Homework1

Homework1 - Section 2.3 Exercise 4(d Show that the...

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1 Mathematics 1c: Homework Set 1 Due: Monday, April 5th by 10am. 1. (10 Points) Using the computing site or otherwise, draw the graphs of the following functions: (a) f ( x,y ) = 3( x 2 + 2 y 2 ) e - x 2 - y 2 ; Tip: On the computing site use E [ x ] to take the exponent of x ; there is no need to type a * for multiplication; we suggest taking x and y between - 2 and 2. (b) f ( x,y ) = ( x 3 - 3 x ) / (1 + y 2 ) Indicate some key features of these graphs, such as the location of the maxima and minima, important sections, etc 2. (10 Points) Section 2.1, parts of Exercises 15, 18. Sketch the zero level set of the function f ( x,y,z ) = xy + yz and the level set for c = 1 of the function f ( x,y ) = max( | x | , | y | ) . 3. (15 Points) Section 2.2, Exercise 12. Compute the following limits, if they exist (a) lim x 0 sin 2 x - 2 x x 3 . (b) lim ( x,y ) (0 , 0) sin 2 x - 2 x + y x 3 + y . (c) lim ( x,y,z ) (0 , 0 , 0) 2 x 2 y cos z x 2 + y 2 . 4. (10 Points)
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Unformatted text preview: Section 2.3, Exercise 4(d) Show that the following function is diﬀerentiable at each point in its domain. Determine if the function is C 1 . f ( x,y ) = xy p x 2 + y 2 . 5. (10 Points) Section 2.3, Exercise 8(c). Compute the matrix of partial derivatives of the function f ( x,y ) = ( x + y,x-y,xy ) . 6. (10 Points) Section 2.3 Exercise 10. Why should the graphs of f ( x,y ) = x 2 + y 2 , and g ( x,y ) =-x 2-y 2 + xy 3 be called “tangent” at (0 , 0) ? 7. (15 Points) Section 2.4, Exercise 18. Suppose that a particle following the path c ( t ) = ( e t ,e-t , cos( t )) ﬂies oﬀ on a tangent at t = 1 . Compute the position of the particle at time t 1 = 2 ....
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