Mastery Review for Test 2
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1.
The tile chart shown at right indicates that ... /i)
ii) is a factor of
iii) If , then
a. / i only / c. / iii only / e. / all threeb. / ii only / d. / i and ii only
____ 2. If then
a. / / c. /b. / / d. /
____ 3. If then
a. / / c. /b. / / d. /
____ 4. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown.
Which of the following is most likely to be true?
a. / The instantaneous rate of change at x=-2 is 0b. / The average rate of change between x= -1 and x= 2 is 0
c. / The instantaneous rate of change at x=4 is -4
d. / The average rate of change between x= 4 and x=6 is 3
____ 5. A man kept track of his weight and daily food consumption every day for several months. He drew a graph with food consumption on the vertical axis and time on the horizontal axis. What units could be used for instantaneous rate of change on this graph?
a. / kg/day / c. / cal/dayb. / years/kg / d. / kg/cal
____ 6. The equation of the function shown below could be...
a. / / b. / / c. / / d. /____ 7. The graph of has horizontal asymptote(s) with equation(s)...
a. / y=2 / b. / y=0 / c. / x=2, x=-2 / d. / x=-3____ 8. If and then g(1)=
a. / 7 / b. / -1 / c. / 15 / d. / 5____ 9.
is a point on the unit circle and is on the terminal arm of angle q. Which of the following is true?
a. / cos q = and tan q = - / c. / cos q = and tan q = -b. / sin q = and tan q = - / d. / sin q = and tan q = -
____ 10. The input/output diagram illustrates a number of transformations to y=cos(x).
The point on the base function with x coordinate 0 is transformed to the point...
a. / (–1,2) / b. / (–1,–12) / c. / (1,2) / d. / (1,–12)____ 11. If where 0 q 360, then
a. / q = -45 or q = 135 / c. / q = 225 or q = 315b. / q = 135 or q = 225 / d. / q = 135 or q = 315
____ 12.
a. / / c. /b. / / d. /
____ 13.
a. / / c. /b. / / d. /
____ 14. If , then
a. / / b. / / c. / / d. /____ 15. Evaluate
a. / 1 / b. / 10 / c. / 2 / d. / 0____ 16.
A table of values for a function is shown on the right. A possible equation for this function is... /a. / / b. / / c. / / d. /
____ 17. Which of the following is true?
i) ii)
a. / i) only / b. / ii) only / c. / both / d. / neither____ 18. Which of the following is true?
i) ii)
a. / i) only / b. / ii) only / c. / both / d. / neither____ 19. Evaluate
a. / 0 / b. / 2 / c. / 6 / d. / does not exist____ 20. Evaluate
a. / -2.5 / b. / 1.5 / c. / 0 / d. / does not exist____ 21. Which of the following limits do not exist?
i) ii) iii)
a. / i) only / b. / ii) only / c. / iii) only / d. / i) and iii)____ 22. Evaluate
a. / 0 / b. / -1.5 / c. / 6 / d. / does not exist____ 23. The slope of the tangent to the graph of at x=-2 is
a. / 18 / b. / 25 / c. / 24 / d. / -13____ 24. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown.
Which of the following is most likely to be true?
a. / does not exist andb. / and
c. / and
d. / and does not exist
____ 25. The expression is most likely to be...
a. / the derivative of a function / c. / slope of a secantb. / the value of a derivative / d. / none of the above
____ 26. If then =
a. / / b. / / c. / / d. /____ 27. If then =
a. / / b. / / c. / / d. /____ 28. If then
a. / / b. / / c. / / d. /____ 29. If then
a. / / b. / / c. / / d. / 6t____ 30. Given ; ; ;
If , then
a. / 16 / b. / 13 / c. / / d. /Mastery Review for Test 2
Answer Section
MULTIPLE CHOICE
1. ANS: B
i) The area (the stuff inside) divided by one dimension is the other,
Therefore, ...
ii) The chart shows us that is a factor of because there are no ‘rejects’.
iii) if , then (from the chart)
and when x=–1,
so,
PTS: 1
2. ANS: C
/ 2 / 3/ - / - / - / - / - / - / - / - / - / - / - / - / - / - / - / - / 0 / + / + / + / +
/ + / + / + / + / + / + / + / + / + / + / 0 / + / + / + / + / + / + / + / + / + / +
/ - / - / - / 0 / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / +
/ + / + / + / 0 / - / - / - / - / - / - / 0 / - / - / - / - / - / 0 / + / + / + / +
So... if
PTS: 1
3. ANS: C
If a fraction is equal to zero then the numerator must be equal to zero and the denominator must not be equal to zero. I.e., , but both and are undefined expressions.
In this case, 0, 2, and 0.5 all make the numerator 0, but x=0 also makes the denominator 0, so .
PTS: 1
4. ANS: B
To the left of x=-1, the instantaneous rate of change is the same as the average rate of change anywhere along it because it is a straight line. This rate is 2 (from (-2,0) to (-1,2).
The instantaneous rate of change of f at x=-1 does not exist because the graph is discontinuous there. The actual function value is -1 when x is -1.
Anywhere between x=-1 and x=2, the slope is 0, so both instantaneous and average rate of change is 0. If we approach x=2 along the curve from the right, the slope approaches 0, so there is no corner there. The instantaneous rate of change at x=2 is 0.
At (4,0), the instantaneous rate of change is 1 because the slope of the tangent line is 1 (equation is y=1x-4)
At (6,3), the instantaneous rate of change is 2 because the slope of the tangent line is 2 (equation is y=2x-9)
The average rate of change between ... (-2,0) and (-1,-1) is -1 (rise of -1, run of 1)
between (-1,1) and (2,-1) is 0; between (2,-1) and (4.0) is 0.5 (rise of 1, run of 2); between (4,0) and (6,3) it is 1.5.
PTS: 1
5. ANS: C
slope would be , so the units could be cal/day
PTS: 1
6. ANS: A
With a vertical asymptote at , the bottom of the fraction should equal 0 when to make the fraction undefined. This means the denominator needs to be x - 3.
With a horizontal asymptote at , the base function of has to be translated up (down if negative) 2 units, so the last part of the equation must be + 2.
Alternatively, we could use the calculator to graph each one and compare, or draw the I/O diagram corresponding to the graph and the required translations and then get the equation.
PTS: 1
7. ANS: A
. As , the 6 in the numerator and the -4 in the denominator become insignificant, so . As , , so the horizontal asymptote has equation y=2.
PTS: 1
8. ANS: D
f(1)=3 because (1,3)f
PTS: 1
9. ANS: C
The x coordinate of the point is cos q , so cos q = .
The y coordinate of the point is sin q , so sin q = and tan q
PTS: 1
10. ANS: A
The input/output diagram for the function is shown below.
The base function, y=cos(x) has coordinates . Horizontally, we have to work backwards from the base function, so the first operation that we have to undo is “multiply by 4”, and then we have to undo “add 1”, so the original x coordinate (0) becomes –1
Vertically, we work forward from the base function, and the first operation is “add 5”, and then second operation is “divide by 3”, so the original y coordinate of 1 becomes 2.
PTS: 1
11. ANS: C
The unit circle shows us the coordinates of all points with principal angles 30, 45 and 60. It shows that at points K and P. These points are 45 away from the x axis, so the angles are 225 and 315
PTS: 1
12. ANS: A
xwq
PTS: 1
13. ANS: B PTS: 1
14. ANS: C
The base is x, and the exponent (the logarithm) is y. So... the logarithm OF 2 base x is y ... i.e., .
PTS: 1
15. ANS: C
because . We want the exponent (the logarithm).
PTS: 1
16. ANS: A
Note that the inverse of this function (interchange x and y coordinates) is an exponential function with base 2 (common ratio between terms is 2). The inverse has equation , so the original has equation .
PTS: 1
17. ANS: B
... the rule is but is NOT the same as . (Order of operations)
... this is the composition of a function ( power of 2) and its inverse, so the output will be the same as the input ... x.
PTS: 1
18. ANS: A
The rules are:
; ;
PTS: 1
19. ANS: D
which does not exist
PTS: 1
20. ANS: B
/ Or... as , the -5x becomes insignificant in the numerator, and the +1 becomes insignificant in the denominator, so ...PTS: 1
21. ANS: A
As , x is greater than 3, so will negative, so will be undefined, so does not exist.
As , x is less than 3, so will positive, so will be defined and will approach 0, so
As , from either side, x is less than 3, so will positive, will be defined, and will approach 1, so
PTS: 1
22. ANS: A
PTS: 1
23. ANS: D
At x=-2,
PTS: 1
24. ANS: C
is the limit of the y coordinates as x approaches -1. From both sides, the y-coordinates approach 1.
So... . Similarly, , , and
is the slope of the tangent at x=2, but there is no tangent line at x=2.
So... does not exist. Similarly, , , and .
PTS: 1
25. ANS: A
If , then the expression is of the form which is the derivative.
PTS: 1
26. ANS: B
if then
so
PTS: 1
27. ANS: C
if then
so
PTS: 1
28. ANS: D
if then
PTS: 1
29. ANS: C
if then and
PTS: 1
30. ANS: D
if then
So,
PTS: 1