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This is the Exam of Advanced Calculus which includes Line, Parametric Equations, Plane, Distance, Plane, Line, Surface, Tangent Plane, Parallel, Vectors etc. Key important points are: Differentiable Function, Directional Derivative, Point, Move Towards, Express, Quantity, Terms, Surface, Determine, Value
Typology: Exams
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The University of British Columbia Final Examination - December 9, 2006 Mathematics 226 Section 101 Instructor: C. Lamb
Closed book examination Time: 2.5 hours
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Student Number
Special Instructions:
No notes, books or calculators are to be used. No credit will be given for the correct answer without the (correct) accompanying work. Use the back of the pages if you need extra space.
Rules governing examinations
Total 100
Page 1 of 17 pages
[15] 1. Let z = f (x, y) be a differentiable function on R^2 such that f (1, 2) = 3, f (1. 2 , 2 .3) = 3.4 and f (0. 9 , 2 .1) = 3.2.
(a) Estimate ∂z ∂x (1, 2) and ∂z ∂y (1, 2). [10pt]
(b) Estimate the value of the directional derivative of z = f (x, y) at the point (1, 2) as you move towards the point (2, 3). [5pt]
[10] 2. Let z = f (x, y) be a differentiable function on R^2 , x = (s^2 + t^2 )/2 and y = (s^2 − t^2 )/2. Express the quantity (^) ( ∂z ∂x
∂z ∂y
in terms of s, t, ∂z ∂s and ∂z ∂t
[10] 4. Let the temperature at a point (x, y, z) be given by w = x^3 y^2 z. Find the point on the plane 2x + 2y + z = 24 where the temperature is a maximum. You do not need to justify that your answer actually gives the maximum. Hint: You may assume in your calculations that x 6 = 0, y 6 = 0 and z 6 = 0 since if x, y or z equals 0, then w = 0 and this will not be the maximum temperature.
[10] 5. Evaluate the interated double integral
∫ (^) y=√π
y=
∫ (^) x=√π
x=y
sin(x^2 ) dx dy.
[15] 7. Let W be the 3-dimensional solid defined by the inequalities x^2 + y^2 ≤ 1 and 0 ≤ z ≤ 2 − x^2 − y^2.
(a) Draw a sketch of W. Be sure to show the units on the axes. [5pt]
(b) Use cylindrical coordinates to evaluate
W
zdV. [10pt]
[15] 8. Let z = f (x, y) be defined by
z =
y^5 x^2 + y^2 if (x, y) 6 = (0, 0)
0 if (x, y) = (0, 0).
(a) Use the definition of partial derivatives as limits to calculate ∂z ∂x (0, 0) and ∂z ∂y (0, 0). [5pt]
(b) Is z = f (x, y) differentiable at (0, 0)? If you think that this is the case, then give a formal − δ proof to justify your belief. Otherwise, indicate clearly why you believe that z = f (x, y) is not differentiable at (0, 0). [10pt]
The End