AP Calculus ABBC
Test 3
Name ________________________
October 20 , 2014
I. (35 points) Find the derivative of each function. Answers need not be in simplest form.
1.
f (x) = ex x -4
f ¢(x) = ex x -4 - 4x -5ex
2.
y=
sinx
1- cosx
dy (1- cosx)(cosx) - sin x(sinx)
=
dx
(1- cosx)2
3. g(z) = tan z +12
2
g¢(z) = 2tanz ×sec2 z
4.
y = log5 x + 5 x+2
dy
1
=
+ (ln5)× 5 x+2
dx (ln5)x
3
5. v(t) = sin (2t )
v¢(t) = 3sin2 (2t)× 2cos(2t)
6. y = (1+ sin(2t))
7
dy
6
= 7(1+ sin(2t)) (cos(2t)) × 2
dx
7. x + 2y + 4xy = 6
2
3
dy
dy
+ 4y + 4x = 0
dx
dx
dy -2x - 4y
=
dx 4x + 6y2
2x + 6y2
1
II. (15 points) Find the antiderivative of each function. Answers need not be in simplest form.
1.
f (x) = ex + e- x
F(x) = ex - e-x + C
2.
p(q ) = secq tanq
P(q ) = secq + C
3.
f (x) = 3sinx - cos2x
1
F(x) = -3cosx - sin(2x) + C
2
III. (10 points) Given j(z) = f (2z ) and the following values:
2
-1
1
2
4
6
8
f (x)
1
3
5
7
0
3
f ¢(x)
8
2
4
6
10
9
find j¢(-1) , j¢(1) , and j¢(2) .
j(z) =
j ¢(z) =
j¢(-1) =
j¢(1) =
j ¢(2) =
f (2z 2 )
f ¢ (2z 2 )( 4z)
f ¢ (2)(-4) = -16
f ¢ (2)( 4 ) = 16
f ¢ (8) 8 = 72
2
IV. (20 points) Consider f (x) = xe-3x .
1. Find f ¢(x) .
-3x
f (x) = xe
-3x
-3x
f ¢(x) = e - 3xe
2. Find f ¢¢(x).
f ¢(x) = e-3x - 3xe-3x
-3x
-3x
-3x
f ¢¢(x) = -3e - 3e + 9xe
3. Determine the interval(s) on which f is increasing.
+
−
f ¢(x) = e-3x - 3xe-3x = 0
1
f is increasing on æ - ¥, ù , as f ¢ is positive there.
1- 3x = 0
è
3 úû
1
x=
3
4. Determine the interval(s) on which f is concave up.
−
-3x
f ¢¢(x) = -6e + 9xe
0 = -6 + 9x
6 2
x= =
9 3
-3x
=0
+
f is concave up on
( 23, ¥)
, as f ¢¢ is positive there.
5. Find the location(s) of any local minima or local maxima.
f¢
(13) = 0
and f ¢(x) changes from positive to negative there, so f has a local maximum point
1
at x = .
3
3
V.
(20 points) Many elements exist as different isotopes, and some isotopes are unstable. When
unstable isotopes break down and emit high energy particles or rays, this is radioactivity. The rate at
which an isotope decays is proportional to the amount of isotope present. Let y be the amount of C14
present in an object, such as a fossil or artifact.
a.
Write the differential equation that describes the rate of radioactive decay of C14 in an object.
(You may write the constant as k. It will be found later.)
y¢ = k y
b.
Find a general solution to the differential equation.
y = Aekt
c.
Half of the amount of C14 present initially in an object will be lost in 5730 years. Use this fact to
find k in your DE and its solution.
1
A = Aek(5730)
2
1
5730k
=e
2
Let A = 100%
1
ln
= 5730k
2
1
ln
2 = -ln2 » -1.2096 ´ 10 -4
k=
5730 5730
()
()
y = e-1.2096´10
-4
t
d.
The Shroud of Turin, is reputedly the burial shroud of Jesus of Nazareth. This cloth, which
surfaced in 1356, contains a negative image of a human body that was widely believed to be Jesus. The
British Museum was commissioned by the Vatican in 1988 to date the cloth relic. The British Museum
scientists reported that the fibers contained 92% of the original C14. Use this information to determine
the age of the Shroud of Turin. Was this the burial shroud of Jesus of Nazareth? Justify your opinion.
y = e-1.2096´10
-4
t
The fabric was found to be approximately 689 years old.
-4
-1.2096´10 t
.92 = e
ln(0.92) = -1.2096 ´ 10 -4 t
ln( 0.92)
t=
-4 » 689.286 years
-1.2096 ´ 10
This means that the fabric in the Shroud of Turin can be dated to a period around 1988 - 689 = 1299 AD
This is in the same time period as the discovery of the fabric, and so it is unlikely to have originated in
the time of Jesus.
4