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

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Apr 03, 2023 145 likes •265 views

MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid October 27th, 2020 15 Warm-up question : If cos( ) = 4 , then sin( ) =...? Derivative of cosine Recall the identity sin 2 x + cos 2 x = 1. We will use the

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MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Muhammad Mohid October 27th, 2020 Warm-up question: If cos(θ) = √ 15 4 , then sin(θ) =...?

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Derivative of cosine

Recall the identity sin2 x + cos2 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).

SLIDE 3

Derivative of cosine

Recall the identity sin2 x + cos2 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

solve for d dx cos(x).

SLIDE 4

Derivative of cosine

Recall the identity sin2 x + cos2 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

solve for d dx cos(x).

3 Do you prefer either method? Why?

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A pesky curve

Consider the curve C of equation x3 + y3 − 3xy = 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)?

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A pesky curve

Consider the curve C of equation x3 + y3 − 3xy = 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?

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A pesky curve

Consider the curve C of equation x3 + y3 − 3xy = 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.

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A pesky curve

Consider the curve C of equation x3 + y3 − 3xy = 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.

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A pesky curve

Consider the curve C of equation x3 + y3 − 3xy = 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?

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Implicit differentiation

The equation sin(x + y) + xy2 = 0 defines a function y = h(x) near (0, 0).

graph

Using implicit differentiation, compute

1 h(0) 2 h′(0) 3 h′′(0) 4 h′′′(0)

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Before next class...

Watch videos 4.1 and 4.2.
Download the next class’s slides (no need to look at them!)