Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 50 views

    MATH 1013 Applied Calculus Section D Fall 2012 Test 2/ November 5, 2012

    Uploaded by

    examkiller
    This document contains a math test with 5 problems. Problem 1 has 8 parts asking to differentiate various functions. Problem 2 asks to find the critical points, intervals of decrease, and sketch a graph based on the derivative of a given function. Problem 3 is multiple choice. Problem 4 uses implicit differentiation. Problem 5 uses linear approximation. There is a bonus problem about the rate of change of height in a draining tank.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    MATH 1013 Applied Calculus Section D Fall 2012 Test 2/ November 5, 2012

    Uploaded by

    examkiller
    50 views8 pages
    This document contains a math test with 5 problems. Problem 1 has 8 parts asking to differentiate various functions. Problem 2 asks to find the critical points, intervals of decrease, and sketch a graph based on the derivative of a given function. Problem 3 is multiple choice. Problem 4 uses implicit differentiation. Problem 5 uses linear approximation. There is a bonus problem about the rate of change of height in a draining tank.

    Original Description:

    Original Title

    MATH1013_TT2_2012F.pdf

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    This document contains a math test with 5 problems. Problem 1 has 8 parts asking to differentiate various functions. Problem 2 asks to find the critical points, intervals of decrease, and sketch a graph based on the derivative of a given function. Problem 3 is multiple choice. Problem 4 uses implicit differentiation. Problem 5 uses linear approximation. There is a bonus problem about the rate of change of height in a draining tank.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    50 views8 pages

    MATH 1013 Applied Calculus Section D Fall 2012 Test 2/ November 5, 2012

    Uploaded by

    examkiller
    This document contains a math test with 5 problems. Problem 1 has 8 parts asking to differentiate various functions. Problem 2 asks to find the critical points, intervals of decrease, and sketch a graph based on the derivative of a given function. Problem 3 is multiple choice. Problem 4 uses implicit differentiation. Problem 5 uses linear approximation. There is a bonus problem about the rate of change of height in a draining tank.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 8

    ..

    9 (y):::
    MATH 1013 Applied Calculus Section D Fall 2012
    Test 2/ November 5, 2012
    Student Name:
    IP-No.:
    You have 50 minutes to solve the following problems: Show your complete
    work. The maximal mark on this test, including any bonus points, is 100.
    Permitted aid:
    help sheet of standard letter size with notes and
    formulas.
    NO CALCULATOR (graphical or nongraphical) is allowed on this test.
    Problem 1)
    Differentiate each function with respect to the independent variable:
    (24 marks)
    2
    .s.ec C...r)
    I
    ( x).:::
    r
    L,(!)::
    )4
    e) L,cs.).::. ( s +- 2
    "2 ( 3 2 2
    .f-'0) . sCs'-'2)-1(> "-b)
    s J- "2 c s t 2 )'Z
    g) fc. .x > :: .s V) ?, ex)


    d) f C X) .. --{ ';"X - I
    f) fcx) :::: .e >\. s; Vl( x--)

    >< .5 ; VI ( K) {
    ...e.. L!> ;"? (X) -t
    h) L, (.)() = X '2. c I +- ..e.)()
    L/( X):::. 2 X.( 1 -l- )(")
    -e ..J-K -e
    Problem 2) (20 marks)
    You are given a formula for the derivative f '(x):
    I
    fcx) = 3x+-t
    Use the formula to
    a) locate any/all critical points off (i.e. points x with f '(x) = 0).
    b) determine the intervals over which f is decreasing, and
    c) sketch a graph off based on your answers from a) and b).
    a.)
    r
    (ex) :.- 0
    =->
    X;:: -
    k
    3
    S) cf v '-C. , 5 v< a. creo :"::) er.e..vu-
    'f(_.,t.
    r( o( 1w
    lo:::_jPv;{ {o

    3ropL,
    wf._, ,Q , .C
    67
    t's
    a-t.; ve.
    I

    0

    f (k)

    < for KC:: -
    5
    =.)
    f
    ;s rA ere o.! ) "=..J
    C-eo -
    r 3 .
    fclc')
    c)
    v
    -3
    X

    Problem 3: Multiple Choice: Underline the right answer (20 marks)
    a) Suppose y '(2) > 0. Then the object whose position is y(t) is ...
    i) speeding up
    ii) slowing down
    iii) moving toward the origin
    iv) moving away from the origin
    v) none of the above
    b) Suppose y = sin-
    1
    (x). Then ...
    r -2
    i) 'f = -S , "'? (X)
    I
    ii) 7 == - cscc.x) co fcx)
    ' I
    i v) L-1 - --======--
    / - -.c; {-)( ?_l
    I
    iii) '7 =
    v) none of the above
    c) Suppose (xo, yo) is on the graph of the invertible function f. Then (f"
    1
    )'(yo) =
    -2 I
    i) - f C'7
    0
    ) F (Ko)
    (
    iii) iv)
    v) none of the above
    d) Suppose, 7 .:: (VI ( 3x - I) . Then ...
    ii)
    f j_
    ~ l
    7
    ~
    o ~ I
    3 .X
    i)
    iii)
    I
    I =
    iv)
    7'
    .:::
    3
    IV!(]x-+1)
    v) none of the above.
    Problem 4) (20 marks).
    Use implicit differentiation to show that the x-intercept andy-intercept of any tangent line
    to the curve ~ 1- -J7 ::: -fC' sum up to c, where c is any positive
    constant.
    .0:_ ( -Ck T- -C7{k) 7)
    d
    =
    dx(-(2)
    o(..x
    /
    ~
    _}_ I
    c{'1
    0
    2-fX'
    +-
    2 -iyrx)
    7
    ::::
    o<x
    _a'1
    I
    ::::._) -
    2 ~ _-0;' -
    a ~
    -
    -
    ..[ X.,-
    I
    2 ~
    ey l/IU 1--;oc- of .fw {o :::;Rr.-{ ( ;c..-e_
    p 0 ; VI {- (X 0 I '1 () ) " (...., CA { b (/ ( 0 ~ 5 ..( '0
    I
    fc- a (,.A
    1
    1- /.A;"( cJ..
    (_,e. cc,...- rvll. :
    (
    '( = 'to + '1 CK
    0
    ) (X - ~ }
    r W ; {c__, '1 ( k'o ')
    ,
    0 - Yo f- '( (Yo ) (x - k
    0
    }
    ->
    Problem 5:
    (16 marks)
    Use linearization at the specified a to approximate the following numbers:
    a)
    ~ a .:: b4
    - b)
    0.1
    = 0
    .e
    a
    I I
    cr) f_cy):::
    I
    Lr x'> =
    fc a > t f ()) , ( x - o )
    fca) +
    (ro) {x- o)
    f c 0 ) :: :1.[6<;'
    wL,ve.
    {c x) ::
    --\""
    -
    ~
    .e.
    I
    -
    [I K
    c K) :: e
    fcx) for- fc x) ::-
    3[7
    ,.
    ~
    fc: 0/ :::
    0
    = I
    f ( K) =
    ..=-)
    .e.
    I
    I
    .3
    ,.-t2-"3
    f' {)
    c 6) = .e ::
    I
    f a -
    1 I
    =->
    L( o.t) =
    0 0
    -
    3
    6 4-.2/3

    -12. Co.t-- o)
    -
    l
    '1--
    I 0.1
    -
    -
    1
    -
    3
    ( r; ct t--"1) 2
    -
    f.t
    -
    =
    l
    -
    _I
    "1
    -
    l(o
    c.,.g
    - ~ I I (
    6 f 3 . 6 v . 6 s; -6 'r)
    Lr f I . I
    4'6
    Bonus question (10 points)
    A tank in the shape of a lower hemisphere is 10 feet in diameter. When the fluid in the
    tank is b feet deep, its volume is
    V==
    'i'rb
    6
    (
    . '""::> 2. 2
    . .JQ +b)
    (
    where a is the radius ofthe circular surface ofthe fluid. The tank is drained at a rate of5
    cubic feet per minute from an outlet at the center of its lower base. At what rate is the
    height of fluid changing when there are 12 cubic feet of water in the tank?
    D = tOf .f.
    /
    /:. fi-
    'I..

    ,..QI d..
    ; c...-<;.
    Q

    { C-..t
    (
    f p
    !
    face ; '-. { IV (....1....,
    1)(
    o- 0. r:-, d._
    b
    r
    2
    2
    a_
    +(r-&) =
    fr "'Jd (
    2.
    o-
    a = ( r
    2
    - (o-&


    - ( r- b <.)
    r f<; (S fl) =->
    2 v.d.. p : l V -1 IVI.A- '> o( r- o e.-.ci.. b Ot-J { 7 '
    -==)> X :::- Y. - lto
    0
    I
    '1 ( Xo)
    <
    = c-rc) - c
    (
    V = f:, ( 3 ( 2 & r - t/ ) -1- b
    2
    ) - b C 30 b - 2 S ")
    - s & 2 - i7- s]
    3
    ;'2 )]
    3rJ.. slep: Vc,t.) = l;'))--'(&'v)) - Scf)
    1 .{ v t, c ) ev-. J v d v vt.--1 e V c ) Cue { '- (.... c i-2 De- c 0 I
    -h. '-"LA lL { . 6 '-ff t..-t o '--7 .( ;d_Rr- :
    o{V
    O(..f
    J c1f t '--'1-.?;f '1
    wL,ve ;)
    I
    & c -1 '> a.( tfG.t_.e vc..-e
    o.e-.c;{ /;) V(v) =: 12. {f
    3
    .
    I 2
    =--::> - 5 = &c .f) C t 0 rr be.;) - & c.,;.)) ( )
    !2
    3

    3 0
    IL,;<;. :s. b7 vJ\{ue_ o{ VL.,e_
    VaL v.e_ Ore Vt,., , o s. V (, .ll r-}3(,{ l-,oc- d- s:-ld.e_
    ./r . r r
    COt..-.hL-, vOv{ t'--1 ;t7
    ar-.cf... _:J rr?Q..f V IC, Gil'- /2 L Or .( o S;,.,.. - j)
    r b ::: t . (,.. bo I IV" s. ..e) { (I (
    I
    o,,_,c( f-or- bCJ.): /f. v'C-,tZ 1'-cAie..
    or-/ WC...,lM -1-G-,t. ftv;d {c;,/(s; IN(__,pc-t
    v o{ v t. /:;. f 2 f 3.

    You might also like