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Three Conditions - Calculus - Exam, Exams of Calculus

West Bengal State UniversityCalculus

These are the notes of Exam of Calculus which includes Traditional Problems, Symmetric Matrix, Property, Conditions, Constants, Matrix, Positive, Anti Symmetric etc. Key important points are: Three Conditions, Function Continuous, Continuity, Derivative, Certain Product, Line Tangent, Indicated Derivative, Decimal Places, Simplifications, Logarithmic Differentiation

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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Dawson College Department of Mathematics
inal xam Math 201-NYA-05 Calculus One
FE
Commerce December 13, 2004 2 5 p.m.
Professor: J. Graham
Instructions: There are 13 questions on this exam. Give complete solutions in the space
provided. Be sure to clearly indicate your final answer or conclusion. Use the back side of
the paper if you need more space. Programmable calculators are not permitted.
[MARKS]
[10] 1. Evaluate the following limits
a) b) lim lim
BÄ# BÄ$
B
B#
#$#
$#
"
B# B $B #B'
B )B "'B $
[6] 2. a) Write down the three conditions for continuity of a function at Cœ0ÐBÑ Bœ-Þ
b) Check the conditions for continuity of the following function at Is theBœ$Þ
function continuous at ?Bœ$
for
for
0ÐBÑ œ B$
&B " B $
Ú
Û
ÜÈ
BB'
#B '
#
%
[5] 3. The demand function for a certain product is $/item. Use the: œ "!! B
È
to find the derivative .definition of the derivative .:
.B
[5] 4. Find an equation for the line tangent to the graph of at theCœ%B "
B
point .Ð"ß &Ñ
[20] 5. Calculate the value of the indicated derivative. Round off to four decimal places.
a) find 0ÐBÑ œ ß 0 Ð$Ñ
"!
#& B
È#
w
b) find 1Ð>Ñ œ =/- %> ß 1 Ð!Þ"Ñ
#w
c) find 2ÐBÑ œ B 68ÐB &Ñ ß 2 Ð%Ñ
$w
d) evaluate at?œ Bœ"Þ&/ß
BÎ
## .?
.B
#
#
[10] 6. a) Use implicit differentiation to find if
.C
.B B C BC œ "!! Þ
%$ #
Make only the obvious simplifications.
b) Use logarithmic differentiation to find if
.C
.B CœÐB -9=#BÑ Þ
##B
Make only the obvious simplifications.
pf3
pf4

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Dawson College Department of Mathematics

Final Exam Math 201-NYA-05 Calculus One

Commerce December 13, 2004 2  5 p.m.

Professor: J. Graham Instructions: There are 13 questions on this exam. Give complete solutions in the space provided. Be sure to clearly indicate your final answer or conclusion. Use the back side of the paper if you need more space. Programmable calculators are not permitted. [MARKS]

[10] 1. Evaluate the following limits

a) (^) BÄ# lim (^) b) BÄ$ lim B  #B# $^ # $ #

B  # B  $B  #B  '

B  )B  "'B  $

[6] 2. a) Write down the three conditions for continuity of a function C œ 0 ÐBÑ at B œ - Þ

b) Check the conditions for continuity of the following function at B œ $ Þ Is the function continuous at B œ $? for for

0 ÐBÑ œ B  $ &B  " B $

Ú
Û
Ü È

B  B  ' #B  '

%

[5] 3. The demand function for a certain product is : œ "!!  È B $/item. Use the definition of the derivative to find the derivative .:.B.

[5] 4. Find an equation for the line tangent to the graph of C œ %B  (^) B" at the point Ð " ß & Ñ.

[20] 5. Calculate the value of the indicated derivative. Round off to four decimal places.

a) 0 ÐBÑ œ (^) È (^) #&  B"! #ß find 0 Ð$Ñw b) 1Ð>Ñ œ =/- #^ %> ß find 1 Ð!Þ"Ñw c) 2ÐBÑ œ B $^ 68ÐB  &Ñ ß find 2 Ð%Ñw

d)? œ / B Î#^ #^ ß evaluate. ?.B at B œ "Þ&

[10] 6. a) Use implicit differentiation to find^ .C.B if B C%^ $^  BC #œ "!! Þ

Make only the obvious simplifications. b) Use logarithmic differentiation to find^ .C.B if C œ Ð B #^  -9= #B Ñ #BÞ Make only the obvious simplifications.

[10] 7. A new video game is being advertised on television. The company expects the total number of games sold over the next 10 days to follow the model R Ð>Ñ œ (^) "  *#!! / ! &>Þ thousand games, where! Ÿ > Ÿ "! days. a) Calculate the rate of sales when > œ % days. Round off to the nearest game per day. b) Find the point of diminishing returns ( i.e. the point on the graph where R is increasing and concavity changes from upwards to downwards)

[5] 8. The demand equation for a certain product is B œ Ð"!!  :Ñ # for "! Ÿ : Ÿ (! $/item.

Recall that elasticity of demand I can be expressed as I œ  Ð (^) B: ÑÐ (^) .:.BÑ Þ Determine the unit price : at which demand is unitary. Round off to the nearest penny.

[5] 9. The total cost to build a natural gas pipeline in a certain region is given by

G œ "$ È^ $'  B # &B  %! thousand dollars, where! Ÿ B Ÿ 8 km Þ Determine the absolute minimum cost G over the interval! Ÿ B Ÿ 8. Round off to the nearest dollar.

[5] 10. The total cost of producing and selling a certain product is given by G œ "!!!  #!! B  #!! 68ÐBÑ dollars. Determine the minimum average

cost G œ^ GB Þ Round off B to the nearest integer, and round off the minimum

average cost to the nearest dollar per unit.

[5] 11. A rectangular playground is to be built and enclosed with fence. One side of the playground is along a busy highway and requires a very tall fence that costs $! $ per meter. The fence along the other three sides costs only "! $ per meter. The total area of the enclosed playground must be %& !!! m. Find the dimensions # of the playground that costs the least to enclose.

[7] 12. Sketch the graph of C œ (^) B  #B Þ Be sure to show all intercepts, horizontal

and vertical asymptotes, relative extrema and inflection points. It is not necessary to evaluate limits to verify vertical asymptotes.

[7] 13. a) Solve this indefinite integral: M œ '^ Ð%B  B%  $ =/- #^ 1 B  /#BÑ .B

b) The marginal profit for a certain product is given by .T.B œ '!  !Þ$B # and the total profit from B œ #! items is #&! $. Find the formula for the profit function.

9. B G

G w^ œ È $' B"$B #  & œ! Ê B œ #Þ& ) "$! !!! $

X 2/ 73837?7 -9=> 3= ""# !!! $ #Þ& ""# !!! $

10. G w^ œ #!! Ð 68ÐBÑ  ' ÑB# œ! Í B œ /' ¸ %!$ Þ X /=> >2/ -<3>3-+

@+6?/ +8. 13@/ =97/ 13@/= >2/ +,=96?>/ 73837?7ß G ¸ #!! $ :/< 3>/7 Þ

  1. BC œ %& !!!ß G œ $!C  "!C  "!B  "!B œ %!C  #!B

G œ #!B  Ð%!Ñ %& !!!B ß G w^ œ #!  Ð%!Ñ %& !!!B# œ! Ê B œ $!!

X /=> >2/ -<3>3-+6 @+6?/ B œ $!! +8. 13@/ =97/ 13@/= >2/ +,=96?>/ 73837?7 -9=> Þ X 2/ .37/8=398= +3-+6 +=C7:>9>/ß >2/+6 +=C7:>9>/Þ

C w^ œ BÐ BÐ^ B  # Ñ % Ñ # +8. C ww œ (^) Ð B  # Ñ) $ X 2/ -<3>3-+6 @+6?/= += 90 380 6/->398Þ X 2/ 986C :938>= >9 :69> ++8Ð 1 BÑ  "#/#B  G

b) T œ '! B  !Þ"B $^  "&! $