MCV4U Calculus and Vectors Mastery Test 1

MCV4U Calculus and Vectors Mastery Test 1

Multiple Choice

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

____ 1. Complete the square: is equal to

a. / c.
b. / d.

____ 2. The equation has discriminant

a. / 25 / b. / 1 / c. / 29 / d. / 21

____ 3. The number of distinct real roots for the equatiion is

a. / 2 / b. / 3 / c. / 4 / d. / 5

____ 4. If then

a. / c.
b. / d.

____ 5. Use a TI-83 to find the equation of the parabola of best fit and the value of R2 (rounded to 2 decimal places) for the following data:

a. / y = -0.31x2 + 20.88x - 157.50 and R2 = 0.94
b. / y = -0.17x2 + 13.52x - 62.73 and R2 = 0.96
c. / y = -0.43x2 + 27.06x - 239.32 and R2 = 0.99
d. / y = -0.09x2 + 8.01x + 21.14 and R2 = 0.98

____ 6. - x - 3 is equivalent to (x + 2)+

a. / 1st box: (x - 3) 2nd box: + 3 / c. / 1st box: (x - 3) 2nd box: + 9
b. / 1st box: (x - 1) 2nd box: - 1 / d. / 1st box: (x - 1) 2nd box: + 5

____ 7. If + 2x - 3 then its zero(s) are at

a. / 1 and –3 / b. / 1 and 3 / c. / –1 and 3 / d. / –1 and –3

____ 8.

Which of the graphs shown at right is most likely to be the graph of ?
a. / A / b. / B / c. / C / d. / D

____ 9. The tables below shows the concentration of CO2 in the air in a room over time. The best estimate of the instantaneous rate of change of the CO2 level at t=3 is...

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

____ 10. If and then g(5)=

a. / 7 / b. / -1 / c. / 9 / d. / -3

____ 11.

The diagram shows the unit circle with points equally spaced around its circumference.
If = -0.26, then which of the following points is closest to the terminal ray of ?
a. / L and N / b. / R and T / c. / R and H / d. / N and X

____ 12. If and then q could be approximately

a. / b. / c. / d.

____ 13. The input/output diagram illustrates a number of transformations to y=cos(x).

The new function has ...

a. / period and amplitude 2
b. / period and amplitude
c. / period and amplitude 2
d. / period and amplitude

____ 14. If where 0 q 360, then

a. / q = -60 or q = -120 / c. / q = 120 or q = 240
b. / q = 240 or q = 300 / d. / q = 150 or q = 210

____ 15. is equal to...

a. / c.
b. / d.

____ 16.

a. / c.
b. / d.

____ 17. Evaluate

a. / b. / 108 / c. / 27 / d.

____ 18. Simplify

a. / b. / c. / d.

____ 19. Simplify . Assume

a. / b. / c. / d.

____ 20. Simplify . Assume

a. / b. / c. / d.

____ 21. Evaluate

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

____ 22. Evaluate

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

____ 23. Evaluate

a. / -2.5 / b. / 1.5 / c. / 0 / d. / does not exist

____ 24. The slope of the tangent to the graph of at x=2 is

a. / 18 / b. / 25 / c. / 31 / d. / 14

____ 25. Given the graph of y=f(x) shown below. Tangent lines to the graph are also shown.

If , then the value of is

a. / 6 / b. / 8 / c. / 11 / d. / 2

____ 26. 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. / and
b. / and
c. / does not exist and
d. / and

____ 27. 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 and
b. / and
c. / and
d. / and does not exist

____ 28. 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. / and
b. / and
c. / and
d. / and

____ 29. 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

____ 30. 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

MCV4U Calculus and Vectors Mastery Test 1

Answer Section

MULTIPLE CHOICE

1. ANS: D

. Arrange the tiles into a square ... we need to add in 25 1-tiles to complete the square, and another -25 1-tiles to keep the expression equivalent to the original. We get

PTS: 1 BNK: MCV 01 Review of Algebra

2. ANS: A

The discriminant =

= 25 - 0

= 25

PTS: 1 BNK: MCV 01 Review of Algebra

3. ANS: C

The real roots are , 2, and -2. has no real roots.

PTS: 1 BNK: MCV 01 Review of Algebra

4. ANS: A

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 0.5 also makes the denominator 0, so .

PTS: 1 BNK: MCV 01 Review of Algebra

5. ANS: B

Clear the screen by pressing CLEAR a couple of times. Make sure the diagnostics are on by pressing 2nd 0 to choose CATALOG. Use the down arrow button to scroll down until the arrow is pointing at DiagnosticOn and press ENTER. This should paste the command DiangnosticOn onto your home screen with the cursor flashing beside it. Press ENTER and it should say Done. Now to enter the data.

Use the list editor on the TI-83 (push STAT and then choose 1. Edit. If lists 1 and 2 are not shown, press teh STAT button again and then 5. SetUpEditor. Clear lists 1 and 2 by moving the cursor up to L1, press CLEAR and then the down scroll button (arrow pointing down). If you push delete by accident, you will lose L1 (you can fix this by using 5.SetUpEditor).

Enter the lengths in L1 by pressing ENTER after each one. Enter the heights in L2. Return to the home screen by pressing 2nd MODE (to QUIT). Clear your screen. Press STAT, use the right arrow button to select CALC, press 5:QuadReg to choose line of best fit. Your screen should now say QuadReg with the cursor flashing to the right. Entre L1 by pressing 2nd 1. Push the comma button (,) and then enter L2 by pressing 2nd 2. This tells the calculator where to find the data you entered. Press the comma button again and then VARS and the right arrow button to select Y-VARS. Select 1:Function by pressing ENTER and then ENTER again to select Y1. This tells the calculator where to store the equation of the line. At this point, you should have:

QuadReg L1, L2, Y1

at the top of your screen with the cursor flashing in the next line. Press ENTER.

The following should appear:

y=ax+b

a=-.1704545455

b=13.52272727

c=-62.72727273

=.96003996

The (rounded) equation is y = -0.17x2 + 13.52x - 62.73 and R2 = 0.96.

PTS: 1 BNK: MCV 02 Review of Functions, Transformations & Rates of Change

6. ANS: A

Partially factor the expression ...

so... (x + 2) (x - 3) + 3

PTS: 1 BNK: MCV 02 Review of Functions, Transformations & Rates of Change

7. ANS: A

The easiest way to do this problem is probably to just sub in x=1 ,or x=–1 and then x=–3 or x=3 to find out which ones ‘work’.

On the TI-83, type in the equation into the y= screen, and the look at the graph or table to see what its zeros are.

or factor the expression ...

so... (x - 1) (x + 3), and its zeros are at 1 and –3.

or ... complete the square or use the quadratic formula.

PTS: 1 BNK: MCV 02 Review of Functions, Transformations & Rates of Change

8. ANS: A

There are many ways to decide which one is most appropriate - note that all four graphs have the same zeros - at 0.5, 2, and 3.

Graph A “bounces” off the x-axis at 2, so its equation probably includes the term or so that to the left and right of 2, the sign of the expression is the same.

Graph B “bounces” off the x-axis at 3, so its equation probably includes the term or so that to the left and right of 3 the sign of the expression is the same.

Graph C passes through the x-axis at all of its zeroes, so its equation cannot include an even power of a factor.

Graph D “bounces” off the x-axis at 0.5, so its equation probably includes the term or so that to the left and right of 0.5 the sign of the expression is the same.

In this case, the best choice is graph A.

PTS: 1 BNK: MCV 02 Review of Functions, Transformations & Rates of Change

9. ANS: D

For instantaneous rate of change, we want the limit of slopes of line segments with one endpoint at 3. The best estimate of the limit is 0.

PTS: 1 BNK: MCV 02 Review of Functions, Transformations & Rates of Change

10. ANS: B

f(5)=3 because (5,3)f

PTS: 1 BNK: MCV 02 Review of Functions, Transformations & Rates of Change

11. ANS: D

is the y coordinate of the point on the terminal ray. Points O and W are at 30° away from the x-axis with y-coordinate of -0.5 ruling both of them out. Sin-1(-.26) is approximately -15° which corresponds to point X. Point N will have the same y-coordinate as point T.

PTS: 1 BNK: MCV 08 Review of Trigonometry

12. ANS: A

Since and , then q is in the 3rd quadrant (the coordinates of the point are )

To find the related acute angle, use and

and are both approximately (make sure your calculator is set to degrees)

However, the angle is in the 3rd quadrant.

So ... if we fit the right triangle shown into the right spot, we can see that the actual angle could be 180° + 33° or

PTS: 1 BNK: MCV 08 Review of Trigonometry

13. 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 “add 1”, so the second horizontal transformation is a horizontal translation of 1 units left

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

The second operation is “subtract 6”, so the 2nd vertical transformation is a translation of 6 units down

The base function has period , so after a horizontal stretch of factor , its period is

The base function has amplitude 1, so after vertical stretch of factor 2, the new amplitude is 2.

PTS: 1 BNK: MCV 08 Review of Trigonometry

14. ANS: B

The unit circle shows us the coordinates of all points with principal angles 30, 45 and 60. It shows that at points L and N. These points are 60 away from the x axis, so the angles are 240 and 300

PTS: 1 BNK: MCV 08 Review of Trigonometry

15. ANS: A

cdsa

PTS: 1 BNK: MCV 08 Review of Trigonometry

16. ANS: A

xwq

PTS: 1 BNK: MCV 08 Review of Trigonometry

17. ANS: A

/ We are using the fact that to split up the power.
An exponent of means that you take the 4th root of the base. I.e., so .

PTS: 1 BNK: MCV 10 Review of Logs and Exponentials

18. 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 BNK: MCV 10 Review of Logs and Exponentials

19. ANS: D

/ To divide powers with the same base, subtract the exponents (beware... you may be subtracting a negative)
Or...
/ Multiply the numerator and denominator by something that will ‘get rid of’ the denominator (make the exponents in the denominator 0 because )

PTS: 1 BNK: MCV 10 Review of Logs and Exponentials

20. ANS: D

/ To take the power of a power, multiply the exponents.
Reduce.

PTS: 1 BNK: MCV 10 Review of Logs and Exponentials

21. ANS: B