Calculus III Week 6

Calculus III – Week 6September 27 – October 1, 2021Hey everyone! My name is Savanah Smith, and I am the Master Tutor for Calculus III thissemester. I hope these resources can become a great studying and review tool for everyone whoencounters them. I will also be having weekly Group Tutoring sessions where we will go overthe topics and practice problems covered in these resources, so feel free to check that out asdescribed below! Please don’t hesitate to reach out to me with any questions or comments orwhen in doubt go to the tutoring website! I hope your first round of exams goes well!Group Tutoring: Mondays from 5:15 – 6:15 pmIn Sid Rich Basement, Room 75Reserve a spot at baylor.edu/tutoringThis week classes should be covering the remainder of Chapter 14 which includes sections14.5-14.7.Key Words: Gradient, Directional Derivatives, OptimizationTOPIC OF THE WEEKGradientsThe gradient 𝑓𝑃 of a function 𝑓 is the vector containing thepartial derivatives of 𝑓 in each direction at point P (a, b, c). 𝑓𝑃 𝑓(𝑎, 𝑏, 𝑐) 〈𝑓𝑥 (𝑎, 𝑏, 𝑐) , 𝑓𝑦 (𝑎, 𝑏, 𝑐) , 𝑓𝑧 (𝑎, 𝑏, 𝑐) 〉Or it can be written without the point P: 𝑓 〈 𝑓 𝑓 𝑓〉, , 𝑥 𝑦 𝑧Important Properties:Gradient for 𝑓 𝑥 2 𝑦 2Addition: (𝑓 𝑔) 𝑓 𝑔Scalar multiple: (𝑐𝑓) 𝑐 𝑓, c is a constantProduct Rule: (𝑓𝑔) 𝑓 𝑔 𝑔 𝑓All pictures, tables, and information if property of Calculus Early Transcendentals (4 th Edition)by Rogawski, Adams, and Franzosa.

Chain Rule: (𝐹(𝑓(𝑥, 𝑦, 𝑧))) 𝐹′(𝑓(𝑥, 𝑦, 𝑧)) f F is a differentiable functionSame method where you take the derivative of the outside and then the inside.Ex. Find the gradient of 𝑔(𝑥, 𝑦, 𝑧) (𝑥 2 𝑦 2 𝑧 2 )8OutsideInside𝐹 𝑓(𝑥, 𝑦, 𝑧)8𝑎𝑛𝑑𝑓(𝑥, 𝑦, 𝑧) (𝑥 2 𝑦 2 𝑧 2 )𝐹 ′ 8 𝑓(𝑥, 𝑦, 𝑧)7 𝑓 〈 𝑓𝑥 , 𝑓𝑦 , 𝑓𝑧 〉𝐹 ′ 8(𝑥 2 𝑦 2 𝑧 2 )7 𝑓 〈 2𝑥, 2𝑦, 2𝑧 〉 𝑔 𝐹 ′ 𝑓 𝑔 8(𝑥 2 𝑦 2 𝑧 2 ) 7 〈2𝑥, 2𝑦, 2𝑧〉Another important note: the gradient of a function is perpendicular to itslevel curves and points in the direction of maximum increase of the function.(see picture on the right)HIGHLIGHT #1: CHAIN RULE FOR PATHSA path is a curve in 3 dimensions represented by a vector-valued function 𝒓(𝑡). Recall that if𝒓(𝑡) represents position, 𝒓′ (𝑡) represents velocity or a rate of change.Chain Rule for Paths: dot product between the gradient of f and the derivative of 𝒓(𝑡)𝑑𝑓 (𝒓(𝑡)) 𝑓𝑟(𝑡) 𝒓′(𝑡)𝑑𝑡𝑑 𝑓 𝑓 𝑓 𝑓 ′ 𝑓 ′ 𝑓 ′𝑓(𝒓(𝑡)) 〈 , , 〉 〈 𝑥 ′ (𝑡), 𝑦 ′ (𝑡), 𝑧 ′ (𝑡)〉 𝑥 (𝑡) 𝑦 (𝑡) 𝑧 (𝑡)𝑑𝑡 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧This is used to describe how a variable f changes along thepath r(t). An example is determining how the temperaturechanges as one travels along a path throughout the U.S (seepicture)All pictures, tables, and information if property of Calculus Early Transcendentals (4 th Edition)by Rogawski, Adams, and Franzosa.

The chain rule can also be used in the same way for composite functions of two variables wherex, y, and z are differentiable by the independent variables s and t.𝑓(𝑥(𝑠, 𝑡), 𝑦(𝑠, 𝑡), 𝑧(𝑠, 𝑡))We can either take the derivative of 𝑓 in terms of s or t. 𝑓 𝑓 𝑥 𝑓 𝑦 𝑓 𝑧 𝑠 𝑥 𝑠 𝑦 𝑠 𝑧 𝑠 𝑓 𝑓 𝑥 𝑓 𝑦 𝑓 𝑧 𝑡 𝑥 𝑡 𝑦 𝑡 𝑧 𝑡HIGHLIGHT #2: DIRECTIONAL DERIVATIVESOne application of using the chain rule for paths is directional derivatives.Consider a line through a point 𝑃 (𝑎, 𝑏) in the direction of a unit vector𝒖 〈ℎ, 𝑘〉. The parametric equations would then be:𝒓(𝑡) 〈𝑎 ℎ𝑡, 𝑏 𝑘𝑡〉Directional Derivative: the derivative of 𝑓(𝒓(𝑡)) at 𝑡 0 with respect to uat P. Notation: 𝐷𝑢 𝑓(𝑃) or 𝐷𝑢 𝑓(𝑎, 𝑏) This represents the rate of change of 𝑓 along the path represented bythe point P and the unit vector u.𝐷𝑢 𝑓(𝑃) 𝑓𝑃 𝒓′ (0) 𝑓𝑃 𝒖The directional derivative needs to include a unit vector, u. One mustdivide a vector by its magnitude to result in a unit vector.𝒗𝒖 ‖𝒗‖𝒓′ (𝑡) 〈ℎ, 𝑘〉 𝒖Since there is no t value,𝒓′ (0) 𝒖The angle between 𝑓𝑃 and 𝒖 can be solved for using the equationbelow. Also known as the angle between the gradient and the direction.𝐷𝑢 𝑓(𝑃) 𝑓𝑃 𝒖 ‖ 𝑓𝑃 ‖𝑐𝑜𝑠𝜃Similarly, the angle of inclination, 𝜓 can be found. For example, theangle of inclination can be described as the angle between the groundand the side of a mountain.𝐷𝑢 𝑓(𝑃) tan(𝜓)All pictures, tables, and information if property of Calculus Early Transcendentals (4 th Edition)by Rogawski, Adams, and Franzosa.

HIGHLIGHT #3: OPTIMIZATIONOptimization: process of finding the extreme values of a function, the local and globalmaximum and minimums Local: within a specified region or disk, D Global: within the entire domain of a function (aka absolute)Step 1: Find critical points Critical points: points where the tangent plane is horizontal Point 𝑃 (𝑎, 𝑏) is a critical point if:𝑓𝑥 (𝑎, 𝑏) 0 𝑎𝑛𝑑 𝑓𝑦 (𝑎, 𝑏) 0 𝑜𝑟 𝑑𝑜 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡Step 2: Second Derivative Test D is also called the discriminate Determines the type of critical point using the equation:2𝐷 𝑓𝑥𝑥 (𝑎, 𝑏)𝑓𝑦𝑦 (𝑎, 𝑏) 𝑓𝑥𝑦(𝑎, 𝑏)All extreme values will be local. In order to find global extrema, theinterior and boundaries of the domain must be evaluated for criticalpoints. The highest and lowest critical points will be the globalextrema.The following videos are great resources to use for additional explanationof the topics covered in this resource!Video Series on Gradients and Directional ntand-directional-derivativesChain Rule for -chain-rule/v/multivariable-chain-ruleOptimization (including global):https://www.youtube.com/watch?v Hg38kfK5w4EAll pictures, tables, and information if property of Calculus Early Transcendentals (4 th Edition)by Rogawski, Adams, and Franzosa.

CHECK YOUR LEARNING1. Calculate 𝑓(3, 2,4) where 𝑓 𝑧𝑒 2𝑥 3𝑦 .2. Find the directional derivative in the direction of 𝑣 〈2,3〉. Let 𝑓 𝑥𝑒 𝑦 and 𝑃 (2, 1).3. Calculate 𝑓where 𝑓 𝑥𝑦 𝑧 and 𝑥 𝑠 2 , 𝑦 𝑠𝑡, 𝑧 𝑡 2 . 𝑠4. Find the local extrema of 𝑓(𝑥 2 𝑦 2 )𝑒 𝑥 .THINGS YOU MAY STRUGGLE WITH1. A lot of things in Calculus III can be difficult to visualize since everything is shifted into3 dimensions. Rely heavily on pictures, videos, or online 3D graphs to help be able tovisualize what you are doing.2. Most of these calculations have multiple steps are dealing with 3 variables. Be sure totake your time and ensure you know the purpose of each variable and what to do with it.It’s okay to make a chart or write out a bunch of intermediate steps to help you stayorganized.That’s all for this week! I hope this was a helpful review of Chapter 14.5 - 14.7!Feel free to visit the Tutoring Center Website for more information atwww.baylor.edu/tutoring.Answers:1. 𝑓(3, 2,4) 〈8,12,1〉2. 𝐷𝑢 𝑓(𝑃) 0.82 𝑓3. 𝑠 2𝑠𝑦 𝑥𝑡4. (0,0) is a local min, (2,0) is saddleAll pictures, tables, and information if property of Calculus Early Transcendentals (4 th Edition)by Rogawski, Adams, and Franzosa.

All pictures, tables, and information if property of Calculus Early Transcendentals (4th Edition) by Rogawski, Adams, and Franzosa. The chain rule can also be used in the same way for composite functions of two variables where x, y, and z are differentiable by the independent variables s and t.