C Roettger Name (please print): . . . . . . . . .
Problem 1 Consider the function h ( x ) = (9 x
3 x +1
a) Find h
( 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
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 b) Describe all solutions for x in [0 ,
π , no rounded decimal fractions.
10] of sin(3 x + 1) = 0 .
Problem 4 Consider the function f ( x ) = 4 cos(3 x + 5) .
a) Find the equation for the tangent line to the graph of
Round all numbers to 2 digits after the decimal point.
b) Use the equation found in a) to approximate f (1 .
f ( x ) at x = 2.
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 e) What is the most sensible value for
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
Body Temp 96.3
97 98 99 99.7
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
+ 1 the measure of noise (this means that noise of a given volume gets higher
M -value with increasing frequency). Suppose that that ω changes over time t with
V = 100 is constant, and
( 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 number S
A population of snails is counted every week. Assume that the 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?