Math 1350
Professor Carlson
Test 2 - SHOW ALL YOUR WORK
1.(10 points) Give the precise definition of the derivative f ′ (α) of a
function f at a number α.
Suppose f is defined on an open interval containing α. Then the derivative is
f (α + h) − f (α)
,
f ′ (α) = lim
h→0
h
if this limit exists.
2. (10 points) (a) If g(x) = x3 − 3x + 1, then g(2) = 3. Using the
definition of the derivative, find the derivative g ′ (2).
(b) Find the equation of the tangent line to the graph of g(x) at (2, 3).
(a) We have
(2 + h)3 − 3(2 + h) + 1 − 3
g (α) = lim
h→0
h
′
9h + 6h2 + h3
= lim 9 + 6h + h2 = 9.
= lim
h→0
h→0
h
(b) The tangent line has the form
y = m(x − x1 ) + y1 .
At (2, 3) the tangent line is
y = 9(x − 2) + 3.
3. (20 points) Find the derivatives of the following functions.
a) f (x) = π 2 ,
b) f (x) =
c) f (x) =
√
3 8
x ,
4
x(x − 1),
f ′ (x) = 0,
f ′ (x) = 6x7 ,
√
1
f ′ (x) = √ (x − 1) + x,
2 x
d) f (x) = sin(sin(sin(x))),
f ′ (x) = cos(sin(sin(x))) cos(sin(x)) cos(x).
4. (20 points) Find the derivatives of the following functions.
a) f (x) =
f ′ (x) =
cos(x)
,
1 − sin(x)
− sin(x)(1 − sin(x)) − cos(x)(− cos(x))
(1 − sin(x))2
− sin(x) + sin2 (x) + cos2 (x)
1 − sin(x)
1
=
=
=
.
(1 − sin(x))2
(1 − sin(x))2
1 − sin(x)
b) f (x) = (x2 + 1)3 (x2 + 2)6 ,
f ′ (x) = 3(2x)(x2 + 1)2 (x2 + 2)6 + (x2 + 1)3 6(2x)(x2 + 2)5 .
c) f (x) =
x2 − 2
,
2x + 1
2x2 + 2x + 4
2x(2x + 1) − (x2 − 2)(2)
=
.
f (x) =
(2x + 1)2
(2x + 1)2
′
d) If
y 5 + x2 y 3 = 1 + x4 y,
find dy/dx by implicit differentiation.
5y 4
dy
dy
dy
+ 2xy 3 + 3x2 y 2
= 4x3 y + x4 .
dx
dx
dx
dy
4x3 y − 2xy 3
= 4
.
dx
5y + 3x2 y 2 − x4
5. (15 points) A particle is moving along a hyperbola xy = 8. As it
reaches the point (4, 2), the y-coordinate is decreasing at the rate of 3 cm/s.
How fast is the x-coordinate of the point changing at that instant?
Start with
x(t)y(t) = 8,
dy
dx
y(t) + x(t)
= 0.
dt
dt
From x = 4, y = 2 and dy/dt = −3 we find
dx
= 6 cm/sec.
dt
6. (15 points) At noon, ship A is 150 km west of ship B. Ship A is
sailing east at 35 km/hour and ship B is sailing north at 25 km/hour. How
fast is the distance between the ships changing at 4 : 00 pm?
Start ship A at x = 0, y = 0, and ship B at x = 150, y = 0. Let
(x(t), 0) be the position of A at time t, while (150, y(t)) is the position of
B. If z(t) is the distance between A and B, then
z 2 (t) = (150 − x(t))2 + y 2 (t).
After 4 hours ship A is at (140, 0) and B is at (150, 100). At that time
the distance between A and B is
p
p
z(4) = 102 + 1002 = 10, 100.
Differentiation gives
dz
dx
dy
= −2(150 − x)
+ 2y .
dt
dt
dt
Plugging in the know values gives
2z
dz
2150
= √
≃ 21.5
dt
10, 100
7.(10 points ) Find the linear approximation of the function
√
f (x) = 1 − x
√
at α = 0 and use it to approximate the number .9 (Show your work!!)
First,
−1
, f ′ (0) = −1/2.
f ′ (x) = √
2 1−x
Then
L(x) = f (0) + f ′ (0)(x − 0) = 1 − x/2.
√
To find 0.9, take x = 0.1 and get
√
√
0.1
0.9 ≃ 1 − 0.1 ≃ 1 −
= 0.95
2