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AP Calculus Name: ﬁi 2 Review 4.14.3
Topics to review for test:
0 Absolute extrema on an interval
Mean Value Theorem
Relative extrema
Intervals of increasing and decreasing
Concavity and points of inflection Second derivative test
F, F', F" For each function and interval below, find
(a) critical numbers: (b) values of relative and absolute extrema.
Justify your answers. 1. f(x) =x2(x2 —4) on [1, 2] 2. f(x)=cos7rx on [0%]
x :x‘l— Lix"
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3. f(x)=2"3+5 on [0, 51 4. h(s)= 12 on [0, 1] (D (“M”)
S—
f'lxl: 7: b) x “X3 Wis): “#312 f O
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a) no Lr\'l'*\5 3 0‘) S: 2
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(lib cow 5 (3 ml in mime!) 5. For each graph, find and characterize any extrema on the intervals depicted. 6. Consider the following graph.
Which of the points are absolute maxima? f Wm Q, Which of the points are absolute minima? A Which of the points are relative maxima? 82,5 Which of the points are relative minima?
DE Which of the points are neither a maximum or minimum? aye 7. Show that f(x)= x + 1 satisfies the requirements of the Mean Value Theorem on the inéepa
x MVT: {l0}; tlb) OK) [i— ,.2] Then find the value(s) of c guaranteed by the theorem 3463 b’k
 — _ (71 1
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2 V? I o X : LL
8. Show that g(x)= x3 satisfies the requirements of the Mean Value Theorem on the interval
[0, 1]. Then find the value(s) of c guaranteed by the theorem. Q: j X l3  CWlVWﬁUﬁ gX—Vﬁ £034 1/1 1 ,1; H3
/ ,3 " i . _ .1 l0 l1 51/3 : ll}? ; V?“
.111111111111 5:11 {To 1:553 (0”) :5 2
9 Explain why g(x)— — x3 from the previous question does NOT satisfy the requirements of the
Mean Value Theorem on the interval [ 1, 1] H ~is not dA’C‘QVf‘» {table 9“; X20 10. Use the graph of f to estimate the numbers in the interval [a, b] that satisfy the conclusion of
the Mean Value Theorem.  11. A TesT plane flies in a sTraighT line wiTh posiTive velociTy v(T), in miles per minuTe, where v is a
differenTiable funcTion of T. SelecTed values of v(T) for [0,40] are shown in The Table. +(minutes)10 15 20 25
v(+)(mm) 92 95 70 45 24 Based on The values in The Table, whaT is The smallesT number of insTances aT which The acceleraTion
of The plane could equal zero on Th5open inTerval (0,40)? JusTify your answer. “RV (OHS W (35231)) We Mam VOL“ \/
Q, 2 "1’7 ” O (9.0, wit/l , Vii/(W )lmre
Al l6~0 ' 30’3‘5 “’0 m1 01%ng Two lnélcw es mi»... <2.
«like ACCAU’M‘IW For each funcTion in 12—13, find
1. criTical numbers CO'DlOL eﬂéuﬂk l O 2. inTervals on which The funcTion is increasing and decreasing
3. values of relaTive exTrema
JusTify your answers. 12. f(x)=3x5——5x3—1 13. g(x)=3+sinx,[0,27r]
$‘(yyf’ l5XLl ”HEX/2.50 3‘06): coSX :79“, > \p 9.70
l5X1’(X.z’l)—’O \’> X: g f... 1 in: 37" ﬂ
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at My X:,[ (L? + 4° helm.“ x91: 1L7,” 14. LeT g(x) = x + 2 sin x on The inTerval [—71, 7:]
(a) For whaT values of x does The graph of 9 have a horizonTal TangenT? 3‘04): 1+;cosx;o X: 21:“ ng .L 3 2
COSX‘; "a .2 j (b) On whaT inTervals is The graph of g concave down? JusTify your answer.
ll _ ,. x. .
900/ 1}”er 1;“ Jr.  l comm: damn
W :0 1"“; (o) m l “40 Y: O 15. Let f be the function given by f(x) = x3 — 5x2 +3x+ k, where k is a constant. (a) On what intervals is f increasing? Justify your answer. \
{‘00: 3x1~im<t~320 H .. l '(m/éW/N)
(SXriy)lew3>;<—’ i 3 ‘9 l‘hC blq , ,L
>\ 3)_3 «“70 (b) On what intervals is the graph of f concave downward? Justify your answer. 9‘00: Lox—lo Tc {‘5‘ I § _\ .. f3 3' t; C onccwc .
i’“ 3 3 a W :5 )
(c) Find the value of k for which f has 11 as its relative minimum. ‘ ‘ ‘P H C 0
Val mm M: 337513? 43l3)+l< )(z3 (Part 0O
3:” K5810 16. True or FaISe. If f'(c)=0 and f"(c) < 0, then f(c) is a local minimum. Justify your answer. f’alSQ l «C(Q) :0 and f”{C>4O 44mm 1C“) is a local mllnlfinw Lde wrlbalivﬁ “VSl) 17. Let f(x) be a twice differentiable function defined for all real numbers. f(x) is not constant.
The table gives values of the first and second derivative at several points. The function has no
critical points other than those mentioned in the table. List all relative extrema. Justify your answer.
X: V} X 7 0 X: 9,
W lAl/WC NOV r (l Gil he Mlnl‘mu ,c. Hickﬁve. max
490 il:O «lléO
l“ LO ,‘ll >0 ll‘co 18. The figure shows the graph of f'(x), the derivative of a function, f. The derivative is
continuous and f'(2) < 0. . y.
Which of the following is true? ' I. The function has at least one relative minimum point.
II. The function has at least one relative maximum point.
III. The function has no relative maximum point. (a) Ionly (b) II only (c) III only .nd II (e) Iand III r‘ ,."". “ A i, l ‘2.  ' 5‘“, l K
Since wt OM 91g 0 Pt M! an Irl t ﬂaw a El 6
19. Match each graph (1—6) with that 0 its derivative (AF). bi MG‘KL “a \l ( lNHln Pl; I gm sf“?
E i E E. E
a 33 § 2 fl
if V"
c. E
i i
E E ;
{RN “X
Maw E W
F. E f;
x
I. W" 20. The figure shows the graph of the derivative of a function. The derivative is never negative.
Which statement is false? E I .. ., (a) The function has no relative maximum value.
(b The function has no relative minimum value.
@The function is always concave up.
) The function has exactly three points of inflection.
(e) The function is increasing for all x. 21. Identify which graph is 1‘, which one is f', and which one is f”: i a f
B: {ll l C \7 f) 22. The figure showgggjaph of f', the derivative of a function f. Which of the following
L“
a. statements must be a out 1‘? has a relative maximum at x = 2. f has a point of inflection at x = 2.
(c) f has a critical point at x = 4.
(d) f has a relative minimum at x = 0.
(e) f is concave up for all x < 2. 23. Sketch a function f having these characteristics: f(0) = O
f'(x) > O, for all x
f"(l) = o
f"(x) < 0, for x < 1
f"(x) > O, for x > 1 *u _ 4
W l 24. Given The graph of f', graph f and f". f(x) f'(><) f"(X) 25. A funcTion f is conTinuous on The inTerval [—5, 5] and iTs firsT and second derivaTives have The
values given in The following Table: (1. 2) _—
Nea ive PosiTive a. WhaT are The x—coordinaTes of all of The relaTive maxima and minima of f on [—5, 5]?
JusTify your answer. i
)(r: «v l r Q (asle‘L l“&% 4 C hang (5 + +0 ‘
x : Ll rll alive» Wk M ”f {.l/‘O.K%Z_S "' ‘l0 + b. WhaT are The x—coordinoTes of all poinTs of inflecTion of f on The inTerval [5, 5]? JusTify
your answer. y; a 1?“ Lhangcs Sign‘g c. SkeTch a possible graph for f which saTisfies all of
The given properTies. ”L ...
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 Spring '09
 Sorray
 Calculus

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