text: # Worksheet_5 - f-22,45 4 Consider f x y = p 1-x 2-y 2(a What...
  
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  Discussion – Tuesday, September 14th 1. Consider f ( x , y ) = xy 2 e x 2 . Compute the following: f x f y 2 f x y 2 f y x What is the relationship between your answers for the last two? This is an instance of Clairaut’s Theorem and holds for most functions. 2. Shown are some level curves for the function f : R 2 R . Determine whether the following partial derivatives are positive, negative, or zero at the point P : f x f y f xx f yy f xy = f yx 3. The wind-chill index W = f ( T , v ) is the perceived temperature when the actual tempera- ture is T and the wind speed is v . Here is a table of values for W . (a) Use the table to estimate f T and f v at ( T , v ) = ( - 20,40). (b) Use your answer in (a) to write down the linear approximation to f at ( - 20,40). (c) Use your answer in (b) to approximate
  
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  Unformatted text preview: f (-22,45). 4. Consider f ( x , y ) = p 1-x 2-y 2 . (a) What is the domain of f ? That is, for which ( x , y ) does the function make sense? (b) Describe geometrically the surface which is the graph of f . (c) Find the tangent plane to the graph at ( x , y ) = (1/2,1/2). (d) Consider the vector v = ( 1/2,1/2,1/ p 2 ) which goes from to the point on the graph where we just found the tangent plane. What is the angle between v and a normal vector to the tangent plane? 5. Consider f ( x , y ) = 3 p x 3 + y 3 . (a) Compute f x (0,0). Note: this partial derivative exists. (b) Is f differentiable at (0,0)?...
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  ## This note was uploaded on 02/20/2011 for the course MATH 241 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.
  
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  Worksheet_5 - f-22,45 4 Consider f x y = p 1-x 2-y 2(a What...
  
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explanation: The provided text is a worksheet or assignment for a mathematics course, likely a multivariable calculus class. It contains several questions and prompts for students to solve, covering topics such as partial derivatives, level curves, wind-chill index, and tangent planes. The text does not contain answers to the questions, only the questions themselves.
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distilabel_metadata.content: # Worksheet_5 - f-22,45 4 Consider f x y = p 1-x 2-y 2(a What...
  
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  Discussion – Tuesday, September 14th 1. Consider f ( x , y ) = xy 2 e x 2 . Compute the following: f x f y 2 f x y 2 f y x What is the relationship between your answers for the last two? This is an instance of Clairaut’s Theorem and holds for most functions. 2. Shown are some level curves for the function f : R 2 R . Determine whether the following partial derivatives are positive, negative, or zero at the point P : f x f y f xx f yy f xy = f yx 3. The wind-chill index W = f ( T , v ) is the perceived temperature when the actual tempera- ture is T and the wind speed is v . Here is a table of values for W . (a) Use the table to estimate f T and f v at ( T , v ) = ( - 20,40). (b) Use your answer in (a) to write down the linear approximation to f at ( - 20,40). (c) Use your answer in (b) to approximate
  
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  Unformatted text preview: f (-22,45). 4. Consider f ( x , y ) = p 1-x 2-y 2 . (a) What is the domain of f ? That is, for which ( x , y ) does the function make sense? (b) Describe geometrically the surface which is the graph of f . (c) Find the tangent plane to the graph at ( x , y ) = (1/2,1/2). (d) Consider the vector v = ( 1/2,1/2,1/ p 2 ) which goes from to the point on the graph where we just found the tangent plane. What is the angle between v and a normal vector to the tangent plane? 5. Consider f ( x , y ) = 3 p x 3 + y 3 . (a) Compute f x (0,0). Note: this partial derivative exists. (b) Is f differentiable at (0,0)?...
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  ## This note was uploaded on 02/20/2011 for the course MATH 241 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.
  
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  Worksheet_5 - f-22,45 4 Consider f x y = p 1-x 2-y 2(a What...
  
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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a worksheet or assignment for a mathematics course, likely a multivariable calculus class. It contains several questions and prompts for students to solve, covering topics such as partial derivatives, level curves, wind-chill index, and tangent planes. The text does not contain answers to the questions, only the questions themselves.
  
  Label: multiple-question