C Roettger Name (please print): . . . . . . . . .
Problem 1 Consider the function h ( x ) = (9 x
2 − 33 x − 25) e
3 x +1
.
a) Find h
0
( x ).
b) Find all critical numbers of h ( x ), and tell which of them is a relative maximum and which a relative minimum (see 5.4, eg. p. 163ff).
Problem 2 a) Solve
7 exp( − 0 .
4 t + 3) − 5 = 0 exactly and to 3 digits after the decimal point (see 5.2, eg p. 149-151).
b) Solve (see 5.3, eg p. 156)
5 log( x + 2) = 15 .
Problem 3 a) Find all solutions for x in [0 , 4 π ] of
8 cos( x ) + 4
√
2 = 0
Give exact answers as multiples of π , no rounded decimal fractions.
b) Describe all solutions for x in [0 , 10] of sin(3 x + 1) = 0 .
2 .
Problem 4 Consider the function f ( x ) = 4 cos(3 x + 5) .
a) Find the equation for the tangent line to the graph of f ( x ) at x = 2.
Round all numbers to 2 digits after the decimal point.
b) Use the equation found in a) to approximate f (1 .
97).

Problem 5 This table gives the body temperature of some child with a slight ’temperature’, measured every two hours and using a 24-hour clock.
Suppose the temperature repeats its values every 24 hours, and that it can be exactly described by (see 5.6 for f) and g), but a)-e) not in notes) f ( t ) = A cos( ω ( t − B )) + C.
a) What is the most sensible value for ω ?
b) What is the minimum and maximum temperature in the table?
c) What is the most sensible value for A ?
d) What is the most sensible value for C ?
e) What is the most sensible value for B ?
f) What will the temperature be at 3pm, according to your function?
g) At what time(s) is the temperature 97.6 degrees?
Hour 2 4 6 8 10 12 14 16 18 20 22 24
Body Temp 96.3
96 96.3
97 98 99 99.7
100 99.7
99 98 97
Problem 6 One measure M of hearing damage due to noise involves the logarithm of the frequency of noise. Let ω be the noise frequency, V a measure of its volume, and
M = V ln
ω
1000
+ 1 the measure of noise (this means that noise of a given volume gets higher
M -value with increasing frequency). Suppose that V = 100 is constant, and that ω changes over time t with ω
0
( t ) = 5 at a time when ω = 440. Find the rate of change of M at that moment in time. (See 4.7, eg examples on p.
140 and combine with 5.4)
Problem 7 A population of snails is counted every week. Assume that the number S of snails obeys the law
F ( S ) = 1 .
036 S + 0 .
012 S log( S ) (1) a) If initially, there are 100 snails, how many snails are there after 2 weeks?
b) What is the positive equilibrium for the number of snails?
c) What is the maximum sustainable harvest?
(See examples in 4.5, eg p. 126/7 and 5.5, problem 6-7).
d) What is the breeding population for the maximum sustainable harvest?