Q1 Q2 Q3 Q4 Q5 Total
The Australian National University
Final Examination Solutions - June 2015
MATH 1014 - Mathematics and Applications II.
Book A - Calculus
Questions 1 - 5
Student No:
U
Important notes:
• One A4 sheet hand written on both sides permitted. This sheet is to cover both
calculus and linear algebra. No calculators or books permitted.
• Answer all questions. Calculus in Book A, Linear Algebra in Book B.
• Questions have different weights, the value of each is shown. The maximum grade
on this part of the paper is 50.
• Place your answers in the space provided on the test sheet. If you need more space,
use the back of the pages.
• You must justify your answers and show your work. Please be neat.
• You have 15 minutes reading time.
• You have 3 hours to complete the entire exam.
MATH 1014 - Book A, Page 1 of 14
Question: 1(Calculus: Taylor series) 10P
(a) Determine the centre, radius, and interval of convergence of the power series
1X
n=1
( 1)n
n32n (x + 1)
n :
(5P)
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MATH 1014 - Book A, Page 2 of 14
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MATH 1014 - Book A, Page 3 of 14
(b) Consider the function
f (x) = e
x
e x
2
:
Find T4, the 4th degree Taylor polynomial, for f (x) centred at 0.
(5P)
Begin Your Solution Here
MATH 1014 - Book A, Page 4 of 14
Question: 2(Calculus: parametric curve) 14P
Consider the curve given by the parametric equations for 0< < 2 :
(
x = sin ;
y = 1 cos:
(a) Find the slope of the tangent line at = 4 . (4P)
(b) For what values of is the tangent line horizontal? (2P)
(c) Find the second order derivatived2ydx2 . Is the curve concave up or concave down?
(4P)
(d) Find the area A under the curve. (4P)
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MATH 1014 - Book A, Page 5 of 14
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MATH 1014 - Book A, Page 6 of 14
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MATH 1014 - Book A, Page 7 of 14
Question: 3 (Calculus: polar curve) 6P
Consider the polar curve
r ( ) = 2(sin + cos ); 0 = 2:
Find the arc length L of the curve.
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MATH 1014 - Book A, Page 8 of 14
Question: 4 (Calculus: partial derivatives) 6P
Show that
f (x; y; z) = 1p
x2 + y2 + z2
satisfies the Laplace equation
@2f
@x2 +
@2f
@y2 +
@2f
@z2 = 0
for all (x; y; z) 6= (0; 0; 0).
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MATH 1014 - Book A, Page 9 of 14
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MATH 1014 - Book A, Page 10 of 14
Question: 5 (Calculus: maxima and minima) 14P
Consider the function
f (x; y) = x3 + y2 3x 2y:
(a) Find the critical points of f . (5P)
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MATH 1014 - Book A, Page 11 of 14
(b) Classify the critical points in Part (a) as local maxima, local minima, or saddle
points. (5P)
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MATH 1014 - Book A, Page 12 of 14
(c) Consider the parametric curve
(
x = t2;
y = t3 5:
Using the chain rule, find dfdt , where f is the same function as before:
f (x; y) = x3 + y2 3x 2y:
Express your answer as a function of t. (4P)
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MATH 1014 - Book A, Page 13 of 14
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MATH 1014 - Book A, Page 14 of 14