F13 MATH 1205 – Test 2 30 Oct 2013 NAME:
CRN:
Use only methods from class. You must show work to receive credit.
1. (35 pts) Find derivatives of the following functions. Do not simplify.
(a) (6 pts) f (x) =
5x
x+1
f 0 (x) =
(x + 1)(5x ln(5)) − (1)(5x )
(x + 1)2
(b) (5 pts) g(t) = tan−1 (4t)
g 0 (t) =
4
1 + (4t)2
(c) (6 pts) z(x) = sec(x) · (log3 (x) + e)
z 0 (x) = (sec(x) tan(x))(log3 (x) + e) + (sec(x))
(d) (10 pts) r(x) = cos(x)
1
x ln(3)
(x3 +1)
0
r (x) =
cos(x)
(x3 +1)
− sin(x)
2
3
(3x )(ln(cos(x))) + (x + 1)
cos(x)
2/3
(e) (8 pts) u(t) = ln(7) · (cot(3t) + t)
−1/3
u0 (t) = ln(7)(2/3) (cot(3t) + t)
(− csc2 (3t) · 3 + 1)
2. (18 pts) Suppose that the functions f and g and their derivatives with respect to x have the values at x = 1, x = 2,
and x = 3 given by the following table, suppose that g(x) is defined for all real numbers and g 0 (x) is always negative.
x f (x) g(x) f 0 (x) g 0 (x)
1
2
3
6
4
√
3
3
1
0
-3
π
7
-2
-8
-9
(a) (5 pts) Find the derivative of f (g(x)) at x = 2.
d
(f (g(x))
= 24
dx
x=2
(b) (6 pts) Find the derivative of g −1 (x) at x = 1.
(g −1 )0 (1) = −
1
8
(c) (7 pts) Find the derivative of x2 · f (x) at x = 3.
√
d 2
(x · f (x)) = 6 3 + 63
dx
3. (23 pts) A particle travels along the s-axis with its position, at time t given by the equation
s(t) =
t3
− 2t2 + 3t
3
where s is in meters and t in seconds.
(a) (3 pts) What is the average velocity between t = 0 and t = 4?
Vavg =
1
m/s
3
(b) (3 pts) What is the instantaneous velocity at t = 2?
v(2) = −1 m/s
(c) (7 pts) At what times between t = 0 and t = 4 does the particle turn around?
The particle turns around at t = 1s and t = 3s.
(d) (4 pts) What is the total distance traveled between t = 0 and t = 4?
Total Distance = 4
(e) (6 pts) At time t = 1.5 is the particle speeding up or slowing down?
Speeding up
4. (12 pts) Find the equation of the normal line to the curve x2 + 2y 2 − 4y = 7 at the point (1, 3).
y − 3 = 4(x − 1)
or y = 4x − 1
5. (12 pts) Suppose two boats start at a common point O. One boat heads due north and the other due east, both at a
constant speed of 3m/s. If, at a particular instant of time, the area of the isosceles triangle formed by O and the two
boats is growing at a rate of 6m2 /s, what is the area of the triangle at this instant?
Area = 2m2