Tutorial 7: SOLUTIONS
Analysis of Functions
1. For each of the following functions, we are asked to nd (a) the intervals on which
f is increasing, (b) the intervals in which f is decreasing, (c) the open intervals on
which f is concave up, (d) the open in
Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 4
Due: at the end of the tutorial session Tuesday/Thursday, 16/18 February 2016
Name and student number:
1. Use a double integral to find the volume under the surface
z = x2 + 2x3 cos y
and ove
Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 3
Due: at the end of the tutorial session Tuesday/Thursday, 9/11 February 2016
Name and student number:
1. Consider the function
f (x, y, z) =
p
4y 2 sin(2x 3z) ,
and the point P (0, 1, 0) .
(a
Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 5
Due: at the end of the tutorial session Tuesday/Thursday, 23/25 February 2016
Name and student number:
1. Consider the portion of the surface (y 1)2 + z 2 = 8 that is above the rectangle
R =
Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 1
Due: at the end of the tutorial session Tuesday/Thursday, 26/28 January 2016
Name and student number:
Consider the vector function (with values in R3 )
r(t) = ln(2 t) i + (1 + t) j
(t 2)2
k
Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 2
Due: at the end of the tutorial session Tuesday/Thursday, 2/4 February 2016
Name and student number:
1. Sketch the level curve z = k for the specified values of k
z = x2 2x + 4y 2 4y ,
k = 2,
UNIVERSITY OF DUBLIN
XMA2E011
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Engineers
SF MSISS
SF MEMS
Trinity Term 2014
Module 2E011,
?, ?
Final Exam
?
9.30 11.30
Dr. Sergey Frolov
ATTEMPT QUESTION 1 and FOUR OT
UNIVERSITY OF DUBLIN
XMA2E011
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Engineers
SF MSISS
SF MEMS
Trinity Term 2012
Module 2E011,
?, ?
Final Exam
?
9.30 11.30
Dr. Sergey Frolov
ATTEMPT QUESTION 1 and FOUR OT
Tutorial 2: SOLUTIONS
Limits
1. We are given the following graphs
1
5
3
6
4
(a) From the rst graph, it is clear that
lim f (x) = ,
x5
lim f (x) = ,
lim f (x) = ,
x5
x5+
f (5) = 1.
(b) For the second graph above,the only value of x0 for which the limit lim
Tutorial 6: SOLUTIONS
Derivatives 2
1. For the following functions, we must compute dy/dx:
(a) y = sin(1/x2 ).
Applying the chain rule gives
dy
2 cos(1/x2 )
.
= cos(1/x2 ) (2x3 ) =
dx
x3
2
1+csc(x
(b) y = 1cot(x2) .
)
We may re-express the function by not
Tutorial 5: SOLUTIONS
Derivatives 1
1. We are asked to nd the equation of the tangent line to the following curves:
(a) f (x) = 2x2 x + 1 at x = 1. We require two things, a point on the line
and the slope of the line at x = 1. Well clearly (x = 1, y = f (
Tutorial 9: SOLUTIONS
Integration
1. We are asked to evaluate the following indenite integrals:
(a)
(b)
(c)
2 19/2
x
+C
19
10 y + 4 y
4
16
dy = 40y 1/4 y 5/4 + y 3/4 + C
3/4
5
3
y
x17/2 dx =
1
(x3 sin x) dx = x4 + cos x + C
4
2. We have the following init
Tutorial 4: SOLUTIONS
Continuity of Trigonometric Functions
1. We are asked to compute the following limits
1
(a) lim cos
. Since the cosine function is continuous, we can bring the limit
x
x
inside to get:
1
cos lim
= cos(0) = 1.
x x
x
. Again since the
UNIVERSITY OF DUBLIN
XMA1E011
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
JF Engineering
JF MSISS
JF MEMS
Trinity Term 2012
Module MA1E01 Engineering Mathematics I
Dr. P. Taylor
SOLUTIONS SOLUTIONS SOLUTIONS SOLUT
UNIVERSITY OF DUBLIN
XMA1E011
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
JF Engineering
JF MSISS
JF MEMS
Trinity Term 2013
Module MA1E01 Engineering Mathematics I
Dr. P. Taylor
SOLUTIONS SOLUTIONS SOLUTIONS SOLUT
Tutorial 1: SOLUTIONS
Functions
1. Were given the piecewise function
x1
3
x < 1.
Then g(3) = 3 + 1 = 2, g(1) = 3 and g() = + 1. To obtain g(t2 1)
is a bit trickier since we need to take the cases when t2 1 1 and t2 1 < 1
separately.
g(x) =
x+1
If t satise
Tutorial 8: Solutions
Applications of the Derivative
1. We are asked to nd the absolute maximum and minimum values of f on the given
interval, and state where those values occur:
(a) f (x) = 2x3 + 3x2 12x on [1, 4].
The absolute extrema occur either at a
Tutorial 10: SOLUTIONS
Applications of the Denite Integral in Geometry
1. Find the area of the region enclosed between y = 2x2 1 and y = 2x + 3 and on
the sides by x = 2 and x = 3.
Graphing these curves, we obtain The x-coordinates of the intersection poi
Module 2E02 (Frolov), Multivariable Calculus
1
Tutorial Sheet 1
Consider the vector function (with values in R3 )
r(t) = ln(2 + t3 ) i + (3 + t3 ) j
(t3 + 2)2
k
4
1. Find the domain D(r) of the vector function r(t).
Solution: The domain D(r) of r(t) is t