a ) ( x − a ) x → a f ( x ) g ( a ) − f ( a ) g ( a ) + f ( a ) g ( a ) − f ( a ) g ( x ) = lim g ( x ) g ( a ) ( x − a ) x → a   � f ( x ) − f ( a ) � g ( a ) − f ( a ) g ( x ) − g ( a ) 1   = lim x − a x − a g ( x ) g ( a ) x → a   1 � � = f ′ ( a ) g ( a ) − f ( a ) g ′ ( a ) g ( a ) g ( a )

Let α ∈ R , and consider the function | x | α f ( x ) = x 2 + 1 For what values of α 1 is f continuous?

Let α ∈ R , and consider the function | x | α f ( x ) = x 2 + 1 For what values of α 1 is f continuous? 2 is f differentiable?

Let α ∈ R , and consider the function | x | α f ( x ) = x 2 + 1 For what values of α 1 is f continuous? 2 is f differentiable? 3 does f have a corner?

Let α ∈ R , and consider the function | x | α f ( x ) = x 2 + 1 For what values of α 1 is f continuous? 2 is f differentiable? 3 does f have a corner? 4 does f have a vertical asymptote?

Let α ∈ R , and consider the function | x | α f ( x ) = x 2 + 1 For what values of α 1 is f continuous? 2 is f differentiable? 3 does f have a corner? 4 does f have a vertical asymptote? 5 does f have a vertical tangent line?

Before next class... • Watch videos 3.10 and 3.11. • Download the next class’s slides (no need to look at them!)