MCV4U Mastery Test 5

MCV4U Mastery Test 5

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____1.

The tile chart shown at right indicates that ... /

ii) is a factor of

iii) If , then

a. / i only / c. / iii only / e. / all three
b. / ii only / d. / i and ii only

____2.The tables below shows the concentration of CO2 in the air in a room over time. The best estimate of the average rate of change of the CO2 level over the first 5 seconds is..

a. / -12.6 / b. / -6.3 / c. / -2.1 / d. / 0 / e. / 8.4

____3.The input/output diagram illustrates a number of transformations to y=sin(x).

The correct sequence of vertical transformations applied to y=sin(x) is...

a. / a vertical stretch of factor , followed by a vert. translation of 1 units down
b. / a vertical translation of 1 units down followed by a vert. stretch of factor
c. / a vertical stretch of factor 2, followed by a vert. translation of 1 units up
d. / a vertical translation of 1 units up followed by a vert. stretch of factor 2,

____4.

a. / / c. /
b. / / d. /

____5.Given that ... in the diagram. Which of the following must be true?

a. / AB=AD / c. / BD=CE
b. / AC=BD / d. / BC=AB

____6. The period of the function graphed below is closest to ...

a. / 4.5 / b. / 1.5 / c. / 7.5 / d. / 2

____7.Simplify

a. / / b. / / c. / / d. /

____8.Evaluate

a. / 0 / b. / 2 / c. / 6 / d. / does not exist

____9.Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown.

The value of is...

a. / 1 / b. / 0 / c. / -4 / d. / 4

____10.The expression is most likely to be...

a. / the derivative of a function / c. / slope of a secant
b. / the value of a derivative / d. / none of the above

____11.The expression is most likely to be...

a. / the derivative of a function / c. / slope of a secant
b. / the value of a derivative / d. / none of the above

____12.Which of the following limits do not exist?

i) ii) iii)

a. / i) only / b. / ii) only / c. / iii) only / d. / i) and iii)

____13.Evaluate

a. / 0 / b. / -1.5 / c. / 6 / d. / does not exist

____14.If then =

a. / / b. / / c. / / d. /

____15.If then =

a. / / b. / / c. / / d. /

____16.If then

a. / / b. / / c. / / d. /

____17.Given ; ; ;

If , then

a. / 16 / b. / 13 / c. / / d. /

____18.Given ; ; ;

If , then

a. / 16 / b. / 13 / c. / / d. /

____19.Given the graphs of y=f(x) and y=g(x) shown below. Tangent lines to the graphs are also shown.

If , then the value of is

a. / 0 / b. / -1 / c. / 4 / d. / 2

____20.Given the graphs of y=f(x) and y=g(x) shown below. Tangent lines to the graphs are also shown.

If , then the value of is

a. / -20 / b. / -3 / c. / 15 / d. / -4

____21.Given the graphs of y=f(x) and y=g(x) shown below. Tangent lines to the graphs are also shown.

If , then the average rate of change of k(x) between x=2 and x=6 is ...

a. / -1.5 / b. / 0 / c. / -3 / d. / 2

____22.Given the graphs of y=f(x) and y=g(x) shown below. Tangent lines to the graphs are also shown.

If , then the average rate of change of k(x) between x=2 and x=4 is ...

a. / -1.5 / b. / 0 / c. / -3 / d. / 1.5

____23.Given ; ; ;

If , then the value of is

a. / -8 / b. / -1 / c. / 4 / d. / 11

____24.Given and .

If , then the domain of y=k(x) is ...

a. / / c. /
b. / / d. /

____25.If , then

a. / / c. /
b. / / d. /

____26.Given the graphs of y=f(x) and y=g(x) shown below. Tangent lines to the graphs are also shown.

If , then the value of is

a. / 0 / b. / 9 / c. / 0.25 / d. / -1

____27.The number lines for y, , and are shown below. All zeros are shown.

-4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5
y / - / - / - / 0 / - / - / - / - / - / - / - / - / - / - / - / 0 / + / + / + / + / +
/ + / + / + / 0 / - / - / - / - / - / - / - / 0 / + / + / + / + / + / + / + / + / +
/ - / - / - / 0 / - / - / - / 0 / + / + / + / + / + / + / + / + / + / 0 / - / - / -

The x-coordinates of all points of inflection are

a. / -3, -1 and 4 / b. / -1 and 4 / c. / -3 and 1 / d. / -3 and 3 / e. / none

____28.The number lines for y, , and are shown below. All zeros are shown.

-4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5
y / - / - / - / - / - / - / - / 0 / + / + / + / + / + / + / + / 0 / - / - / - / - / -
/ + / + / + / 0 / + / + / + / + / + / + / + / 0 / - / - / - / - / - / - / - / - / -
/ - / - / - / 0 / + / + / + / 0 / - / - / - / - / - / - / - / 0 / + / + / + / + / +

The x-coordinates of all relative minimums are

a. / -3, -1 and 3 / b. / -1 and 3 / c. / -3 and 1 / d. / -3 and 3 / e. / none

____29.The number lines for y, , and are shown below. All zeros are shown.

The x-coordinates of the absolute maximum is

a. / -3 / b. / 3 / c. / -1 / d. / 1 / e. / none

____30.The number lines for y, , and are shown below. All zeros are shown.

-4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5
y / + / + / + / + / + / + / + / 0 / - / - / - / - / - / - / - / - / - / - / - / - / -
/ - / - / - / 0 / - / - / - / - / - / - / - / 0 / + / + / + / + / + / + / + / + / +
/ + / + / + / 0 / - / - / - / 0 / + / + / + / + / + / + / + / 0 / - / - / - / - / -

The x-coordinates of all relative minimums are

a. / -3 and 1 / b. / -3, -1 and 3 / c. / -3 / d. / 1 / e. / none

MCV4U Mastery Test 5

Answer Section

MULTIPLE CHOICE

1.ANS:A

i) By adding up the interior of the ‘rectangle’ with the ‘rejects’, we get the equation

One way to get something that looks like the equation in i) is to divide both sides by x + 1, to get

which simplifies to

Alternatively (probably a better way in the long run), we could push the rejects (4) into an extra column in the chart...

/ / / ?
x / / / /

? is the expression that multiplies by x + 1 to get 4. This would have to be .

I.e. , so...

x / / / /

The area (the stuff inside) divided by one dimension is the other,

Therefore, ...

ii) The ‘remainder’ is the ‘stuff’ that doesn’t fit nicely into the chart without fractions. i.e., it is the reject pile. In this case this is 4.

The remainder when is divided by is 4, and

iii) if , then (from the chart)

and when x=–1,

so,

PTS:1

2.ANS:C

For average rate of change, we want the slope of the line segment from 0 to 5.

slope

PTS:1

3.ANS:A

The input/output diagram for the function is shown below.

Horizontally, we have to work backwards from the base function, so the first operation that we have to undo is “multiply by 5”, so the first horizontal transformation is a stretch of factor .

The second operation we have to undo, working away from the base function, is “subtract 6”, so the second horizontal transformation is a horizontal translation of 6 units right

Vertically, we work forward from the base function, and the first operation is “divide by 2”, so the first vertical transformation is stretch of factor .

The second operation is “subtract 1”, so the 2nd vertical transformation is a translation of 1 units down

PTS:1

4.ANS:D

xwq

PTS:1

5.ANS:B

PTS:1

6.ANS:A

The graph crosses the y-axis very close to 3 and then it appears that one full cycle is graphed between here and (4.5,3), so the period is 4.5.

PTS:1

7.ANS:D

/ To multiply the fractions, just multiply all the numerators and all the denominators.
The powers of x have the same base, so add the exponents.
/ The power of a fraction is the same as the pwoer of the numerator divided by the power of the denominator.
The power of a product is the same as the product of the powers.
To find the power of a power, multiply the exponents.

PTS:1

8.ANS:A

PTS:1

9.ANS:A

is the slope of the tangent at (4,0). This slope is 1 (comparing y=x-4 to y=mx+b.)

PTS:1

10.ANS:A

If , then the expression is of the form which is the derivative.

PTS:1

11.ANS:B

If , then the expression is of the form which is the value of the derivative when x is 2 (i.e. the slope of the tangent at one point).

PTS:1

12.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

13.ANS:A

PTS:1

14.ANS:A

if then

PTS:1

15.ANS:D

if then

PTS:1

16.ANS:D

if then

PTS:1

17.ANS:B

if then

So,

PTS:1

18.ANS:D

if then

So,

PTS:1

19.ANS:C

,
so / Note: because the slope of the tangent to f at x=4 is 1, and because the slope of the tangent to g at x=4 is 0.

PTS:1

20.ANS:C

,
so / Note: because the slope of the tangent to f at x=6 is 2, and the derivative of x (with respect to x) is 1.

PTS:1

21.ANS:B

Average rate of change
/ Note: average rate of change is the slope of the line segment joining the points with the given x-coordinates.

PTS:1

22.ANS:D

Average rate of change
/ Note: Average rate of change is the slope of the line segment joining the points with the given x-coordinates.

PTS:1

23.ANS:A

,
so

PTS:1

24.ANS:D

, then
so

For k(x) to be defined,

The domain of y=k(x) is

PTS:1

25.ANS:B

, so

PTS:1

26.ANS:A

, then
so
so / Note: because the slope of the tangent to the graph of f at x=4 is 1. because the slope of the tangent to the graph of g at x=4 is 0.

PTS:1

27.ANS:B

At a point of inflection, the second derivative will change sign, (it changes from concave up/down to concave down/up), so they are at x=-1 and x=4.

PTS:1

28.ANS:E

There are no relative minimum points. For a relative minimum, we need the function to be decreasing on the left (neg. derivative) and increasing on the right (positive derivative).

PTS:1

29.ANS:D

To the left of x=1, the function is increasing or stationary (the derivative is positive or zero),and to the right of x=1, the function is decreasing (derivative is negative), so there is an absolute maximum at x=1.

PTS:1

30.ANS:D

To the immediate left of x=1, the function is decreasing (the derivative is negative), to the immediate right of x=1, the function is increasing (derivative is positive), so there is a relative minimum at x=1.

PTS:1