MAT 137 — LEC 0601 Instructor: Alessandro Malusà TA: Muhammad Mohid October 27th, 2020 √ 15 Warm-up question : If cos( θ ) = 4 , then sin( θ ) =...?
Derivative of cosine Recall the identity sin 2 x + cos 2 x = 1. We will use the relation to compute the derivative of cos( x ) in two different ways. 1 Solve for cos( x ) in the identity, so as to express it in terms of sin( x ). Then differentiate to find the derivative of cos( x ).
Derivative of cosine Recall the identity sin 2 x + cos 2 x = 1. We will use the relation to compute the derivative of cos( x ) in two different ways. 1 Solve for cos( x ) in the identity, so as to express it in terms of sin( x ). Then differentiate to find the derivative of cos( x ). 2 Use implicit differentiation: derive both sides of the identity and then d solve for d x cos( x ).
Derivative of cosine Recall the identity sin 2 x + cos 2 x = 1. We will use the relation to compute the derivative of cos( x ) in two different ways. 1 Solve for cos( x ) in the identity, so as to express it in terms of sin( x ). Then differentiate to find the derivative of cos( x ). 2 Use implicit differentiation: derive both sides of the identity and then d solve for d x cos( x ). 3 Do you prefer either method? Why?
A pesky curve Consider the curve C of equation x 3 + y 3 − 3 xy = 0. If you already know this curve, please don’t post spoilers about its name or its shape in the chat!!! 0 Does C pass through the origin O (0 , 0)?
A pesky curve Consider the curve C of equation x 3 + y 3 − 3 xy = 0. If you already know this curve, please don’t post spoilers about its name or its shape in the chat!!! 0 Does C pass through the origin O (0 , 0)? 1 What is the slope of the tangent line at O ?
A pesky curve Consider the curve C of equation x 3 + y 3 − 3 xy = 0. If you already know this curve, please don’t post spoilers about its name or its shape in the chat!!! 0 Does C pass through the origin O (0 , 0)? 1 What is the slope of the tangent line at O ? 2 Find all the points on C with a horizontal tangent line.
A pesky curve Consider the curve C of equation x 3 + y 3 − 3 xy = 0. If you already know this curve, please don’t post spoilers about its name or its shape in the chat!!! 0 Does C pass through the origin O (0 , 0)? 1 What is the slope of the tangent line at O ? 2 Find all the points on C with a horizontal tangent line. 3 Find all the points on C with a vertical tangent line.
A pesky curve Consider the curve C of equation x 3 + y 3 − 3 xy = 0. If you already know this curve, please don’t post spoilers about its name or its shape in the chat!!! 0 Does C pass through the origin O (0 , 0)? 1 What is the slope of the tangent line at O ? 2 Find all the points on C with a horizontal tangent line. 3 Find all the points on C with a vertical tangent line. 4 What can we conclude about the tangent line(s) of C at O ?
Implicit differentiation The equation sin( x + y ) + xy 2 = 0 defines a function y = h ( x ) near (0 , 0). graph Using implicit differentiation, compute 2 h ′ (0) 3 h ′′ (0) 4 h ′′′ (0) 1 h (0)
Before next class... • Watch videos 4.1 and 4.2. • Download the next class’s slides (no need to look at them!)
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