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NEW YORK CITY COLLEGE OF TECHNOLOGY PEER LED TEAM …Feb 01, 2016 · New York City College of...
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MSEI
P IN
COM
PUTI
NG
NEW YORK CITY COLLEGE OF TECHNOLOGY
PEER LED TEAM LEARNING
MAT 1475/D604: CALCULUS I Prof. Boyan Kostadinov Lecture: Mondays and Wednesdays from 10-11:40 (Namm 718) Workshop: Mondays from 12-1 PM (Namm 505A)
PEER LED TEAM LEARNING MAT 1475 WORKSHOP SCHEDULE
DATE MODULE February 1, 2016 Review Pre Calculus and begin Module I February 8, 2016 1 February 15, 2016 No workshop February 22, 2016 2 February 29, 2016 3 March 7, 2016 4 March 14, 2016 5 March 21, 2016 6 March 28, 2016 7 April 4, 2016 8 April 11, 2016 9 April 18, 2016 10 April 25, 2016 No workshop – Spring Break
May 2, 2016 11 May 9, 2016 12
May 16, 2016 Review for the Final Examination
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[MODULE1:FOUNDATIONS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Sketch a graph of the distance travelled by a student over time from the first floor to the seventh floor of City Tech’s elevator. Assume that the elevator stopped only on the 4th and 6th floors and spent approximately 3 minutes at each stop. Carefully state the assumptions you made. Does your graph represent a function?
2. Fill in the appropriate domains and ranges for each function in the table below.
Function Domain Range
a) 21 x−
b) 2
11 x−
c) 1sinx
⎛ ⎞⎜ ⎟⎝ ⎠
d) 21xx−
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[MODULE1:FOUNDATIONS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
3. For the function ( ) 22 1f x x x= − + − , carefully evaluate and simplify( ) ( )f x h f x
h+ −
, for 0h ≠ .
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[MODULE1:FOUNDATIONS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
4. Plot the graphs of ( ) ( )1tan sin ( )f x x−= and ( )21
xg xx
=−
separatelyandthenonthesameaxes
overappropriateintervals.Makeaguessontherelationshipbetweenthemandseeifyoucanproveyourconjecture.Identifyallasymptotes?
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[MODULE2:LIMITS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Evaluate ( )4 2xf x
x=
+ − at several points near to 2x = and use the result to estimate the limit
0lim
4 2x
xx→ + −
.
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[MODULE2:LIMITS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. Simplify 4 2
4 2 4 2x x
x x+ +
+ − + + and then use the simplified expression to find the limit
0lim
4 2x
xx→ + −
. What is the technique called? Can you think of one other way to evaluate
this limit.
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[MODULE2:LIMITS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
3. If a fair coin is tossed x times, the probability that a tail is not tossed is ( )0.5 x . How likely to
never toss a tail is equivalent to finding ( )lim 0.5x
x→∞
. Estimate this limit by using several large
numbers.
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[MODULE2:LIMITS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
4 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
4. Applications discussed in problem 3 above also arise in predicting lottery outcomes
and in financial mathematics to name a few. Estimate the limit 10 7lim 2 1x
x x→∞
⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ that
occurs when considering compound interest.
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[MODULE3:CONTINUITYLIMITSANDSUCH] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Discuss whether utility bills and income tax rates are functions with discontinuities. You should create examples to justify your claims.
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[MODULE3:CONTINUITYLIMITSANDSUCH] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. For the functions listed below, determine their points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left-or right continuous.
a) ( ) 2
1f xx
=
b) ( ) 1/5 52 4f x x x−= −
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[MODULE3:CONTINUITYLIMITSANDSUCH] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
3. Evaluate the limits algebraically if it exists. If not, determine whether the one- sided limits exist (finite or infinite).
a) 2
2
3 2lim2x
x xx→−
+ ++
b) 0
5 5limh
hh→
+ −
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[MODULE3:CONTINUITYLIMITSANDSUCH] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
4 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
4. Is the function ( ) 29f x x= − continuous at 3x = ? Carefully explain your answer.
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[MODULE4:POWEROFDERIVATIVES] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. The position of a metal bolt falling from a skyscraper has the position function ( ) 216 19s t t= − +in feet and time measured in seconds. Find the instantaneous velocity of the metal bolt when
the time is two seconds by evaluating the limit: ( ) ( )2
2lim
2t
s s tt→
−−
.
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[MODULE4:POWEROFDERIVATIVES] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. Which derivative is approximated by
2cos 0.000000124 20.00000012
π⎛ ⎞− − +⎜ ⎟⎝ ⎠ ?
3. State and use the limit definition of derivatives to compute ( )/f a and find the equation of
the tangent line to ( ) 22f x x x= − + at 1a = − .
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[MODULE4:POWEROFDERIVATIVES] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
4. Find the first derivatives of the functions below using the power rule and appropriate properties of derivatives.
a) ( ) 23 e xf x x x e e= − + +
b) ( ) ( )24 25 1f x x x−= −
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[MODULE5:PRODUCT,QUOTIENT,RATESOFCHANGES,ANDHIGHERDERIVATIVESALLINAMIX] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Reading assignment: For a practical application of the product rule of derivatives see the following link. Computing the speed of model rockets with the product rule. https://en.wikibooks.org/wiki/Calculus/Product_and_Quotient_Rules
2. State and use the Product Rule to calculate the derivative
( ) ( )( )2 2
9, 1 3 1
x
df f x x x x xdx
−
== − + − −
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[MODULE5:PRODUCT,QUOTIENT,RATESOFCHANGES,ANDHIGHERDERIVATIVESALLINAMIX] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
3. State and use the Quotient Rule to calculate the derivative ( )2
31
5 2,4 1x
df x xf xdx x=
− −=+
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[MODULE5:PRODUCT,QUOTIENT,RATESOFCHANGES,ANDHIGHERDERIVATIVESALLINAMIX] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
4. Find the rate of change of the Volume V of a cylinder with respect to its radius if the height is
twice the radius.
5. Find the rate of change of the fifth root 5 x with respect x when 1, 32x = and 243.
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[MODULE5:PRODUCT,QUOTIENT,RATESOFCHANGES,ANDHIGHERDERIVATIVESALLINAMIX] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
4 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
6. Find the n -th derivative of the function ( ) kf x x= , for the following three cases: k n< , k n=
and k n> .Assumethat k isapositiveinteger.Theanswersforthethreecasesare:0, !n and( )
!!
k nk xk n
−
−
respectively.Itisbestifyoupickappropriatevaluesfor n and k toseeeachcase.
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[MODULE6:DERIVATIVES:TRIGONOMETRIC,EXPONENTIAL,LOGARITHMIC,ANDINVERSEFUNCTIONS] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Compute the derivatives of the functions below. a) ( ) 2sin secxf x x e x−= +
b) ( ) ( ) 33 1 sinf x x x= − +
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[MODULE6:DERIVATIVES:TRIGONOMETRIC,EXPONENTIAL,LOGARITHMIC,ANDINVERSEFUNCTIONS] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
c) ( ) ( )3 32 5 5 1xf x e x−= + −
2. Compute the higher derivative: 5
5
1ddx x
⎛ ⎞⎜ ⎟⎝ ⎠
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[MODULE6:DERIVATIVES:TRIGONOMETRIC,EXPONENTIAL,LOGARITHMIC,ANDINVERSEFUNCTIONS] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
3. Evaluate the first derivatives of the following functions:
a) ( )1xf x x=
b) ( ) ( ) ( )2 1 2tan 2 7ln 1f x x x x−= − +
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[MODULE6:DERIVATIVES:TRIGONOMETRIC,EXPONENTIAL,LOGARITHMIC,ANDINVERSEFUNCTIONS] NewYorkCityCollegeofTechnology
MAT1475PALWorkshops
4 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
c) ( ) ( )( )
115 2
3
2 1 2 73 1
xx xf x e
x+− +
=−
. Hint: Do not use the quotient rule.
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[MODULE7:IMPLICITDERIVATIVESANDRELATEDRATES] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. The figure below shows a portion of the graph 4 22 3 20x xy y− + = . Find the equation of the tangent lines at the points ( )1,6 and ( )1, 3− .
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[MODULE7:IMPLICITDERIVATIVESANDRELATEDRATES] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. Find dydx
given that ( )sin ( ) 0.x yπ + = Sketchthegraphofthisequation.
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[MODULE7:IMPLICITDERIVATIVESANDRELATEDRATES] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
3. The radius R and the height H of a circular cone change at a rate of 3cm/s. How fast is the volume of the cone increasing when 5R = and 15?H =
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[MODULE7:IMPLICITDERIVATIVESANDRELATEDRATES] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
4 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
4. The base of a right triangle increases at a rate of 5 /cm s , while the height remains constant at 25cm . How fast is the angle between its base and its hypotenuse changing when the length of its base is 25 ?cm
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[MODULE8:LINEARIZATION,EXTREMESANDMONOTONICITYALLTOGETHER]
NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Estimate the following using linearization: a) 15.999
b) 1
15.999
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[MODULE8:LINEARIZATION,EXTREMESANDMONOTONICITYALLTOGETHER]
NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. Find the minimum and maximum of the functions below on the given intervals. a) ( ) xf x xe−= on [ ]0,5
b) ( )2
2
11
xf xx
+=−
on [ ]5,5−
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[MODULE8:LINEARIZATION,EXTREMESANDMONOTONICITYALLTOGETHER]
NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
3. Find the critical points and the intervals on which ( ) 3 22 9 12 24f x x x x= + + + is increasing or decreasing. Use the first derivative test to decide whether the critical point is a local maximum or minimum or neither.
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[MODULE9:CONCAVITY,LIMITS,ANDCURVESKETCHINGINATWIST]
NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Find the intervals on which the function ( ) ( )2 2 xf x x e−= − for 0x > is increasing, decreasing,
concave up and concave down. Identify its inflection point, and any extrema(local max/min) and then make an appropriate sketch. check that the horizontal axis is an asymptote to this curve.
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[MODULE9:CONCAVITY,LIMITS,ANDCURVESKETCHINGINATWIST]
NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. Evaluate the limits below: a) ( )2lim 2 x
xx e−
→∞−
b) 2
2
5 24lim6 18
x
xx
x ex e
−
−→∞
− +− +
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[MODULE9:CONCAVITY,LIMITS,ANDCURVESKETCHINGINATWIST]
NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
c) 29 5lim
x
xx→∞
+
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[MODULE10:OPTIMIZATIONISTHEWAYTOGO] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Find the dimensions of an open box with minimal surface if its volume is 312m and it has a square base.
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[MODULE10:OPTIMIZATIONISTHEWAYTOGO] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. Find the points on the arch with graph 22y x= − that are closest to the point ( )0,1 .
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[MODULE11:ANTIDERIVATIVESANDDEFINITEINTEGRALS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Evaluate the indefinite integrals below:
a) 2xe x dxx
⎛ ⎞− + +⎜ ⎟⎝ ⎠∫
b) 3/24 2x dxx
⎛ ⎞−⎜ ⎟⎝ ⎠∫
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[MODULE11:ANTIDERIVATIVESANDDEFINITEINTEGRALS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
c) ( ) ( )sec tanx x dx∫
d) 16 99 16x x dx
−⎛ ⎞+⎜ ⎟
⎝ ⎠∫
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[MODULE11:ANTIDERIVATIVESANDDEFINITEINTEGRALS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
3 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
2. By drawing an appropriate graph of the signed area, evaluate the integrals below:
a) 5 5
5x dx
−∫
b) 4
61x dx
−+∫
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[MODULE11:ANTIDERIVATIVESANDDEFINITEINTEGRALS] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
4 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
c) 3 2
09 x dx−∫
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[MODULE12:THEFUNDAMENTALTHEOREMENDSITALL] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
1 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
Name:___________________________________ Points:______
1. Calculate the derivatives below by using the fundamental theorem:
a) 41
1xd dtdx t
⎛ ⎞⎜ ⎟⎝ ⎠∫
b) ( )( )sin 2
51
xd t dtdx
−∫
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[MODULE12:THEFUNDAMENTALTHEOREMENDSITALL] NewYorkCityCollegeofTechnologyMAT1475PALWorkshops
2 PreparedbyProf.SatyanandSinghSupportedbytheMSEIPGrant
c) 4
1x
x
d t dtdx
⎛ ⎞+⎜ ⎟⎝ ⎠∫