Download Math 121, Quiz Two

Math 121, Quiz Two. 4 points each; score is out of 25.
1. What best describes the path of a particle whose position is descried by the following parametric
equations: x(t )  cos( 2t ), y  sin( 2t ),0  t  2
a)
Travels in a counterclockwise direction around a circle twice, starting at (1,0).
b) Travels in a counterclockwise direction around a circle once, starting at (1,0).
c)
Travels in a clockwise direction around a circle once, starting at (0,1).
d) Travels in a counterclockwise direction half-way around a circle, starting at (1,0).
(answer: A; we go around once as “t” goes from 0 to 1 because the “t” in the argument is multiplied by
2 ).
2. Under ideal conditions a certain bacteria population is known to triple every 4 hours. Suppose that
there are initially 100 bacteria. How many bacteria are there in 9 hours?
9
a)
b)
100 * 2 4
100 * 2
4
9
3
9
c)
100 * 2
d)
100 * 3 4
e)
100 * 3 9
9
4
t
Answer: D; the formula for the population at time t is P(t )  100 * 3 4 as one can readily check by
plugging in 0 for t (the initial population) and by plugging in “4” for t (the time to triple).
0  a  1 . Which best describes the graph of y  a x ?
3.
Suppose
a)
b)
c)
d)
The graph is increasing everywhere and has neither origin nor y-axis symmetry
The graph is increasing everywhere and has y-axis symmetry
The graph is decreasing everywhere and has origin symmetry
The graph is decreasing everywhere and has neither origin nor y-axis symmetry
Answer: D; because a is less than one the graph decreases (example: if a < 1, then a  a ) and since
2
a x is always positive, origin symmetry is not possible and because a x  a  x 
graph cannot have y-axis symmetry.
3
1
(unless a = 1) the
a
4.
Let f be the inverse function for the function described in problem 3. Note that f ( x)  log 1 ( x).
2
Which of the following graphs is the graph of f ? Circle the correct letter
Answer: the graph for f has to be decreasing and negative for x > 1. So the upper left hand graph (A) is
correct.
5.
a)
b)
If f ( x )  3 x  1 , find the inverse function
x 1
3
1
f 1 ( x) 
3x  1
f 1 ( x) 
f
1
c)
f 1 ( x) 
d)
f
1
x
1
3
does not exist because f is not a one-to-one function.
Answer: (a) again! A straight forward algebra calculation.
6.
a)
b)
c)
d)
Which function most accurately describes the following graph?
y  5 * (3 x )
y  3 * (2 x )
y  3 * (2  x )
y  2 * (3 x )
Answer: (b). (c) and (d) are eliminated as the graphs of these functions decrease. (a) is eliminated as the
y-axis intercept is less than 5.
7. (a challenge) What best describes the path of a particle whose position is described by the following
parametric equations: x(t )  2 cos( 2(t 
a)


)), y (t )  3 sin( 2(t  )) , 0  t  2
2
2
Travels twice in a counterclockwise direction around a circle centered at the origin starting at the point
(2,0) .
b) Travels twice around in a clockwise direction around an ellipse centered at the origin starting at the
point (0,3) .
c) Travels twice around in a counterclockwise direction around an ellipse centered at the origin starting at
the point (2,0) .
d) Travels halfway in a counterclockwise direction around a circle centered at the origin starting at the
point (2,0) .
x
2
y
3
Answer: (c) : note that ( )  ( )  1 so the equation describes an ellipse centered at the origin. The
2
2
start point is easy to compute (plug in t = 0) and, because t is multiplied by 2, we go around twice.