Sample Calculus Quiz

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Problem #1

The following table gives the windchill W  (° F) at various temperatures T  (° F) and wind speeds v  (mph).

Windchill Factor
T = 15 T = 10 T = 5 T = 0
v = 15 -11 -18 -25 -31
v = 20 -17 -24 -31 -39
v = 25 -22 -29 -36 -44

  1. Use the table to find approximate values for dW/dv  and dW/dT  when T = 0 and v = 25.

  2. Using the table and your answers to a), give (an approximation of) the local linearization for W  around T = 0 and v = 25.

  3. Use the local linearization to approximate W  when T = -5 and v = 25.

Problem #2

Suppose you are given two infinite wires which are parallel to the z -axis, passing through the points (1,0) and (-1,0) in the xy -plane which are homogeneously and oppositely charged. The potenial energy P  of a unit charge in the xy -plane is given by

P =  (1/2)· ln ((x  - 1)2 + y  2) - (1/2)· ln((x  + 1)2 + y  2)
  1. What is the gradient of the potential energy at (3,3)?

  2. What is the rate of change of P  in the direction u = (1/2)·i - (31/2/2)·j  at (3,3)?

  3. An equipotential line  is a curve along which the electrical potential is constant. What is the direction of the equipotential line passing through (3,3)?

  4. Another particle with a unit charge is moving up the y -axis at a rate of 5 m/sec. What is the rate of change of the potential energy of this particle?

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Created: January 1, 1997; Modified: January 11, 1997
Comments to:   alexsuciu@neu.edu
http://www.math.neu.edu/~suciu/quiz/htmlquiz.html