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Unformatted text preview: MACZ311 MG3 (2.1, 227, 2.8) Name: 50 i M 0 V) S Fall 2011 Section: (circle yours) MWF 9200 MWF 10:00 Directions: Use all available resources, including the math lab, Teague’s office hours, and fellow
students. To justify answers andearn partial credit, show work in the spaces provided on this document. MG3 is due at the start of ciass on Friday 9/23/11. Late papers will be penalized on a
case—byncasc basis, but typically a one letter grade (10%) penalty per day. Derive 6 Remarks: Once an expression in one variable [like I or x] has been authored, it can be assigned
function notation as follows. Rightclick the expression, Copy, click the Author menu, choose
Function Definition, type the desired function notation [like s{!) or f(x)] in the Function Nameand . Arguments field, rightclick in the Function Definition field, and paste the expression there. When you
click OK the function notation and expression appear with the := symbol instead of a regular : symbol.
From this point forward in your Derive session, the software be able to simply 01' evaluate function
notation expressions that you author, like f(2) or f(2+h). Also be aware that you can get the decimal form for an exact numerical result by highlighting it and clicking the icon. Tangent Line Slopes; Instantaneous Velocity; ri‘he Derivative For a function given by y = f (x), the average rate of change of f over the interval a 5 x5 39 is
f (19) “*f (a)
b~a
(17, f (52)) . In physical terms, let s (2‘) represent the position of a moving object at time t, measured given by . Graphically, this is the slope of the line through points (a; f (a)) and from an established origin. The object’s average velocity over the interval 2‘, S t S t, is given by change in postion _ 8U, ) H S (:1)
t, — tt . 7 _.
1 (IVE change in time ﬁr“. Consider function f The derivative of f at x is i) (conceptual) the instantaneous rate of change of fat .15.
ii) (graphical) mm, the slope of the tangent line at (x, f (2.)). iii) (algebraic) given by f’(x):1im_w , provided this iimit exists.
h—)D h d [f The derivative has a variety of notations, including f’(x) , hf ,
f I The derivative function is deﬁned to he f’(x) = lim W, provided this limit exists. Iii—)0 h FunCtion f is differentiable at x only if the derivative exists as a finite real number. Said another
way, f is differentiable at all x«values that are in the domain of f ’. As it turns out, a function
will fail to be differentiable wherever f has points of discontinuity, cusps, or vertical tangents.
We say... f is nondifferentiable at points of discontinuity, cusps, or vertical tangents. 1. Suppose a ball is dropped from the top of the CN Tower in Toronto, 450 meters above the ground.
3(1) = 450— 4.9;2 gives the ball’s position (height) above the ground at time t seconds, ignoring air resistance. We wish to first estimate, and eventually find exactly the instantaneous velocity at time I = 5 seconds.
Average Velocity
{seconds} (meters per second)
55% ~»ihaa
[5.5.051 ~ may  , m
Samflt micalahm: Overﬂgﬁﬂj \ng W— h ' «00%? b) Let v(r) represent instantaneous velocity at time I seconds. The velocity at time t = 5 seconds is
s(5+h)—s(5) a) Since the ball is falling, its position relative to the ground is
decreasing, so average velocity is a negative quantity. Complete the table to show average velocity over smaller time intervals.
{No work required. Record all signiﬁcant ﬁgures shown by default technology settings.) Based on the results in part a), we might estimate the instantaneous
velocity at time t: 5 seconds to be We ‘2“ ? meters per second. 2 SfS'. l) “'3 (5)_mgf2 given by 12(5) = llli‘i . Calculate this limit in the space below by hand, showing all h——)D h I
algebraic work. Conﬁrm with Derive. ’
Y 9(5’+L\3»~5(5>
We) a w k a .
he»: 0 Z
1: lim [450 w khan; k
‘. LlS’o « 9.01 (25’HOL HF“) “23237.5 'h “$0 {4
— lam “"30 “‘22.?” " tilt “ LS.ch WW.WMWWMWWHK~M __.. r h
1*: hm “(#qu V3615”: WW Mm weanca keno [q
H, Um (w, Win? “A :3 WIN melee; Jacarath
m Wee ' a home Derive Remarks: Follow directions on 9.1 to set Derive to associate 3(1‘) as the function notation for
s(5+h)—s(5) It
lim W. Note: When you highlight ms (5 “1)— S (5). h—w ;; h
Derive’s “variable” ﬁeld. Unlike previous limit Derive work, where expressions always had just one variable, Derive will recognize two variables here, A: and It. Click the drop—down arrow in the Variables
' ' field and select it to make sure you’re asking Derive to make 11 approach 0, not At. 450 — 4.9t2 . Then author diﬁerence quotient and use Detive’s limit feature to find and click the icon, the ﬁrst field is 2. Consider f(x)r:x2~x~6. "2: [Lani “£9 “$3 “@ '9a‘mi’ (\jwé’) a) Graphically speaking, f ’(1) is the slope of the tangent line to the graph off at the point (I, f (1)) , and this value can be found by evaluating f’(1)= limw = . Find this limit by hand, I h—iD h
showing all algebraic details in the space below. Circle the ﬁnal answer. = f,(1)=ﬁmf(1+h)—f(l)= Hm (I+1_«52(\+mé~ta:l;a2w
h MH/WLWJlWJF‘FdACﬂ tan it“san b) A portion of the graph of f is shown. Write an equation for
the tangent line at (I, f (1)) , and sketch this tangent line. Y“ RUﬂjC'OiOLel)
Yet*6): Mm!)
)Netg exﬂ Answer: x 3? K W “7 c) The derivative function for f can be found by evaluating this limit: lim f(x+h)“f(x). Find 5—H} ;;
this limit, showing all algebraic details in the space below. Circle the final answer. f(x+h)f(x)_ “M btwehl;éa WCXFW Kma... f’(x):limW— _ _ mo 1; \Nﬁb ‘5‘ 2“
«54a “ X +‘
.. \;M “Li m was in .3 \CM Laws9 b1 ml‘m 2 Him!
a...“ >
szxsoel' TM. L~w.ﬁ$L§MNJWmW £00 "'3 2*: m a 3. portion of the graph of sinxis shown. It is a fact that the tangent lines at xaci) x = 11) and x: 211 nuke 45° angles with the Xnaxis. With this in mind, estimate the tangent line slopes along'the sin x curve at x z 0 , x 2123—, x = 7: , x = E and x = 277: . Then use the second figure‘to plot these slopes as if they 2
were y—values, and connect the points with a smootheurve. Finally, In as if they were the yvalues of
a new function. Then connect these points with a smooth curve to see the graph of the derivative
function sin x. From your graph, guess the formula for the derivative of sin x. Answer: For y = sin :6 , the derivative function is EL: CD'S X '61): You can check yOur guess in Derive as follows: Author the expression sinx in Derive. Then, with sin .1;
still highlighted, click the El icon and click Simplify. ‘ The f  graph below is made from a‘line segment and a quartercircle. Sketch f ’ . Your sketch should use open or solid dots as needed, and asymptotic behavior should be represented using a
dashed line labeled with its equation. Graph of f Graph of f ’ S. Considerf(x)=x/E. “Fora :dngr “.3: J7?» , a) Set up the limit that gives f ’(4), and find this limit by hand. Show all algebraic details. (rm £(hthlk3nmggf2 n4): WWW
h‘ﬁp .
:lim 43AM
twee h
e. w 4W «new: ,\.
h‘ﬁo h + l b) A portion of the graph of f is shown. Write an
equation for the tangent line at (4, f(4)), and sketch ' .7 
this tangent line on the graph. www' (M)
y w ‘ X “r l c) Set up the limit that gives derivative function f’(x) , and find this limit by hand. Show all
algebraic details. gm? W WM VMmWAumwwWﬂmsr Mao an” a.
a tw Aemcyawgnﬁn
knee in r; a \m J??? 459*“ <5???“ “i” E: We h .W+J§?§ m. \*M ﬂx«t‘m w‘) ~ “it magnate
\ .( * * hm . W 524‘ (1) Over what x—axis interval is f differentiable? Asked another way, what is the domain of f’(x)? Answer: (“Bong V5 6. Consider the cube root function f = x% . Author the function x A (1/ 3) in your calculator and view the graph. Then author the function in Derive, click the Options menu, select Mode Settings,
click the down arrow next to the “Branch” ﬁeld, and change the setting from “Principal” to
“Real”. Open a 2D plot window and plot this function. I a) With it“ highlighted in Derive, use the E] icon to find the derivative function. f ’(x) = z .3 x 1’3 b) With the derivative function highlighted in Derive, use the icon to ﬁnd the one—sided limits
of the derivative as it approaches 0. Answers: iim f’(x)=ﬂand Hm f’fﬂzﬂ xaﬁ' 190+ c) The results of part b) tell us the behavior of the tangent line slopes on the graph of x“ as
x approaches 0, so they also provide information about the derivative graph as x approaches 0. i
With this in mind, make sidebyside sketches of 1A and its derivative. Graph of f ’(x) d) Of course, f 2 1% is deﬁned for all real numbers. However, f is not differentiable for all real numbers. After sketching f and f’,complete this sentence: f is defined for all real numbers,
but f is differentiable only for C.“ may) u £0 ﬁrm) cm X $ 0 e) Optionai Extra Credit In the last part text example 5 on pp.157158, Stewart uses the limit
deﬁnition of derivative to show that f =le, is not differentiable at x = 0. In similar fashion, attach the limit work that shows f z 1% is not differentiable at x = 0. Show all steps. ¥ 7; Arﬁﬁwmiaz‘aéig My “Jaw ﬁr; AME?!”
Jr a, le'k meal? mwméﬂgﬁml (gem H Swims EM.) 9: p dxwga (Hm: ref” (g§ 7 ...
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 Spring '11
 Teague
 Calculus

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