If your solution to stated problem does not match the given solution, you should self-critique per instructions at

   http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution.  If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

At the end of this document, after the qa problems (which provide you with questions and solutions), there is a series of Questions, Problems and Exercises. 

11.4.

Question:  `q001. 

The surface described by the graph z = f(x, y) = 16 - 4 x^2 + y^2 includes the point (2, 3, 9). 

• If the surface is 'sliced' by the plane x = 2, what is the resulting z vs. y function, and what is its slope at the y = 3 point?
• What is a vector in the x = 2 plane which is tangent to the surface at the given point? 
• If the surface is 'sliced' by the plane y = 3, what is the resulting z vs. x function, and what is its slope at the x = 2 point?
• What is a vector in the x = 2 plane which is tangent to the surface at the given point?
• What is a vector perpendicular to both of the tangent vectors?
• What therefore is the equation of the tangent plane?
• What are the coordinates of the graph point above the point (2.1, 3.2) in the xy plane?
• What are the coordinates of the point on the tangent plane that lies directly above the point (2.1, 3.2) in the xy plane?
• How far is it from the graph to the tangent plane, directly above the point (2.1, 3.2) in the xy plane?
• What do the derivatives f_xx, f_xy and f_yy have to do with how accurately the tangent plane approximates the actual graph in the vicinity of the point (2, 3, 9)?

Your solution: 

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Question:  `q002.  The surface described by the graph z = f(x, y) = 16 - 4 x^2 + y^2 includes the point (2, 1, 1).  Show how to use the derivatives f_x and f_y at this point to find the equation of the tangent plane.  What are the values of f_xx, f_xy and f_yy at this point, and what do they tell you about how the graph 'pulls away' from the tangent plane as you move in various directions.

Your solution: 

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Question:  `q003. 

If x is the length and y the width of a rectangle, then

• If the length changes by `dx, by how much does the area change?
• If the width changes by `dy, by how much does the area change?
• If the length changes by `dx and the width by `dy, by how much does the area change?
• If `dx is 3% of x, and `dy is 2% of y, then what fraction of the area change is represented by the product `dx * `dy?
• How accurate is it to therefore say that `d ( x y ) = x * `dy + y * `dx?

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Question:  `q004.  If z = f(x, y) = x * y, where x and y are both functions of t, then what is the expression for dz / dt?  What therefore is the expression for dz?

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Question:  `q005.  If w = f(x,y, z) = x^2 * cos(y) * e^z, where x, y and z are all functions of t, then what is the expression for dw/dt?  What therefore is the expression for `dw?

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Question:  `q006. 

It is easy to verify that when x = 3 and y = 4, the value of the function f(x, y) = x^2 * sqrt(y) is 6.  It is also easy to verify that df =  2x sqrt(y) dx + x^2 / (2 sqrt(y)) dy. 

• If x = 3 and y = 4, with dx = .1 and dy = .2, then what is the value of df?
• What therefore would you predict to be the value of f(3.1, 4.2), without actually evaluating the function at (3, 4)?
• Go ahead and evaluate the function at  x = 3.1 and y = 4.2, and compare with your predicted value.
• Use the same type of analysis to predict f(2.2, 8.9), and compare the prediction with the actual value of the function.

Your solution: 

Confidence rating:
 

Given Solution: 

Self-critique (if necessary):

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