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Unformatted text preview: MATH 2200, Fall 2009
Practice Final 5mm RN (1) Compute the following derivatives: (a) (2) Compute the following:
(a) Suppose f(:z;) = sin(x)cos(:c). What is f”(:c)? HA : 5?er (—574 251— cosx' Cos X 3' Sz‘n} X fkaaZK “(A 5 rZsMxmsx + stx (—8?“ ¥\ ’0 (b) Suppose y = f is a function such that 3y2x + 2mg — sin(a: + y) = Vim f;ng 1;ij ~SI"1[&1"3>\ )
L \a . d .
Flnd a in terms of m and y. (éiﬂi’b.x+$UZQ + 4— W33  cosCx+3\ (\+%2§:~0 ig‘ CoS(X1°S\Xb>'$‘3 xx. 0” u +— .. S fov ‘
(3) Consi er the function 6 x CO me) = ex Use linear approximation to estimate the value of f (1.1). M 2: 6’: C awe“ amen itlrhﬁimﬂ‘mﬂbh
:: Q + 8(‘\\”\\ z (4) Suppose y = f is a function Which satisﬁes the equation
362 — 43/3 = 5 and such that f(3) = 1.
(a) Find the slope of the tangent line to the graph of f at the point (3, 1). a! z. 3: — _,Z><_><
Mr M )4 m 4 M (max, {3.1
L
2"”Ziﬁ'ﬁ’0 / 9%? «Mm: 31E
(b) Use linear approximation to estimate the value of f (3 + 6 ' 1L“ 2 4 ((3 +73) 2: U3) + 1%) = “lairL) (5) Calculate the following indeﬁnite integrals: (a) R‘Wéra  r ’3
I
/(2$4—[email protected]:x (6) Suppose y is a function of a: such that (7) Suppose y is a function of :r such that dy —=4 2—2
ydcc m and y(1) = 2. Find y. (8) Suppose y is a function of ac such that iv._ (y + 3) y sin(23:)
z... dzc ,__
and 31(0) 2 3. Find 3/. ‘9 J jigs—Q 71%ka ‘3
ﬂ+13€§ 4‘3 3 Wyliiyvz‘x
(3+3ln‘3 ='7i_cost +C_ é— ("loo ° ‘3 (9) Suppose y is a function of a: such that 5 \V‘L Q“ SUV
——4(y—3)3~sin(w+y—7)+3 4 dy _
clan and y(4) = 3.
(a) Find the slope of the tangent line to the graph of the function y at the point
(4, 3) (b) Use linear approximation to estimate the value of y(4 + 3mm) 3m +3‘(~5( wt; ~‘t\
:: 3' 3+1}; TEE (10) Consider the function 7 . = lnx + E — 3 on the interval [1, 00) (a) Find all critical points of f on this interval (or write ‘none’). Critical points: 4m: 3‘;  gL : x: 0 a‘l’X‘zl’“ (b) Construct sign chart for the derivative of f in the space below, illustrating
where the function f is increasing and decreasing. r
.\,4/’ Show your work below: (c) Find the minimum value obtained by f on the given interval: (11) A farmer would like to set up a rectangular enclosure of fencing at a ﬁxed cost of
$100. If three of the sides of the enclosure cost $2 per linear foot of fencing, and
the fourth side costs $5 per linear foot, ﬁnd the dimensions of the enclosure which
maximize the total area at this cost. (a) What is the quantity which you are trying to maximize? mg z ‘ Write an equation for the quantity to be maximized, as a function of only one
variable, and state what the meaning of this variable is (i.e. radius, diameter,
height, etc.). Equation : A :1 I” < [00 , q. I‘ Use the space below to show your work. AJN £7133” “’ “"M‘ C”; :100 : 1w 4.2,,» +ZL+STQ
: 4w 1—7‘L 8"
“,5 walk
Lt— 0
(b) What is the interval on which this function is deﬁned? / L9— 1 You may justify your answer below (only for partial credit if above answer if
incorrect). W tit, H1 (c) Find all critical points of this function on the interval you have found. Use the space below to Show your work. (d) Find the dimensions of the fence (length, width) which maximize the total area. Dimensions: lee ’r 7 M'zl'f’L: Use the space below to Show your work. {1750/7 Jo A: : wd’gz’é (12) Suppose the demand of a given commodity is related to its price by the equation 1
D1n(D/10) + 1 = I; Where D is the demand and p is the price, and recall that the revenue may be
expressed as R=Dp. (a) Find the rate of change of the revenue with respect to price when p = 10 and
D = 10 AEUHD/Io\ +igs—LF3 \J (b) Use linear approximation to estimate the revenue when the price is 1 D: to ‘93“)
ML
K 73 + —— ll—lb AD (11) 1
’l’zn R‘kw AP i?:w( \ I6 *J’ZEE
0
5100 + 52 ’”Z:
/\ , v / at? (am?
:: “O ’ " 3104.1?)
El ' 4g, 4g? +D
(kw/0:10 ‘ l . —Z— .to +10 (13) Consider the following graph of a function f In the spaces below, write either a ’+’ or a ’—’ to describe the intervals on which the
ﬁrst and second derivatives of f are positive or negative. ...
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 Fall '08
 KAZEZ
 Calculus

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