Download Due: 03/10/2016 in class Instructor: Janina Letz MATH 1210, Spring

Due: 03/10/2016 in class
Instructor: Janina Letz
MATH 1210, Spring 2016
Lab #8
Name:
Question:
1
2
3
4
Total
Points:
10
15
32
13
70
Score:
Instructions: Please show all of your work as partial credit will be given where appropriate, and there may
be no credit given for problems where there is no work shown. All answers should be completely simplified,
unless otherwise stated. No calculators or electronics of any kind are allowed.
1. (10 points) The illumination at a point is inversely proportional to the square of the distance of the
point from the light source and directly proportional to the intensity of the light source. If two light
sources are s feet apart and their intensities are I1 and I2 , respectively, at what point between them
will the sum of their illuminations be a minimum?
1
MATH 1210, Spring 2016
Lab #8
2. (Maximization of arithmetic operation)
(a) (5 points) Let a be a fixed positive number. Find b, c such that a = b + c that maximizes the
product bc.
Answer:
(b) (5 points) Find two numbers a, b whose product is −12 and the sum of whose squares is minimal.
Answer:
√
(c) (5 points) For what number a does the principal square root a exceed eight times the number
by the largest amount?
Answer:
2
Due: 03/10/2016 in class
Instructor: Janina Letz
MATH 1210, Spring 2016
Lab #8
Name:
3. (Analyze and sketch functions) Analyze the function and sketch the graphs of the following functions.
3x4 − 20x3
(a) f (x) =
12
i. (2 points) Find the limits for x → ±∞.
ii. (2 points) Find all horizotal and vertical asymptotes.
iii. (2 points) Find all zeros of f .
iv. (4 points) Find all intervals, where f is increasing and decreasing. Fill out the table.
increasing/decreasing
interval
v. (4 points) Find all the intervals, where f is conave up and concave down. Fill out the table.
interval
concave up/down
vi. (2 points) Sketch the graph.
200
150
100
50
0
-4
-2
0
-50
-100
3
2
4
6
8
MATH 1210, Spring 2016
Lab #8
√
x(x − 3)2
(x − 3)(5x − 3)
√
f 0 (x) =
4
8 x
i. (2 points) Find the limits for x → ±∞.
f 00 (x) =
(b) f (x) =
3(5x2 − 6x − 3)
√
16x x
ii. (2 points) Find all horizotal and vertical asymptotes.
iii. (2 points) Find all zeros of f .
iv. (4 points) Find all intervals, where f is increasing and decreasing. Fill out the table.
interval
increasing/decreasing
v. (4 points) Find all the intervals, where f is conave up and concave down. Fill out the table.
concave up/down
interval
vi. (2 points) Sketch the graph.
12
10
8
6
4
2
0
-4
-2
0
-2
-4
-6
4
2
4
6
8
10
Due: 03/10/2016 in class
Instructor: Janina Letz
MATH 1210, Spring 2016
Lab #8
Name:
4. (Sketching a graph) Sketch the graph of a function f that has the following properties.
iv. f 0 (x) > 0 for x > −3
(a) (5 points) i. f is continuous everywhere.
ii. f (−3) = 1.
iii. f 0 (x) < 0 for x < −3
v. f 00 (x) < 0 for x 6= −3.
(b) (8 points) i. has a continuous first derivative; vi.
ii. is decreasing and concave up for x < 3;
iii. has an extremum at (3, 1);
vii.
iv. is increasing and concave up for 3 < x < 5;
v. has an inflection point at (5, 4);
viii.
5
is increasing and concave down for
5 < x < 6;
has an extremum at (6, 7);
is decreasing and concave down for x > 6.