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Unformatted text preview: MAC2311 MG6 (3.10, 4.1, 4.7) Name:
Fall Section: (circle yours) 9:00 Directions: Use all available resources, including the math lab, Teague’s ofﬁce hours, and fellow
students. T o justify answers and earn partial credit, show work in the spaces provided on this document. MG6 is due in Teague’s unit by 3:00PM on Monday 10/24/11. Late papers will be
penalized on a casebycase basis, but typically one letter grade (10%) penalty per day. “Linear Approximation; Differentials; MaxﬂVIin Values; Optimization” )‘A Let y : f , and consider the speciﬁc point on its graph, (0, f ((1)). Linear approximation function L(x) is given by L(x)=fv(a)+f'(a)(x—ﬂa) . At any particular xuvalue near x : a, L(x) approximates f . 1. Let L(x) be the linear approximation function for f = at a = 8. 5/3 L00"; {3(a) "i" ‘JCIMNXFOQ ow dag) and {3(a) 5: ?C  a) Find the formula for L, sketch L (x) on the graph shown, and complete the sentence below.
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Rx) 7* X 3 Low W) + News?) ' as} Y . .n .3 ' x... g) UK) W \E + 3%: ( 5? a) .2; “;LM .3 d? H L00 *3: 2, “t— 7: (Xe or _ .1... 5’s. ((2.?
L (>0 =~’ :2. X l" a 5»)
For xvalues near 8, the function f = is
approximated by the linear function ...L.. .i
L(x)= WarX4» 3‘: L00 ll} 12 14 ‘0) Use L(x) to approximate «3/7186 . The result will be a repeating decimal, so record it as such. (For instance, the repeating decimal 6334949494949. .. is written 68319 .) Then use the calculator to evaluate €77.86 directly, recording all digits shown by the calculator. (Of course, your ﬁrst answer
should be close to the second one!) By linear approximation formula L(x) , {77.8 m , but V3 7.86 = loqggg’é’qé’og . I.
WK
1 i“ "m ’2' a vain ‘0 isw if” 2. Consider3223A. ‘ 7:: 83* dx a» C”; 2” J amt as, a) Find the differential expression dy . Then evaluate dy for x : 4 with dx 2 0.02, recording all 10
sig ﬁgs shown by the default setting of your calculator. I (Q)  [ﬁtﬂ Lev?“ O o m r) AIJ
Answers: City: 2” X,and dy= 2... a ' 2‘" “‘ngng b) Calculate Ay as x varies from x = 4 to at x = 4.02, again recording all 10 sig ﬁgs shown by the default setting of your calculator. (Of course, your answer from part a) should be close to this
answer because differential dy is supposed to approximate thekexact change Ay when Ax is small.) “ﬂog/2. L372. AF 6 m e 2: .‘Otﬁraalaaaam Let y .2 f (x) , and recall that its derivative can be denoted by (j—y
(x or f‘(x). From % : f’(x) , it follows that dy=f’(x)dr . ,1?
Consider points P(x, f and Q(x+Ax, f (x +A‘c))? and let dx 2 Ar.
The exact change in y is Ay = f (JC+ Ar) — f . As shown in the ﬁgure, dy approximates exact change Ay as long as Ax is relatively small. 3. The surface area of a sphere of radius r is given by S = 47er . Suppose the radius of a particular
sphere is measured to be 8 cm using a measuring device with a maximum error of 0.03 cm. a) Calculate the difference in surface area between a sphere of radius 8 cm and a sphere of radius
8.03 cm. Record the exact answer (the one with 7: in it), and also record the decimal form of this
answer using all sig figs shown by the default setting of your calculation. Answer: When Ar: 0.03, AS: W7 cm2 or Q. cmz. exact form decimal form with 10 sig ﬁgs
is e an“ (9.033% Lin“ (3)”: are (sets e m 82’) b) Use differentials to estimate the maximum error in surface area when radius is measured to be 8
cm using a measuring device with a maximum error of 0.03 cm. Record the exact answer (the one
with 75 in it), and also record the decimal form of this answer using all sig ﬁgs shown by the
defauit setting of your calculator. (Of course, your answer should be close to both answers in part
a) because differential dS approximates the error in surface area when radius is measured to be 8
cm with maximum error of i 0.03 cm. A Answer: When dr = 0.03 , dS : W cm2 or Cm 0?” 8 5W] cmz.
S m (31? raw“ W ( exact form decimal form with 10 sig ﬁgs
e m 0.03) 0) Calculate the corresponding relative error, and percentageerror in the surface area associated
with measuring the radius to be 8 cm using a device with maximum error 0.03 cm. . a 6 relative en'or = e 0 9 W25 percentage error = v %
W w Bur” die Def. (p.274) A critical number of a function f is a number 6 in the domain of f such that either
f ’(c) = O or f '(c) does not exist.  Extreme Value Theorem (paraphrased from p.272)
If y = f is continuous on closed interval [a,b] , then there is at least one x—value in [a,b] where f reaches an absolute maximum and at least one xvalue in [a,b] where f reaches an absolute minimum. Incl'easingmecreasing Test p.287
a) If f'(x) > 0 on open interval (0,1)), then f is increasing on that interval. b) If f '(x) < 0 on open interval ((1,1)), then f is decreasing on that interval. The First Derivative Test (for Local Extrema) p.288
Suppose c is a critical number of a continuous function f . a) If f’ changes Sign from positive to negative at c , then f has a local maximum at c.
b) if f ’ changes sign from negative to positive at c , then f has a local minimum at c.
c) If f ’ does not change sign at c , then f has no a local maximum or minimum at c. 4. Find the absolute extreme values of each function on the indicated interval. No sketch is required, but
your work should show the derivative function and how you found critical numbers. Express answers
in exact form. Use the follbwing phrasing to state answers: Absolute maximum value 7 occurs at x =
Absolute minimum value . occurs at x =
1 3
a) : 4x—»3«x on [0,4] m r: 441
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\WWM Abs mag V‘otime ﬁ .33& W” % “Ci” K “’2, .
A“ We Value anger)? w wig mi” vain 7C 00 a; LMWYZa— W)?" m 7 W MM ‘ '2 . . L
I “ZQV‘X” (5M1)! “"1505 K) 5m2’x +505): 2:2, 8—x 29/x(12»x)2 [#114]. Express answers in exact form. Also sketch the portion of the graph over the interval 5. Consider f(x)=w;—312’.x2—x3 with f'(x)= . Find the absolute extreme values on [~2,14]. Your sketch should show the correct graphic behavior at points of nondifferentiability.
Use the following phrasing to state ﬁnal answers: t 3 “W” Absolute maximum value occurs at x = Absolute minimum valuew‘lﬁﬁ occurs at x = j i . r. \
. 7. “ti ..,,stt:JW
____hm=: v7xﬂﬁgﬁﬂ.‘M.¢Tu,:7M#sw~9nx~ h»sz .. "a .. 2%}; (AM X T“: g ‘ rte" O
I 7‘ t:
llm {:00 '2: +09
Me’s 0+
“Mr. W, ..._ {a .. 4“
I a ‘t 6. Civil Engineering A road from point A to point B
must cross ﬁrst through public land and then through
private land. (See figure.) The cost of building the road over public lani $80,000 per mile, and over private land it’ ~ ' I $96 00 0 You work for a civil engineering firm and your boss needs a cost analysis that determines the cheapest
route for this road. The table organizes the possibilities: u. 1 F 1min
pn+al CO§+ '3; +‘ $6000 ( a", [35 >
x Total Cost (rounded to nearest doilar) I A to J, and then I to B
A to D, and then D to B a) The ﬁrst three possibilities do not require calculus. Determine total cost for these routes and
record answers in the table. In the space below show how you are getting these answers. b) Let x represent the distance from point K to point D. Write a formula that expresses total cost as a
function of mileage x, and state the closedinterval domain for this cost function. Sketch of the
cost function over this domain, “Baha‘i your gal/e 5‘ ' ca): Domain: :0; 3: I .‘1 veneers? c) Use calculus and DERIVE to determine the critical xvalue for the cost function in part b). Then
use all signiﬁcant digits reported by DERIVE for this critical xvalue to determine total‘cost.
Report both answers in the table above, rounding the x—Value to the nearest hundredth of a mile
and total cost to the nearest dollar. Also label the sketch in part b) at the point where cost reaches
its minimum with both coordinates. » it. me as.  L... W“ + Mm) Kai4;”)
C (X) .3; 9 W... X . is “firmness.. _ W \Iﬁ'i—l « ah (ax44o 9 ‘3‘? 9  x:1.ﬁoe‘t3é2.5’7m mo car at at”;
Xxlfil micrograms?) 2’ autism cm a \o‘l’
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 Spring '11
 Teague
 Calculus

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