Download 1 - Penn Math

Sample Final Examination
1.
Consider the function y(x) defined implicitly by x2  3xy3  3y  16 . Find the equation of the line
tangent to the graph at the point (4, 0). Use your equation to find an approximate value for y when x =
4.1.
2.
The graphs of function f, its antiderivative, F, and its second derivative, f are shown in this plot:
Write labels here (f, F, or f )
Identify each curve and briefly justify each of your choices. {N.B.: Do not justify your third choice by
saying something like “well...that’s the only choice left.” Indicate why your choice is correct.}
3.
A psychological study of memory and memorization skills tested how quickly the “average” person can
memorize a list of words. They found that the number of words, W, that can be memorized in t minutes
is given by:
0.02t 2  t
W(t) 
t 1
At t = 5 minutes, what is the rate (in words/minute) at which the average person in the study can
memorize words?
4.
A holiday turkey is placed in a 325°F oven to roast. The turkey is initially at room temperature, 70°F.
(a) Assume that the turkey takes 5 hours to reach a temperature of 180°F. Newton’s Law of Heating
gives the temperature of the turkey at any time t after it has been placed in the oven as
T(oven) – T(turkey) = Cekt .
Find the constants C and k. and write the equation.
(b) Use the function found in part (a) to determine at what rate the turkey’s temperature is changing one
hour after it is put into the oven.
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5.
A cyclist is on a long straight road leading from her home to a nearby lake. Assume that the speed of
travel toward the lake is taken as positive (and speed toward home is negative) The graph below shows
the cyclist’s speed as a function of time. She reaches the lake at the point labeled A, stops for lunch, and
leaves for home at the point labeled B.
(a) determine the distance from the cyclist’s home to the lake; and,
(b) determine the total distance the cyclist traveled during her trip.
6.
The table below shows the values of two different functions. One is a power function and one is an
exponential function.
x
f(x)
g(x)
1.2 5.184
0.977
1.4 8.232
1.095
1.6 12.288
1.224
(a) Find an equation for each of the functions.
(b) Find an equation for the function h(x) shown in the graph:
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7.
Consider two functions f and g. The graphs of their derivatives are given below. For each given
statement answer true or false (circle the appropriate word) and include a very brief explanation.
f' ( x)
g' ( x)
1. f has no local maxima
2. f  40  5
3. f has no points of inflection
4. g is invertible
5. g , g and g all have points of inflection
6. g , gand g  all have local minima to the right of the y axis
TRUE
TRUE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
8.
The regular airfare between Philadelphia and Phoenix is $500. One airline flies the route using 747’s
with a capacity of 380 passengers. The airline does a study and finds that their average flight carries 300
passengers. The study also revealed that for every $20 reduction in the airfare, 20 more passengers
would take each flight.
(a) If x is the number of passengers, write an equation for the revenue, r(x)
(b) Use your equation from (a) and some calculus to find the fare that maximizes the revenue (solution
by other means will receive no credit).
9.
Evaluate the integrals:
2
a)

b)

0
0
2x 
 /2

2x dx
x cos(x 2 ) dx
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10.
An aspiring rocket scientist launches a model rocket from ground level at time t =0. The rocket goes
straight up. The graph below shows the velocity of the rocket as a function of the time since the rocket
was launched:
t3
t2
t1
• Sketch a graph of the acceleration of the rocket as a function of time
t1
t2
t3
• Sketch a graph of the height of the rocket as a function of time (h = 0 at launch
t1
t2
t3
• Let v(t) be the rocket’s velocity as a function of time. From the graph of the rocket’s velocity
t2
(above), which of these quantities is larger:  v(t)dt or
0
• What do you know about the sign of
t3
t
t3
t
v(t)dt
2
v(t)dt ? What does this mean physically?
2
• What happens to the rocket at t1 ? at t2 ? at t3 ?
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11.
Find the area of the region enclosed by the curves y = x2 – 2 and y = 2x +1. Write an appropriate
integral and show your work in evaluating the integral (a numerical result produced by a calculator will
receive no credit).
1 2 .
Find t h e
v o l u m e
o f
t h e
s o l id
obtained
b y
r o t a t in g
t h e
region in
t h e
p l a n e
b o u n d e d
b y
t h e
c u r v e s y  x  x2
a n d
y
=
0
a r o u n d
t h e
l in e
x
= 2 .
1
dx .
0 1  x2
trapezoid rule with three subdivisions to compute an approximation for arctan(3). .
1 3 .
1 4 .
W e
h a v e
ar c t a n ( 3 )
=

3
Use the
T h e
g r a p h
b e l o w is
t h e
g r a p h
o f
g
( x )
( t h e
derivative o f
t h e
function g ( x ) ) .
Suppose
w e
also
k n o w
t h a t
g ( 0 )
=
1 0 .
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
( a )
x
g ( x
( b )
g
d
( c )
c
( d )
g
( e )
1 5 .
Complete t h
0
)
1 0
W h a t
( x ) f o r
o
t h e y
W h e r e
o n c a v e
F o r
w
r a p h
o
S k e t c
L e t
e
1
following
2
t a b l e :
3
4
5
6
are t h e maximum and minimum
v a l u e s
o f
0 ≤x≤6 ,
a n d
w h e r e (i.e., for what values of x)
occur ?
is t h e
g r a p h
o f
g ( x )
u p ?
h a t
values
o f
x
d o e s
t h e
f
g ( x ) h a v e a n inflection point ?
h
t h e
g r a p h
o f
g ( x ) .
g(x) 
ln x
f o r
x
x
>
122
0 .
( a )
F o r
w h a t
values
o f
x
d o e s
g ( x )
h a v e
a
( l o c a l )
maximum, minimum or inflection point?
( b )
C o m p u t e
t h e range
o f
g ( x ) .
( c )
F o r
w h a t
values
o f
c
d o e s
t h e
equation
l n
x
=
c x
h a v e
a t
least
o n e
solution ?
( d )
F o r
w h a t
values
o f
c
d o e s
t h e equation
l n
x
=
c x
h a v e
m o r e
than
o n e
solution ?
( e )
F o r
t h a t
values
o f
a > 0
d o e s
t h e
x
equation a
=
x
h a v e
a
solution ?
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