Math 120 A - Autumn 2016Midterm Exam Number Two
November 17th, 2016
Name: Student ID no. :
Signature: Section:
1 142 153 154 16
Total 60
• This exam consists of FOUR problems on FIVE pages, including this cover sheet.
• Show all work for full credit.
• You may use a TI-30X IIS calculator during this exam. Other calculators and electronicdevice are not permitted.
• You do not need to simplify your answers.
• If you use a trial-and-error or guess-and-check method when a more rigorous method isavailable, you will not receive full credit.
• If you write on the back of the page, please indicate that you have done so!
• Draw a box around your final answer to each problem.
• You may use one hand-written double-sided 8.5” by 11” page of notes.
• You have 50 minutes to complete the exam.
–
1. Happy Thursday! I bought you this graph.
-3 -2 -1 0 1 2 3 4 5
-2
-1
0
1
2
3
4
5
x
f(x)
(a) [4 points] Compute f(f(f(4))).
(b) [5 points] Find the domain and range of f�1(x).
(c) [5 points] Let g(x) = f(2x+ 1) + 1. Find the domain and range of g(x).
fff -H4D)=f(ffiD=f( 3) =D
Domain of f"
= range of f
=[-2,l]U(2§Range off "=domain of f =f3,DU[3,§
Domain Range
Original ( -3 ,DU[ 3.5 ] [ '
2.1 ]U( 2,5 )
Shift up1 ( '
3,1 )U[ 3,5 ] [ 1,2 ]u( 3,6 )
Shift left 1 ( ' 4,0) 42,4 ] [ 1,2 ]U( 3,6 )
Scatgyhorijs. -2,4€Et2]u@
2. [15 points] Gomba is on a diet. His weight is a linear-to-linear rational function of time.
Right now, Gomba weighs 21 pounds.
In 1 month, he will weigh 20.5 pounds.
In 7 months, he will weigh 18.75 pounds.
In the long run, what will Gomba’s weight approach?
(Assume Gomba will live forever.)
*- a××+thd f(o)=hf=21 - b=
21dg%omba's weight f( D= 9+4=20.5 - atb
¥20.5+20.5A
lwetixjnjil,
" * :# .pt?IIiIiI:P7 (20.5-0.51)+211=131.25+18.75115.6€
-12.25=1.251
D= 9.8
b=21( 9.8 )b=205.8
a=20.5--5/9.8)a=
15.6
3. [5 points per part] The rent for a one-bedroom apartment in Beattle is growing exponen-tially. (Even though the city is filled with bees.)
(a) In the year 2000, the rent in Beattle was $1020, and it increases by 2.3% per year.
Write a function f(t) for the rent in Beattle t years after 2000.
(b) The average monthly rent in Tickoma is also growing exponentially.
In the year 2007, the rent in Tickoma was $500 less than the rent in Beattle.
In the year 2016, the rent in Tickoma is $1000.
Write a function g(t) for the rent in Tickoma t years after 2000.
(c) When will the rents in Beattle and Tickoma be equal?
(Round your answer to the nearest year.)
⇐=A.btb.lt#
-10204.0235¥= Aobt
g(7)=A .
b7= fl 7) - 500=696
g( 16)=Aob' 6=1000 edit
r%÷¥17;¥:] ...
.gs#5mmb
1020 ( 1.0235=525 ( 1.04109 )t
In ( 1020 ) + In ( 1.025) = In ( 525 ) th ( 1.04109T)
In ( 1020 ) t £ In ( 1.023 )=h( 525 ) t th ( 1.04109)
+ ( In ( 1.023) - In ( 1.04109))= In ( 525 ) - In ( 1020 )
t.int?oIYHYoI?m=3s#
4. A polar rectangle is the region bounded by two concentric circular arcs and two raysthrough the center of those arcs. Okay, fine, here’s a picture:
y
x
✓
(a) [4 points] Write a formula for the area of this polar rectangle.(Your answer will involve x, y, and ✓. Let ✓ be measured in radians.)
(b) [4 points] Write a formula for the perimeter of this polar rectangle.
(c) [8 points] Suppose you have 24 meters of fencing, and you want to construct a fencein the shape of a polar rectangle with central angle ✓ = 1.2 radians. What is themaximum possible area of your fence?
),
A=tsox2-£oy÷
Inkiest. ¥iII
24=(2+1.2)×+4+1.2)y
t
24=3.2×-0 .8y
y=4¥30 A= 's(1.3×3 - £4.2)( 4×-393
A = 0.6×3 - 0.646×2-240×+900)
A =- 9×2+144×-540
fsfaph !
Area ,
maxareayy.gg-540 .
YEH -36cg
>x