· .. - .. ··Ada •.... • eli.ruc •.• Page I 7 Student Notes Derivatives and Graphs of Functions Jt.'lu lti Choice Ouestions Part A. No calculator. -2. ..• \ 2.. Graph of t' 1. The graph of [" the derivative of the function [, is shown above. Which of the following statements is true about [? (A) [is decreasing for -1 :::;x :::; l. (B) [is increasing for -2 :::; x :::; O. (C) [is increasing for 1 :::;x :::; 2. (D) [has a local minimum at x = O. (E) [is not differentiable at x = -1 and x = 1. 2. Let [ be the function with derivative given by t' (x) = X2 -.:.. On which of the following intervals is x [ decreasing? (A) (-00,-1] only (B) (-00,0) (C) [-1,0) only (D) (0, V2] (E) [\12,(0)
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·..-..··Ada •....• eli.ruc •.•
Page I 7
Student Notes Derivatives and Graphs of Functions
Jt.'lu lti Choice Ouestions
Part A. No calculator.
-2. ..• \ 2..
Graph of t'
1. The graph of [" the derivative of the function [, is shown above. Which of the following statements
is true about [?(A) [is decreasing for -1 :::;x :::;l.(B) [is increasing for -2 :::;x :::;O.(C) [is increasing for 1 :::;x :::;2.(D) [has a local minimum at x = O.(E) [is not differentiable at x = -1 and x = 1.
2. Let [ be the function with derivative given by t' (x) = X2 -.:.. On which of the following intervals isx
[ decreasing?(A) (-00,-1] only(B) (-00,0)(C) [-1,0) only
(D) (0, V2](E) [\12,(0)
Student Notes Derivatives and Graphs of Functions
3. Let [ be the function given by [(x) = 2xex. The graph of [ is concave down when(A) x < -2 (B) x> -2 (e) x < -1 (D) x> -1 (E)x < 0
I g'~X) I -; I -: I ~ I ~~ I -~ I -~ I ~ I ~ I ~4. The derivative g' of a function 9 is continuous and has exactly two zeros. Selected values of g' are
given in the table above. If the domain of 9 is the set of all real numbers, then 9 is decreasing onwhich ofthe following intervals?
(A) -2:::; x:::; 2 only
(B) -1:::; x :::;1only
(C) x 2: -2
(D) x 2: 2 only
(E) x:::; -2 or x 2: 2
5. The function [ is given by [(x) = X4 + X2 - 2. On which of the following intervals is [ increasing?
(A) (- Jz, 00)
(B) (- Jz, Jz)(C) (0,00)
(D) (-00,0)
(E) (-00, - Jz)
.-.-..·-.dUne•• ~aTIIII:.'"
Page I 8
?..-.=
Student Notes
Part B. Graphing calculator allowed.
Derivatives and Graphs of Functions
6. Let [ be the function with derivative given by t' (x) = sin(x2 + 1). How many relative extrema
does [have on the interval 2 < x < 47
(A) One (B) Two (C) Three (D) Four (E) Five
7. For all x in the closed interval [2,5], the function t has a positive first derivative and a negative
second derivative. Which of the following could be a table of values for [?
(A)x ((x)
2 73 94 125 16
...-..·-Un ...• SlEaT.E • .,
(B)x f(x)
2 73 114 145 16
(C)x ((x)
2 163 124 95 7
(D) (E)x [(x)
2 163 144 115 7
x [ex)2 163 134 105 7
Page I 9
3
Student Notes Derivatives and Graphs of Functions
tree- .\' estions,'"1S~;'
2007 AB 4 Form B
fV;) r-: ::;
(2.2)
(5, -1)
------ ."
Graph of f'
Let [be a function defined on the closed interval -5 ::; x ::;5 with [(1) = 3. The graph of r. thederivative of [, consists of two semicircles and two line segments, as shown above.
(a) For -5 < x < 5, find all values of x at which [ has a relative maximum. Justify your answer.
(b) For -5 < x < 5, find all values of x at which the graph of [has a point of inflection. Justifyyour answer.
(c) Find all intervals on which the graph of [is concave up and also has a positive slope. Explainyour reasoning.
(d) Find the absolute minimum value of [(x) over the closed interval -5 ::; x ::;5. Explain your
reasoning .
...••••• ""ne• CI_TUC:."
Page I 3
J l'-J
Student Notes Derivatives and Graphs of Functions
2001 AB 4 and Be 4
;i_~::. .-.',~ ',i1> <'~~-~
Let h be a function defined for all x '*°such that h( 4) = -3 and the derivative of h is given by
X2 -2h'(x)=-- for all x,*O.
x
(a) Find all values of x for which the graph of h has a horizontal tangent, and determinewhether h has a local maximum, a local minimum, or neither at each of these values. Justify
your answers.
(b) On what intervals, if any, is the graph of h concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of h at x = 4.
(d) Does the line tangent to the graph of h at x = 41ie above or below the graph of h for x > 4 ?
Why?
2001AB 5
i'i"-.-," ~_~_.{ '~-~}i
A cubic polynomial function f is defined by
f(x) = 4x3 +ax2 +bx+k
where a, b, and k are constants. The function f has a local minimum at x = -1, and the graph off has a
point of inflection at x = -2.
(a) Find the values of a and b.
(b) If s: f(x) dx = 32, what is the value of n
...•••••••••••• 1I1_ras:."Page 14
~~
it;Ap® CALCULUS AB
2007 SCORING GUIDELINES (Form B)
(a) For -5 < x < 5, find all values x at which fhas arelative maximum. Justify your answer.
(b) For -5 < x < 5, find all values x at which the graph of fhas a point of inflection. Justify your answer.
(c) Find all intervals on which the graph of f is concave upand also has positive slope. Explain your reasoning. Graph oft'
(d) Find the absolute minimum value of f(x) over the closed interval -5 ~ x ~ 5. Explain your reasoning.
Question 4
Let f be a function defined on the closed interval -5 s x s 5with f(l) = 3. The graph of f', the derivative of J, consistsof two semicircles and two line segments, as shown above.
v
(5. -I)
(~. 2).,
• I I I I ( i" . .12 -I ()
-I:'\
_"l •.
(a) f'(x) = 0 at x = -3,1,4f' changes from positive to negative at -3 and 4.Thus, f has a relative maximum at x = -3 and at x = 4.
(b) f' changes from increasing to decreasing, or vice versa, atx == -4, -1, and 2. Thus, the graph of f has points ofinflection when x = -4, -1, and 2.
(c) The graph of f is concave up with positive slope where f'is increasing and positive: -5 < x < -4 and I < x < 2.
(d) Candidates for the absolute minimum are where f'changes from negative to positive (at x = 1) and at theendpoints (x == -5,5).
f-S Jr
f( -5) = 3 + 1 f'ex) dx = 3 - '2 + 2Jr > 3
f(l) = 3
f5 I 3·2 1
f(5)=3+ If(x)dx=3+T-"'2>3
The absolute minimum value of f on [-5,5] is f(l) = 3.
2 : { 1 : x-values1 : justification
{I: x-values2:1 : justification
2 : { 1 : intervals1 : explanation
{
I: identifies x == 1 as a candidate3: 1: considers endpoints
1 : value and explanation
© 2007 The College Board. Allrights reserved.Visit apcentral.collegeboard.com (forAP professionals) and www.collegeboard.comlapstudents (forstudents and parents).
-----------------
Ap® CALCULUS AB2001 SCORING GUIDELINES
Question 4
Let h be a function defined for all x 7: 0 such that h(4) = -3 and the derivative of h is given
x2 - 2by h'(x) = -- for all x 7: O.
X
(a) Find all values of x for which the graph of h has a horizontal tangent, and determine
whether h has a local maximum, a local minimum, or neither at each of these values.
Justify your answers.
(b) On what intervals, if any, is the graph of h concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of h at x = 4.
(d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for
x ;» 4? Why?
7
(a) h'(x) = 0 at x = ±-J2
h'(x) 0 + und 0 +-1 I - Ix --J2 0 -J2
Local minima at x = --J2 and at x = -J2
(b) "( ) 2h x = 1 + 2"" > 0 for all x 7: O. Therefore,x
the graph of h is concave up for all x 7: O.
(c) h'(4) = 16-2 _ 74 -"27y+3=-(x-4)2
(d) The tangent line is below the graph because
the graph of h is concave up for x > 4 .
1 : x = ±-J21 : analysis
2 : conclusions< -1 > not dealing with
discontinuity at 0
4:
3, 11 : hl/(x)
1 : hl/(x) > 0
1 : answer
1 : tangent line equation
1 : answer with reason
Copyright © 2001 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
5
Ap® CALCULUS AB2001 SCORING GUIDELINES
Question 5
A cubic polynomial function f is defined by
f(x) = 4x3 + ax2 + bx + k
where a, b, and k are constants. The function f has a local minimum at x = -1 , and the graph
of fhas a point of inflection at x = -2.
(a) Find the values of a and b.
(b) If fol f(x)dx = 32, what is the value of k?
8
(a) f'ex) = 12x2 + 2ax + b
1"(x) = 24x + 2a
1'(-1) = 12 - 2a + b = 0
f"(-2) = -48 + 2a = 0
a = 24
b = -12 + 2a = 36
(b) 11(4x3 + 24x2 + 36x + k) dxo
= x4 + 8x3 + 18x2 + kx I::~= 27 + k
27 + k = 32
k=5
1: f'ex)
1 : 1"(x)
5 : ~ 1 : 1'(-1) = 0
1 : 1"(-2) = 0
1 : a, b
4:
2 : antidifferentiation
< - 1 > each error
1 : expression in k
1 : k
Copyright © 2001 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
6
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