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Unformatted text preview: MACZ311 MGl Algebra/Trig'RevieW
Fall 2011. Name: Section: (circle yours) 9 MWF 10:00 Directions: Use all available resources, including the math lab, Teague’s ofﬁce hours, and fellow students. To justify answers and earn partial credit, show work in the spaces provided on this
document. MCI is due at the start of class on Tuesday 8/30/11, and the ‘flate watch” starts at the
startofclass deadline. Papers submitted less than 24 hours late are penalized one letter grade
(10%), and papers are not acceptable more than 24 hours late. 1. An open box with square base x meters on each side must have volume 40 m3 (cubic meters). b) Find a function formula for S (x) , the single—variable function that expresses
surface as a function of base dimension x. Answer: S(x) = Suppose material for the sides of this box costs $3 per square meter, and material for the bottom
costs $5 per square meter. Find a function formuia'for C(x) , the single—variable function that expresses total cost of materials as a function of base dimension x. Answer: C(x) = 2. Recall that the standard equation for a circle of radius r centered at (ink) is given by
(ac—[’02 +(y—k)2 : r2. a) b) Write the equation for the full circle centered at the origin with radius 3. Answer: Let y = 500 = —\/9 — 3c2 . Solve the equation in part a) for y to Show that the graph of s is the‘
non—positive semi—circle centered at the origin with radius 3. Sketch s andstate its domain and
range. . . . . 4 . Domain: Range: 0)! Complete the formula for the piecewise—deﬁned ﬁmction shown. The left and right pieces are
linear and the middle piece is the semi~circle of radius 2 centered at the origin. {3:3 f(x)= if—2Sx32 2
3. Consider r(x) = xmg. a)' Rewrite r in radical form. Answer: r(x)= b) Each of the non—zero xvalues below produces a rationalnumber output. Complete the table
without using technology, expressing function values in exact form. Sketch the graph.
Note: It will be virtually impossible to make a handdrawn sketch to scale, so don’t try to do
this... just sketch the graph without establishing ﬁxed scales on either axis. c) State the domain and range of 1". domain:  rangef 3x+l 4. Consider f (x) x for x at 0. x
a) Simplify the formula for f by carrying out the division. Answer: if = b) As x tends toward +00 and —00 , the function values tend toward a speciﬁc numerical value,
called a “limit”. Determine this limit by using technology to examine numerical tables and the
graph of f . The result should give information about aSymptote in the graph. Conclusion: lim f (x) = lim f (x) : x—>+m x—>—°0 , so the graph has  asymptote ( horizontal or vertical '2 ) ( equation ) As x tends toward 0 from the left 
(negative) side of O, the function values
again get inﬁnitely large numerically.
Determine Whether the function values
tend toward +00 or ~00. c) As x tends toward 0 from the right
(positive) side of O, the function values
get inﬁnitely large numerically.
Determine whether the function values
tend toward +00 or —oo. Answer: lim f x—) 0' Answer: lim f‘(x): x4 t)+ Conclusion: The graph has asymptote ( horizontal or vertical ? ) ( equation) d) Sketch the graph of f . If the graph is asymptotic to a line other than a coordinate axis, sketch this asymptote using a dashed line, and label the asymptote with its linear equation. Also label
the x—intercept. f(x+h)—f(x). 5. Definition: For ﬁmction f (x) , the difference quotient at inputx is given by. h a) Let f (x) z 3 — x2 . Set up the difference quotient and simplify it completely. b) Let f (x) = Set up the difference quotient at x = 2 and showthat it Simpliﬁes t0 —¥i~
x 2+h' 6. Without using technology, find the exact value of each expression or write “undeﬁned”. Express ' answers'for parts k)Fn) as exact radian measures. . 27? _ I
a) sm[?]— 7. a) Sketch at least two periods of the
function y = cos(3x). Label enough tick markson each axis to indicate their
scales. b) Sketch the function y = 2 sin‘1 x. Label the endpoints with exact x and ycoordinates,
and label enough tick marks on each axis to
indicate their scales. 3‘ 8. Solve each equation, ﬁrst over [0, 27F), and then over (—oo,+oo) . Express answers in exact radian
measure form. When solving over [0, 27:) you will, of course, get a ﬁnite number of solutions. The solutions, over (e00,+oo) include all solutions over [0, 27:) and all angles coterrninal with these, so
this will be an inﬁnite set. a) 2cosx —1 m 0 ' l b) sin(2x) m cosx
Answers: . ,
Over [0,271) : ' Over [0,271.) :
Over (—oo,+oo) : 7 Over (—oc,+oo) : 9. Consider f (x) : xef". a) Simplify the ﬁinction formula to express f without the negative exponent, and state the domain.
Answer: f (x) 2 domain: b) As it tends toward +00 , the function y = ex grows much faster than 3} = x . Consequently, the
. as xtends toward +00. That is, lim f (x) = x—)+nc function values for f tend toward 10. Consider f (x) = i .
in x a) State the domain. Answer: b) As x tends toward +00 , the function y = x grows much faster than y = in x. Consequently, the function values for f tend toward as x tends toward +00. That is, lim f (x) = x—)+oo lnx—l . c) The function f ’(x) = 2 is the derivative function for f . Find the exact values of x, if any, (in x)
that satisfy f'(x) 2 0. Show the work. Answer: 36 = ...
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 Spring '11
 Teague
 Calculus

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