# Math 181 - Practice Exam 3

C Roettger Name (please print): . . . . . . . . .

# Math 181 - Practice Exam 3

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 b) Describe all solutions for x in [0 ,

π , no rounded decimal fractions.

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

Round all numbers to 2 digits after the decimal point.

b) Use the equation found in a) to approximate f (1 .

97).

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

C

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

Body Temp 96.3

96 96.3

97 98 99 99.7

16

100

18

99.7

20

99

22

98

24

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 that ω changes over time t with

V = 100 is constant, and

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 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?