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Math 206 Section A Test 2: Solving Geometric and Calculus Problems, Exams of Mathematics

Birla Institute of Technology and ScienceMathematics

The instructions and problems for a university-level mathematics exam focusing on geometry and calculus. Students are required to sketch sets, determine points of discontinuity, find critical points, apply the chain rule, analyze the relationship between function inputs and outputs, sketch vector fields, calculate line integrals, and find directions of steepest descent. The document also includes a problem involving a supermarket's beef and chicken purchasing function.

Typology: Exams

2012/2013

Uploaded on 03/07/2013

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Math 206 Section A
Test 2
75 points
Name:
Show all your work to receive full credit for a problem.
There are eight questions. Questions are printed on both sides of a page.
1. (8 points) Sketch the following set. Determine if the set is open, closed, or neither. Use
the definition of open sets and closed sets to illustrate this with your sketch. Also find the
boundary and complement.
A={(x, y, z)∈R3|x2+y2>4}
pf3
pf4
pf5
pf8

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Math 206 Section A

Test 2

75 points

Name:

Show all your work to receive full credit for a problem.

There are eight questions. Questions are printed on both sides of a page.

  1. (8 points) Sketch the following set. Determine if the set is open, closed, or neither. Use the definition of open sets and closed sets to illustrate this with your sketch. Also find the boundary and complement.

A = {(x, y, z) ∈ R^3 | x^2 + y^2 > 4 }

  1. (9 points) Determine the point(s) of discontinuity in the following function. Explain why the point(s) are points of discontinuity. Are the discontinuities removable? Explain.

f(x, y) =

3 x^2 y x^4 + y^2

  1. (10 points) Let g(x, y) = x^2 − 3 y^2 , f(x, y) = (xy, x + y^2 ), and ~a = (2, 1). Let h = g ◦ f.

(a) Use the chain rule to write a formula for Dh(~a).

(b) Use h(~a) and Dh(~a) to find an approximation for h(1. 99 , 1 .01).

  1. (9 points) The quantity of beef, Q (in pounds per week) purchased in a supermarket is a function of the price of beef, b, and the price of chicken, c, (where b and c are in dollars per pound.) So we have Q = f(b, c). Suppose fb(1. 99 , 3 .99) = −400 and fc(1. 99 , 3 .99) = 300. Use this information to answer the following questions.

(a) Explain in words the meaning of the statement fb(1. 99 , 3 .99) = −400.

(b) Find the differential dQ at the point (1. 99 , 3 .99).

(c) Use your answer in part (b) to estimate the change in the quantity of beef purchased in the supermarket if the price of beef increases by $0.50 per pound and the price of chicken decreases by $0.50 per pound.

  1. (10 points) The surface of a mountain is modeled by h(x, y) = 25 − 2 x^2 − 4 y^2. All distances are in miles. A hiker is walking on a path on this mountain. It begins to rain when she is at the point with x = 1, y = 1.

(a) In what direction should she head to descend the mountainside most rapidly? (In other words she would like to take the path which has the steepest descent.)

(b) Find the equation of the line tangent to the level curve through the point (1, 1).

(c) Instead of descending most rapidly, the hiker decides to head off in the direction ~i + ~j. Find the rate of change in elevation in this direction.

  1. (9 points) Evaluate the line integral

C F~ · d~x, where F~ (x, y) = (−y, x) and C is the closed path that consists of the line segment from (− 2 , 0) to (2, 0) followed by the semicircle of radius 2 centered at the origin in the upper half plane.