Math 161 - Solutions To Sample Exam 2 Problems Worksheet (Page 8 of 8) in pdf

Math 161 - Solutions To Sample Exam 2 Problems Worksheet Page 8

Solutions To Sample Exam 2 Problems – Math 161

iii. f (1.7) = 2.3 is a local maximum, and f (4.5) =

2.5 is a local minimum.

(b) Note that, in this part of the problem, we are given the graph of the DERIVATIVE of g.

i. The function g is increasing where g (x) > 0, i.e., where its derivative lies above the x-axis.

Similarly, g is decreasing where its derivative lies below the x-axis. Therefore:

g is increasing on

[0, 3]

g is decreasing on

[3, 8]

ii. The function g is concave up where g (x) > 0. But g (x) > 0 when g (x) is increasing. Therefore, g

is concave up where the graph of g is increasing. Similarly, g is concave down where the graph of g

is decreasing. Therefore:

g is concave up on

(0, 1.7) and (4.5, 8)

g is concave down on

(1.7, 4.5)

iii. g has a local maximum value at x = 3 because g (x) changes sign from positive to negative at x = 3.

g has no local minimum values.

i. The function h is concave up where h (x) > 0, i.e., where the graph of h lies above the x-axis.

Similarly, h is concave down where the graph of h lies below the x-axis. Therefore:

h is concave up on

(0, 3)

h is concave down on

(3, 8)

9. The speed of a car slowing down for a red light is given by the data in the table below.

t (seconds)

v (ft/sec)

(a) Find the maximum possible distance and the minimum possible distance that the car could have traveled

during the 10-second time interval shown in the table.

(b) Give your best estimate of the distance that the car actually did travel during the 10-second interval.

Solution:

(a) Since the car is slowing down, the maximum speeds occur at the left sides of each 2-second subinterval,

and the minimum speeds occur at the right sides. Therefore, we have:

Maximum Distance = 2(50 + 44 + 29 + 18 + 9) = 300 feet

Minimum Distance = 2(44 + 29 + 18 + 9 + 6) = 212 feet

(b) To obtain a reasonable estimate, we will average the maximum and the minimum distance traveled in

part (a) to obtain and estimate of

300 + 212

= 256 feet traveled.

Math 161 - Solutions To Sample Exam 2 Problems Worksheet Page 8