MTH 151 HB January 20, 2006 Test 1
Show all non-trivial work; I am more concerned with how you got the answer than the
answer itself. I don’t give points for guessing. You may use a graphing calculator – in
fact you will have to use a graphing calculator on several of the problems. Do your work
on these sheets; use the back as scratch paper. I won’t grade anything on the back unless
you tell me to; I grade everything on the front!
1. Given the graph of y f ( x) :
-2 -1 1 2 3 4
Find the intervals where f is increasing. (Remember, these intervals are x
intervals)
4 x2 if x 1
2. Given f ( x) 3 x 3
2 2 if 1 x 3 , find
x3 if x 3
a) f (0) b) f (1)
c) f (2) d) f (2)
1
3. Given f ( x) , find
24 x
a) f (8) f (15)
b) f ( x 4)
c) f ( x2 )
d) [ f ( x)]2
e) f ( x h)
f) the domain of f
g) the range of f
4. Given f and g defined by the following table:
x -3 -2 -1 0 1 2 3 4 5
f ( x) 6 0 -4 -1 4 1 6 2 0
g ( x) -2 3 7 2 8 -5 -4 5 -3
Find
a) ( f g )(1) b) ( f g )(2)
f
c) ( f g )(2) d) (3)
g
e) ( f g )(0) f) (f f )(0)
g) ( g f )(1)
5. Find functions f and g such that ( f g )( x) tan x
6. Your graphing calculator in “Y =” mode can not graph the hyperbola
2 y 2 x 2 1. Find two functions which when graphed together will give the
graph of this ellipse.
7. Given the graph of y f ( x) :
2
1
-2 -1 1 2 3 4 5
-1
-2
a) List the domain and range of this function.
List the domain and range of the following transformations:
b) y f ( x) 2 .
c) y f ( x 2) .
d) y 2 f ( x) .
e) y f (2 x) .
f) y 2 f ( x) .
8. Suppose a Corrie decides to wash her dog in the laundry tub. She fills the tub
2/3 full with warm water, puts the dog into the tub and shampoos it, removes
the dog from the tub to dry it off, then pulls the plug to drain the tub. Let t be
the time in minutes, beginning when she starts to fill the tub, and let h(t ) be
the water level in the tub at time t. If the total time for filling and draining the
tub and washing the dog was 40 minutes, sketch a possible graph of h(t ) .
9. If f ( x) 4 5x2 and g ( x) sin 2 x , find
a) f ( g ( x))
b) ( g f )( x)
10. Express the perimeter of a square as a function of its area.
f ( x h) f ( x )
11. If f ( x) x 3 5 x 2 3x , simplify as much as possible.
h
12. Observations of the stern waves that follow a boat at right angles to its course
have disclosed that the distance between the crests of these waves (their wave
length) increases with the speed of the boat. The following table shows the
relationship between wave length and the speed of the boat.
Wave Speed
length (m) (km/h)
0.20 1.8
0.65 3.6
1.13 5.4
2.55 7.2
4.00 9.0
5.75 10.8
7.80 12.6
10.20 14.4
12.90 16.2
16.00 18.0
18.40 19.8
a) Find a power regression equation for the data in this table, where x is the
wave length and y is the speed of the boat.
b) Using this model, calculate the speed of the boat when the wave length is
11 m.
c) Now find a linear regression model for this data and use this equation to
predict the boat’s speed when the wave length is 11 m.
d) Plot the data points and both models on the same graph (You don’t have to
copy anything onto this test). Which regression model gives the better fit?
So you don’t just guess, tell me how many of the data points each model
passes through.
13. Use your graphing calculator to find all points of intersection of the curves
y 2 x and y x 2 .
Extra Credit Problems
a) CG&E has a power plant on the Ohio River at a point where the river is straight
and 800 feet wide. They want to lay a new cable to Dayton, Ky, a city 2 mile
downstream on the opposite side of the river. It costs $180 per foot to lay the
cable across the river and $100 per foot along the shore. If they lay a cable from
the plant to a point x feet downriver from the point directly opposite the plant (see
figure), find an expression for the total cost of laying the cable from the plant to
the city.
b) You are sitting in a classroom next to the wall looking at the blackboard at
the front of the room. The blackboard is 12 ft long and starts 3 ft from the
wall next to which you are sitting (see figure). Show that your viewing
15 3
angle is tan 1 tan 1
x x
c) Give a convincing argument, using the figure, to convince me that
tan 1 1 tan 1 2 tan 1 3 .