Question 1:
(1). f (x) =
p
sin(x3 ), find f 0 (x).
3x2 cos(x3 )
f 0 (x) = p
2 sin(x3 )
(2). f (x) =
x2 −1
,
x2 +1
find f 00 (x).
f 00 (x) =
4 − 8x2 − 12x4
(x2 + 1)4
(3). y 3 x+sin(xy) = x3 +1. Assume this equation determines a function y = f (x).
dy
Find dx
.
3x2 − y cos(xy) − y 3
dy
=
dx
x cos(xy) + 3xy 2
(4). f (θ) = cos θ + sin θ, find f (11) (θ).
f (11) (θ) = − cos θ − sin θ
1
2
Question 2:
Use linear approximation (the differential) to estimate
Let f (x) =
√
4.25.
√
x. Then
√
4.25 = f (4.25) = f (4 + 0.25) ≈ f (4) + 0.25 · f 0 (4)
by linear approximation.
√
Here f 0 (x) = (2 x)−1 . Therefore,
√
√
1
4.25 ≈ 4 + 0.25 · ( √ ) = 2.0625
2 4
3
Question 3:
Airplane A is heading north with speed 550 km/h. At the same time, airplane B
is heading east with speed 600 km/h. Assume they took off at the same airport.
How fast is the distance between the two airplane changing 1 hour after they took
off?
Let L be the distance between A and B. Then
p
L(t) = (550t)2 + (600t)2
Therefore,
dL
5502 t + 6002 t
=p
dt
(550t)2 + (600t)2
Att = 1, we have
5502 + 6002
dL
=p
= 813.9
dt
(550)2 + (600)2
4
Question 4:
Using calculus to draw the graph of function
f (x) = x4 − 4x3 + 10
on [−4, 4] and figure out where the function is increasing, decreasing, concave up
and concave down. Also find all the critical points as well as all the local minimum,
local maximum , global minimu and global maximum values of the function.
f 0 (x) = 4x3 − 12x2
Since f 0 (x) > 0 for x > 3 and f 0 (x) < 0 for 0 < x < 3 or −4 < x < 0, we have
f (x) is increasing on [3, 4] and f (x) is descreasing on [−4, 3].
f 00 (x) = 12x2 − 24x
Since f 00 (x) > 0 for x > 2 or x < 0, and f 00 (x) < 0 for 0 < x < 2, we have f (x) is
concave up on (−4, 0) ∪ (2, 4) and f (x) is concave down on (0, 2).
Critical points:
stationary points: x = 0, x = 3;
singular points: none;
end points: x = −4, x = 4.
Since f (0) = 10, f (3) = −17, f (−4) = 522, f (4) = 10, together with the monotonicity, we know that f (3) is a local minimum and global minimum, f (−4) is a
local maximum and global maximum, and f (4) is a local maximum.