150 l 6 l 8 l 7 l 6 l 7 l 8
Code: 4725 DEPARTMENT OF MATHEMATICS IJniversity of Toronto MAT 135Y Term-Test ffL Tuesday, Novemb er 7, 2006 Time allowed: 90 minutes
PleasePRINT in INK or BALL-POINT PEN: (PleasePRINT full name and UNDERLINE surname): NAME OF STUDENT: STUDENT NO.: SIGNATURE OF STUDENT (in INK or BALL-POINT PEN): TUTORIAL
CODE (".g. M4A, RsD, etc.):
TUTORIAL
TIME (".e. T4, R5, F3, etc.):
NAME OF YOUR T.A.: NOTE:
FOR
1. Beforeyou start, checkthat this test has 12 paees.There are NO blank pages. 2. This test has two parts: PART A [50 marks]: 10 multiple choicequestions PART B [50 marks]: 7 written questions Answers to both PART A and PART B are to be given in this booklet. No computer cards will be used. 3. No aids allowed. No calculators! DO NOT TEAR OUT ANY PAGES.
MARKERS
QUESTION PAHI
A
MARK
150
B1
l6
B2
/8
B3
l8
B4
l7
B5
l6
B6
l7
B7
l8
TOTAL
Page L of.L2
ONLY
/ 100
Code: 4725 PART A [50 marks] P1easeread carefuilv: PART A consistsof 10 multiple-choicequestiors,eachof which has exactly one correct answer. Indicate your arxiwerto eachquestionby completely fflling in the appropriate circle with a dark pencil. MARKING SCHEME: 5 marks for a correct answer,0 for no answeror a wrong answer.You are not required to justify your answersin PART A. Note that for PART A, only your final answers (as indicated by the circles you darken) countl your computations and answers indicated elsewhere will NOT count. DO NOT TEAR OUT ANY PAGES. 1. lim 0--+3
12
-r-6
4
o
; o
@
0
@
1 _,
@ @
2.
1 2- 2 r - 3
1
t undefined
14+2r3+2 Iim "*L5-12-3ra
1
@
; o
@
; c
2
@
- 52
C
- 51
@
undefined
Page 2 of L2
Code: 4725
3 "ULtfr sin(#) : G ) o
4.
@
*oo
@
some finite number between 0 and
O
some finite number greater than 1
o
1
The graph of
@
x) a - L + 2 s r z ( f o rr ) 0 ) has a horizontal tangent line at r
1
B
,.
@
B 1
o ; 9
3 %
@ o?
5 . I f f ( * ) - arctan(2*'), then f 'e;) :
@
o
3
@ i @ i o ; @
;
Page 3 of 12
Code: 4725
thenf-'(r) = 2 ,- 3 r ' L*2r *gr L*2r
6. rt f (r): vt
a rr= I
It2r -5+3r
a
I-2r b+3r L-2r
-
5
r
. ^ o-Jr
7. Let
r(*):{:._i' li; : 1,
where c is a negative constant. Find the value of c so that f is continuous everywhere.
g ^ - 5 @ - 6
c ) - 8 @ ) - 7
@ - e
Page 4 of t2
Code:4725 I - 5r -'-t. ,l l"' Suppose firnctionsuchthat g(8):2. Let h(z) : f(v(ffr))) that g is a differentiable lr'G) : L4, what is the r,alueof 9 /(8) ?
8. Let /(r) :
@ @
not determinable dueto insu-fficient data ,. -11
@ #
@ +
o-+
Page 5 of L2
. ff
Code: 4725
9. Let 1@ @) denote the n -th derivative of / at a . It f (r):
@
27r
e
272
@
274
o
273
@
275
s i n 4r - c o s 4 r , t h e n 7 3 s )( # )
:
Page 6 of L2
Code:4725 10. The orthogonal trajectories of the family of curves r" - a" = & a're the curves
@ . . r "+ 2 u ' = C @ 3 r z+ g 2 : c @
s "+ a 2 : C r
O
xY:c
@
a: Cs'2
Page 7 of.L2
Code: 4725 PART B [50 marks] Pleaseread carefullv: Presentyour completesolutionsto the following questionsin the spacesprovided, in a neat and logicai fashion,showingall your computationsand justifications. Any answerin PART B without proper justification may receivevery little or no credit. Usethe back of eachpagefor rough work only. If you must continueyour formal solutionon the back of a page,you shouldindicate clearly, inLARGEIetters,..SoLUTIoNcoNTINUEDoNTHEBACKoFPAGE you may get credit for what you write on the back of that page,but you may also be penalized for mistakeson the back of that page. MARKS FOR EACH QUESTIONARE INDICATED BY II, DO NOT TEAR OUT ANY PAGES. t. Let f (x) : iD2. Find //(o) from first principles(i.e. by usingonlv the definition of the derirative).
t6l
Page 8 of 12
Code: 4725 ? ' dY for each of the following. There is no need to simplify d" your final answers for this question.
2. Use any suitable method. tc) nno
(u)
a - e2'sin 3r .
t4l
(b) a-
tanr L*12
t4l
3 . For this question, simpli$r your final answersas much as possible.
(u) rt f (r) - \m,
find f'(4).
t/,l
(b) It f (r)
T -
).t
)
findf '(r)
t/,1
Page 9 of 12
Code: 4725 ft Find the generalexpression
4. Supposethat ngs * 2y2*rgI2rs:0. valueof
dtt
fi
af the point where c:
H.
AIso,find the
1.
t7l
5. A ball is being thrown upward from the top of a building so that its height (aboveground) I secondsafter it is tbrown is (90 + 64t - 1612)feet. Find the velocity of the ball when it is 10 feet above ground on its way down.
t6l
Page 10 of 12
Code: 4725 6. Find the line passingthroughthe point 10,-l; x> 0) at somepoint.
*a tangentto the curve g:4xa
t7l
Page tL of 12
(wherc
Code: 4725 7' (Note: This is not an easy question and will be marked very strictly.) Let if ec +5 tBl
Page 12 of 12
PleasePRINT in INK or BALL-POINT PEN: (PleasePRINT full name and UNDERLINE surname): NAME OF STUDENT: STUDENT NO.: SIGNATURE OF STUDENT (in INK or BALL-POINT PEN): TUTORIAL
CODE (".g. M4A, RsD, etc.):
TUTORIAL
TIME (".e. T4, R5, F3, etc.):
NAME OF YOUR T.A.: NOTE:
FOR
1. Beforeyou start, checkthat this test has 12 paees.There are NO blank pages. 2. This test has two parts: PART A [50 marks]: 10 multiple choicequestions PART B [50 marks]: 7 written questions Answers to both PART A and PART B are to be given in this booklet. No computer cards will be used. 3. No aids allowed. No calculators! DO NOT TEAR OUT ANY PAGES.
MARKERS
QUESTION PAHI
A
MARK
150
B1
l6
B2
/8
B3
l8
B4
l7
B5
l6
B6
l7
B7
l8
TOTAL
Page L of.L2
ONLY
/ 100
Code: 4725 PART A [50 marks] P1easeread carefuilv: PART A consistsof 10 multiple-choicequestiors,eachof which has exactly one correct answer. Indicate your arxiwerto eachquestionby completely fflling in the appropriate circle with a dark pencil. MARKING SCHEME: 5 marks for a correct answer,0 for no answeror a wrong answer.You are not required to justify your answersin PART A. Note that for PART A, only your final answers (as indicated by the circles you darken) countl your computations and answers indicated elsewhere will NOT count. DO NOT TEAR OUT ANY PAGES. 1. lim 0--+3
12
-r-6
4
o
; o
@
0
@
1 _,
@ @
2.
1 2- 2 r - 3
1
t undefined
14+2r3+2 Iim "*L5-12-3ra
1
@
; o
@
; c
2
@
- 52
C
- 51
@
undefined
Page 2 of L2
Code: 4725
3 "ULtfr sin(#) : G ) o
4.
@
*oo
@
some finite number between 0 and
O
some finite number greater than 1
o
1
The graph of
@
x) a - L + 2 s r z ( f o rr ) 0 ) has a horizontal tangent line at r
1
B
,.
@
B 1
o ; 9
3 %
@ o?
5 . I f f ( * ) - arctan(2*'), then f 'e;) :
@
o
3
@ i @ i o ; @
;
Page 3 of 12
Code: 4725
thenf-'(r) = 2 ,- 3 r ' L*2r *gr L*2r
6. rt f (r): vt
a rr= I
It2r -5+3r
a
I-2r b+3r L-2r
-
5
r
. ^ o-Jr
7. Let
r(*):{:._i' li; : 1,
where c is a negative constant. Find the value of c so that f is continuous everywhere.
g ^ - 5 @ - 6
c ) - 8 @ ) - 7
@ - e
Page 4 of t2
Code:4725 I - 5r -'-t. ,l l"' Suppose firnctionsuchthat g(8):2. Let h(z) : f(v(ffr))) that g is a differentiable lr'G) : L4, what is the r,alueof 9 /(8) ?
8. Let /(r) :
@ @
not determinable dueto insu-fficient data ,. -11
@ #
@ +
o-+
Page 5 of L2
. ff
Code: 4725
9. Let 1@ @) denote the n -th derivative of / at a . It f (r):
@
27r
e
272
@
274
o
273
@
275
s i n 4r - c o s 4 r , t h e n 7 3 s )( # )
:
Page 6 of L2
Code:4725 10. The orthogonal trajectories of the family of curves r" - a" = & a're the curves
@ . . r "+ 2 u ' = C @ 3 r z+ g 2 : c @
s "+ a 2 : C r
O
xY:c
@
a: Cs'2
Page 7 of.L2
Code: 4725 PART B [50 marks] Pleaseread carefullv: Presentyour completesolutionsto the following questionsin the spacesprovided, in a neat and logicai fashion,showingall your computationsand justifications. Any answerin PART B without proper justification may receivevery little or no credit. Usethe back of eachpagefor rough work only. If you must continueyour formal solutionon the back of a page,you shouldindicate clearly, inLARGEIetters,..SoLUTIoNcoNTINUEDoNTHEBACKoFPAGE you may get credit for what you write on the back of that page,but you may also be penalized for mistakeson the back of that page. MARKS FOR EACH QUESTIONARE INDICATED BY II, DO NOT TEAR OUT ANY PAGES. t. Let f (x) : iD2. Find //(o) from first principles(i.e. by usingonlv the definition of the derirative).
t6l
Page 8 of 12
Code: 4725 ? ' dY for each of the following. There is no need to simplify d" your final answers for this question.
2. Use any suitable method. tc) nno
(u)
a - e2'sin 3r .
t4l
(b) a-
tanr L*12
t4l
3 . For this question, simpli$r your final answersas much as possible.
(u) rt f (r) - \m,
find f'(4).
t/,l
(b) It f (r)
T -
).t
)
findf '(r)
t/,1
Page 9 of 12
Code: 4725 ft Find the generalexpression
4. Supposethat ngs * 2y2*rgI2rs:0. valueof
dtt
fi
af the point where c:
H.
AIso,find the
1.
t7l
5. A ball is being thrown upward from the top of a building so that its height (aboveground) I secondsafter it is tbrown is (90 + 64t - 1612)feet. Find the velocity of the ball when it is 10 feet above ground on its way down.
t6l
Page 10 of 12
Code: 4725 6. Find the line passingthroughthe point 10,-l; x> 0) at somepoint.
*a tangentto the curve g:4xa
t7l
Page tL of 12
(wherc
Code: 4725 7' (Note: This is not an easy question and will be marked very strictly.) Let if ec +5 tBl
Page 12 of 12
