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ch03Test-Version B - Key 2020 Flipbook PDF
ch03Test-Version B - Key 2020
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AP Calculus AB Chapter 3 Test B - 2020
key
NAME_________________________________
1) Matching. Use the choices on the right to match with the problems on the left. Write your legible letter answer on the line provided for each problem. Note: there are more answers than problems, so not every letter points will be used. You may use a letter more than once if you choose.
9Leach
____1.
B
d x [e ] dx
A. cos( x)
A
d [sin( x)] dx
B. e x
____3.
E
d [cot( x)] dx
C.
____4. L
d [csc( x)] dx
D.
____5.
d [cos( x)] dx
E.
____6. H
d [tan( x)] dx
F. 0
____7.
M
d [sec( x)] dx
G.
____8.
F
d [c], where c is a constant dx
H. sec2 ( x)
____9. G
d tan dx
____2.
J
1
x
sec( x) tan( x)
1 1 x2
csc2 ( x)
1 1 x2
I. sec( x)
J.
sin( x)
x K. xe
L.
1
csc( x) cot( x)
M. sec( x) tan( x)
Show/explain answers for all problems. 2) Differentiate the following functions with respect to x. No simplification needed unless indicated otherwise.
y x
a.
sin 3 ( x) 3 sin x
4points
sin x 3
1stterm 7353112
3sin x
scos ypg.gg
3 eymyyxs 3sin2Xcosxt3simh3cosxt3204 t
ot
answer
y x
b.
2points
Perk
y
U 44 V I
ln 4 x µ 3 x V
13x I x_ 1312 ex
3
out e
zpoints
in X
uperepor
y1 1 3 2 y x
d.
cos
3
1
3x 2 1
2
Y'A
3) Find dy / dx if x
solved
i
cot 3x
csc43xty 3ty
forage1 35013 9
2
e
32
1 132114
I
in 3Xty
y'csc43xty
12 or 36 2 1 3 174
24
Alternatesolving approaches
y .
l
Csec
Coty
cs
csc43Xty 3ty
yT 3tY
1t3csc43Xty yi e.sc y
y'csc43xtu
92 y 1 37 345 or 1305043
74
3 x xbhl4X X 3X
2132Dlex
outootc
sameasy
l
thx
arefine
out cosC mid 12 2C in 3 2 1 lex
µpefpor
toolkit
incl 4x 4
Cscxcotx
gpoints
zpoints
L
44hm nochainneeded
Noteanyoftheanswersbelow
13X
csc x
C
3rdmsin
1
3
inl4xKD
y x
c.
2nd 3 In3.3 term sinx
Sinxscosx
csc43xty
ok
sin43xty
3
3ok sin73xtyJ
cot X 3Xty y got X 3X
y
2
3
dy dx
4) Suppose a function y has
Y
2points
product rule
2X2y't4y
d2y in terms of only x and y. dx 2
Cft
2 x 2 y as its derivative. Determine
Y 24229 t4xy subfory y 4x4yt4xy
1 toffinative
5) Determine the equation of the line tangent to the graph of y
9
c z cjk 1 1342 out in x i zpoints slope
1
point
reqn
y IH115
txt
Y3
L4 3x
f X _3 f x _3 12 2 3
0 1 fsolve 3 122 0 1 3 122 3
O
cos
I
h h
0
d cos x at x dx
C) cos
fix
12 1
o
fx h
A)
12 have a horizontal tangent line? x
2
7) One could evaluate lim
point
1point 3yes 13,2 2 Slope Y133
y flx 3
6) At what value(s) of x does the function f x
zpoints
x 1 at x 3 .
1
by recognizing that it is equivalent to the expression
, so the limit is 0
, therefore the limit is 1 . nitionotden.VN e
8) Suppose we know f ' 3
0 ; therefore the limit does not exist. 0 domore 0 0 1 , the limit is 1. D) , and since 0 0
1
B)
nottrue
1 . Circle the letters of ALL the statements below that MUST be true.
A) f is increasing at x 3 fits o B) f is differentiable at x 1 weknowitisatx 3 0 C) f is continuous at x 3 D) f is decreasing at x 3 O fits 0notLO isoml UMMA E) f is differentiable at x 3 bicthederivexists F) f is continuous at x 1weontyknowatt3 I O don't MOSH G) The slope of the line tangent to f at x 3 is 1. H) f 3 1we 2 all 0 aYYorPfIn okay TEEfor all values of x. J) f has a horizontal tangent line at x 3 wrong I) f is differentiable
zpoints
Fitmptfireensticaobnitihinthity
1
weomyknowatk3
1350
9) Given the table below and assuming the functions f, g, and h are all differentiable functions of x, determine each of the following, if possible. If the calculation is not possible, explain why. Show work.
a.
p ' 1 if p x
x
f x
f' x
h x
h' x
1 3
−2 1
−3 −5
3 2
−1 5
f x h x
2 x3
productruleEknownfunction
P X fixhlx thlx flx t6X2 16112 p i Hth'llthlDf'll f2X1 3 f3 16
zpoints
devil
f
2 916
10 b.
q ' 3 if q x
h f x
hC he Chainrule out in xg
sfyx qlxj
hlflxD.fix
zpoints
hYH3Dfl3 q3 h1 f 3 fly 5 c.
g ' 3 if g x
h
1
answer
tdlatifgwed
x
2points found
i
9Y3
time1 g 3 T
10) The position of an object moving along a straight path at time t is given by s t a. Determine the average velocity of the particle over the time interval 0
t
wet
t 3 2t 3 . centimeters
3 . Show work.
51313321333
5133701 20 37 33 1
27 6 3 30
8seconds
2points
1 difference quotient
b. Determine the velocity of the particle at the time t
sets.ua
3tzIs
1 answer
3 . Show work. fungi'ints
70
Vl3 3137 2 27 2 29
c. Determine the acceleration of the particle at the time t
vet
vtttaaft.EE _l8me
fifth'The
del 3 . Show work.
3
2points 1 deriv 1 value
11) The graph of the function y
f x is
shown. Give all x-values on the open interval 2 x 5 at which the Esper vault function f is not differentiable, and give a one or two-word justification for each gpet value listed. reason
3points
x o verticaltangent continuous x 2 fix isnot lines x 4 infinite tangent drawn
canbe
12) Suppose a taconite processing plant in northern Minnesota has a conveyor belt that dumps 25 cubic yards of taconite pellets per minute onto a stockpile in the shape of an inverted cone (i.e. pointed upward) in such a way that its diameter is always half its height. How quickly will the radius of the base of the stockpile be changing when the height of the stockpile reaches 8 yards? Include units in your answer. Recall that the volume of a right circular cone with radius r and height h is given by V
ht
y Td I
d Eh
2r Ih
4r h
d 25 h 4r
want
whenh 8 AKAwhen r
3
r 2h .
2
gpoints V tzTr2h
V tzTr 4r
v
4gtr3
DV_4xr2dI
at
1 relationship 1 differentiation I Sublsolve I Answer I units
dt
25 411122drat 25 16 drat dr ybmd.TN dt
ffy
2.497391 min
NoteHenteredincorrectly inthecalculator 4.9087yd
min
f nosubstitution ismadebeforedifferentiation y
ph
willneedthefollowinginfo productruleis requiredAND
did 15hr2d t ZzFrhdrat
25 25
IT 22 4drat t ETI 2 11 ydrat t 31 Tdrat 3 3
8 drat
25 161Tdrat seeabovefor solutionfrom here
4r h go4drat dhqµhff r 2