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Unformatted text preview: Math 171 ' R. Jones
Fall 2005 December 20,2005 Final Exam Name: TA’s Name: Section Time: No calculators, notes, or books are allowed. You must show all your work, and
explain your reasoning to receive credit for your answers. Work the problems in an order that will maximize your score, and check your
answers Whenever possible. Good luck! Math 171 Exam 2 2 1. (a) [2 points] Harry’s mustache grows at a constant rate of two inches per month.
What kind of function describes how the length of the mustache changes over
time? (b) [4 points] Suppose that Harry’s mustache is 12 inches long this month. Write
a function that describes the length of the mustache at month t, Where t = O
is this month. (0) [3 points] Draw a graph of the function from part ' Math 171 Exam 2 . 3 2. (a) [2 points] In January in Calculusville, AK the temperature smoothly rises and falls in a cycle each day. What kind of function might describe how the tem—
perature changes? (b) [3 points] Let t = 0 be noon. At 2 pm the temperature hits its daily high of
14 degrees Fahrenheit, while at 2 am the temperature hits its daily low of 14 degrees Fahrenheit. Write the amplitude, period, and phase shift of a function
that gives the temperature at time t (where t is measured in hours). (c) [4 points] Draw a graph of the function from part (b) and write down a formula
for the function. Math 171 Exam 2 3. Find all solutions of the following equations or inequalities: (a) [4 points] ’22: — 6 > 2 (b) [4 points] cos 3:1: = ~ cos 2:. Math 171 Exam 2 5 4. Find the following limits. If the limit is unbounded, determine whether it is positive
or negative inﬁnity. , _ 3:132 ~ 6x
(a) [4 pomts] 113% W 3 2 — 7
(b) [4 points] —:—_2—x Math 171 Exam 2 3 cos :3 sin :3
4 'nts 1' ——
(C) [ p01 ] 9:125 9: cos 2:1: Math 171 Exam 7 21:2
(x  2)”
all intercepts of f, and then carefully sketch the graph of f. 5. [10 points] Let f 2 Find the vertical and horizontal asymptotes and Math 171 Exam 2 6. Find the derivative of the following functions. Don’t simplify your answers. (a) [4 points] 433(1 + cosa:
b 4 't —————
( ) [ pom S] $5—sin1: Math 171 Exam 2 (c) [4 points] cos3(6:c + 2) Math 171 Exam 2 10 7. [10 points] The radius of a cone is constant at 4 inches while the height is decreasing at 2 inches per second. How fast is the surface area changing when the cone is 3
inches tall? [Hints see formula sheet for surface area of a cone] Math 171 Exam 2 ‘ 11 8. [10 points] Find all points where the graph of f (:5) = sin2 .7: + coszc has horizontal
tangent line. Math 171 Exam 2 12 9. [10 points] An object moving along a straight line has position given by 3(t) 2
it“ — 4t3 + 18t2 + 3613 + 45. Find all values of 15 when the object is decelerating. Math 171 Exam 2 v 13 10. [10 points] Using the deﬁnition of derivative, ﬁnd the derivative of f = cos x. You may use without proof that ling @ﬁ = 0.
13—» Math 171 Exam 2 14 Trig Formulas CoFuncttons (Complements) sin( — u) = cosu cos(§ — u) = sinu
7T
5 Addition 59’ Subtraction cos(u + v) = cosucosv — sinu sinu cos(u — u) = cosucosu + sinusinu
sin(u + u) = sinucosu + cosusinu sin(u — u) = sinucosv —— cosu sinv _ tan u+tan u _ _ tan 'u—tan u
tan<u + v) — 1—tanutanv tan(u v) _ 1+tanutanv Double—Angle £3 Half—Angle sin2u=25inucosu ' — t v i 1—cosu
 an— = ——
cos 2n = cosgu — Sln2u 2 1 + 0051) =1~25in2u = 1—cosv =QCOS2u—1 Sm”
SlIl’U
2tanu tan 2U : l—tan2 u — 1 + COS 'U Sums 55 Products sina+sinb= 23inﬁgcosa—gé [sin(u + ’U) + Sin(u — v) 2
a+b a—b sinucosv =§ ]
cosusinv = %[sin(u + v) — sin(u — 12)] sina — sinb = 2cos —— sin —
1
5
1
5 2 2 [cos(u + v) + cos(u — v ] cosa + cos b = 2 cos 9%? cos “7—” [cos(u — u) — cos(u + 22)] cos a ~ cos b = —2 sin “7+” sin 9—513 COS ’Lt COS U = sinusin v = Geometric Formulas length of a circular arc: 5 = r6 volume of a sphere: V = §7r7n3
area of a circular sector: A = %r26 surface area of a sphere: S = 47er
volume of a cone: V = énr2h “alume 0f a right CiTCUZGT’ cylinder: V = 2
surface area of a cone: S = 7T’r'\/7‘2 + h2 W" h h . '
surface area of a rzght czrcular cylmder: S = 27rrh P(cos t, sin t) P(x, y) ...
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This note was uploaded on 08/08/2008 for the course MATH 171 taught by Professor Gomez,jones during the Fall '07 term at University of Wisconsin.
 Fall '07
 GOMEZ,JONES
 Calculus, Algebra, Trigonometry

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