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GreenPrint

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Hi, for spring break homework I had to take this "calculus readiness test" thingy and I got a score of 19/24 and I'm trying to go through the test and see which ones I got wrong. I'm however perplexed because I don't see which ones I got wrong. I guess I got five wrong but don't see which ones. I don't even see how I got one wrong. I was hoping someone could point out to me which ones I got wrong and what the right answers are, thanks. Also note that this test took me about 10 minutes to do and you it's a very quick test for someone that already knows calculus so don't be scared by it... I just would like to know which ones I got wrong... below you'll find my answers along with my justifications... thanks for any help

The x-coordinate of the point of intersection of the graphs of x + 2y = 3 and -3x + y = -2

A. 2

B. 0

C. -2

D. -1

E. 1

x + 2y = 3

x - x + 2y = 3 - x

2y = 3 - x

1/2 (2y) = 1/2(3-x)

y = 1/2(3 - x)

-3x + y = -2

3x - 3x + y = -2 + 3x

y = -2 + 3x

-2 + 3x = 1/2(3 - x)

-4 + 6x = 3 - x

-4 + 6x + x = 3 - x + x

-4 + 7x = 3

-4 + 4 + 7x = 3 + 4

7x = 7

(1/7)7x = (1/7)7

x = 1

If f is a function whose graph is the parabola sketched below, then f(x) > 0 whenever

[PLAIN]http://img703.imageshack.us/img703/1189/capturezd.jpg

A. -1 < x < 3

B. x< -1 or x > 3

C. x > -1

D. x < 3

E. x > 3

The answer is not E. because by inspection of the graph provided of the function x values greater than 3 are negative, for example f(6) is clearly negative, hence the answer choice doesn't agree with f(x) > 0.

The answer is not D. because by inspection of the graph provided of the function x values less than 3 are not always positive, for example f(-4) is clearly negative, hence the answer choice doesn't agree with f(x) > 0.

The answer is not C. because by inspection of the graph provided of the function x values greater than -1 are not always positive, for example f(6) is clearly negative, hence the answer choice doesn't agree with f(x) > 0.

The answer is not B. because by inspection of the graph provided of the function x values less than negative one or greater than 3 are negative, for example f(6) is clearly negative as well as f(-4) , hence the answer choice doesn't agree with f(x) > 0.

The only answer choice left is A.

If log base 3 of (2x + 1) = 2, then x =

A. 10

B. 9

C. 4

D. 2/log(3) - 2

E. 5

log base 3 of (2x + 1) = 2

3^2 = 2x + 1

9 = 2x + 1

9 -1 = 2x + 1 - 1

8 = 2x

1/2(8) = 1/2(2x)

4 = x

x^2 - 2x can be represented as (x + a)^2 + b where

A. {a = -1, b = 1}

B. {a = -1, b = -1}

C. {a = 1, b = -1}

D. {a = -2, b = 2}

E. {a = 2, b = 2}

(x - 1)^2 - 1

(x -1)(x -1) - 1

x^2 - x - x + 1 - 1

X^2 -2x

A certain cougar population increases by a factor of 1.2 every year. (for example, if there are 100 cougars now, a year from now there will be 120.) Of the following, which best approximates the factor by which the population will increase over a 15-year period?

A. 18.000

B. 15.000

C. 7.400

D. 15.407

E. 26.986

1.2 = 6/5

15(6/5)

3(6)

18

If a^2 is approximately equal to 3000, then, of the following, which best approximates a^4?

A. 9,000

B. 900,000

C. 9,000,000

D. 3000^4

E. 6,000

3000^2 = 9,000,000

The algebraic quantity x - y is a factor of how many of the following?

{x^2 - y^2} {x^2 + y^2} {x^3 - y^3} {x^3 + y^3}

A. none

B. one only

C. two only

D. three only

E. all four

x^2 - y^2 = (x - y)(x + y)

x^2 + y^2, cannot be factored

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

x^3 + y^3 = (x + y)(x^2 - xy + y^2), cannot be factored further

hence there are only two that have x - y as a factor

Simplify the expression SQRT(9u^6 + 36v^4), if u > 0 and v > 0.

A. 3u^3 + 6v^2

B. 3SQRT(u^6 + 2v^2)

C. 18u^3 v^2

D. Cannot be simplified more.

E. None of these

SQRT(9u^6 + 36v^4)

SQRT(9(u^6 + 4v^4))

3SQRT(u^6 + 4v^4)

I chose E. because this cannot be simplified further and is not one of the other options

A child 4 feet high casts a shadow 6 feet long. At the same time, the length of the shadow cast by a pole is 12 feet. what is the height of the pole in feet?

A. 18

B. 16

C. 8

D. 6

E. 14

tan^-1 (4/6) = Ans

tan(Ans) = x/12

12 tan(Ans) = x

8 = x

In the figure shown below, what is the distance between (-2, 4) and (2, 7)?

[PLAIN]http://img69.imageshack.us/img69/8663/capturellc.jpg

A. 9

B. 8

C. 7

D. 6

E. 5

SQRT( (7 - 4)^2 + (2 + 2)^2 )

SQRT( 3^2 + 4^2 )

SQRT( 9 + 16 )

SQRT( 25 )

5

If P is proportional to T and P = 4 when T = 1/3, then when P = 1, what is T equal to?

A. 1/12

B. 1/3

C. 3

D. 12

E. None of these.

PT = (1/3) 4 = 4/3

1T = 4/3

1/1 (1T) = 1/1 (4/3)

T = 4/3

If ( (3x - 3)(x + 2) )/(x - 2) = 0, then x =

A. 2 or -1

B. -2 or 1

C. 2 or 1

D. -2, 1, or 2

E. -1, 2, or -2

3x - 3 = 0

3x - 3 + 3 = 3

3x = 3

1/3 (3x) = 1/3 (3)

x = 1

x + 2 = 0

x = -2

In the standard coordinate system, the graph of a line with slope 2/3 is

A. a vertical line

B. a horizontal line

C. a line through the origin

D. a line falling to the right

E. a line rising to the right

A line with a first derivative that is positive is "rising to the right" I guess... I thought the term was funny

What is the area of the rectangle shown below (not drawn to scale)?

[PLAIN]http://img225.imageshack.us/img225/97/capturewl.jpg

A. 1.320

B. 1705

C. 4.774

D. 1.023

E. 0.990

1.1 = 11/10

(11/10)^2 + 2(11/10)

121/100 + 22/10

121/100 + 220/100

341/100

341/100(1.4 - 1.1)

341/100(.3)

341/100(3/10)

1023/1000

should be slightly greater than 1 so I chose D

The width of a certain rectangle is 1 inch more than its length. What is the length of the rectangle if the perimeter of the rectangle is 26 inches?

A. 5 inches

B. 8 inches

C. 5.5 inches

D. 6.5 inches

E. 6 inches

26 = 2(L + W)

W = L + 1

26 = 2(L + L + 1)

26 = 2(2L + 1)

13 = 2L + 1

13 - 1 = 2L + 1 - 1

12 = 2L

1/2(12) = 1/2(2L)

6 = L

Definition: A function f has a maximum value at c if f(c) => f(x) for every x if in the domain of f. The domain of the function whose graph is shown below is [-1, 2]. At which of the following numbers does the function appear to have a maximum value?

[PLAIN]http://img189.imageshack.us/img189/8938/capturejg.jpg

A. -1

B. 0

C. 1

D. 1.5

E. 2

I don't like how the defined maximum value of a function but either way based off of observation of the graph the function has a maximum value at would appear to be a number close to 0.

The inequality |x - 6| <= 9 is equivalent to:

A. -6 <= x <= 9

B. -9 <= x <= 6

C. x <= -3 and x => 15

D. -3 <= x <= 6

E. -3 <= x <= 15

x - 6 <= 9

x -6 + 6 <= 9 + 6

x <= 15

6 - x <= 9

6 - 6 - x <= 9 - 6

- x <= 3

1/-1 (- x) => 1/-1 (3)

x => -3

-3 <= x <= 15

If f(x) = x^2 for all x, then f(x+h) - f(x) =

A. h^2

B. f(h)

C. 2hx

D. 2hx + h^2

E. x^2

f(x+h) - f(x) = (x + h)^2 - x^2

= (x + h)(x + h) - x^2

= x^2 + 2xh + h^2 - x^2

= 2xh + h^2

How many distinct numbers in the interval 1 <= x <= 3 are solutions to the equation x(x^2 + 1)sinx = 0?

A. 0

B. 1

C. 2

D. 3

E. 4

x(x^2 + 1)sinx = 0 has only one real solution at x = 0, which isn't in the domain the question provided.

Simplify: ( x^2/( 2 x^(1/3) ) )^-3

A. x^-3/8

B. 8x^-5

C. 8x^3

D. X^3/8

E. None of these

( x^2/( 2 x^(1/3) ) )^-3 = ( 8x )/ x^6 = 8/x^5

Most would consider answer B unsimplified because of the negative exponent.

The equation of the line passing through points (1, 5/2) and (4, 10) is y = mx + b, what is b equal to?

A. 2/5

B. -5/2

C. 5/2

D. 23/10

E. 0

first derivative = (10 - 5/2)/(4 - 1) = (20/2 - 5/2)/3 = (15/2)/3 = 15/6 = 5/2

y = 5/2 x + b

5/2 = 5/2 (1) + b

5/2 - 5/2 = 5/2 - 5/2 + b

0 = b

Of the following, which best approximates the radian measure of the angle [theta] shown? (Note: the picture is not drawn to scale."

[PLAIN]http://img846.imageshack.us/img846/6394/capturecvv.jpg

A. 0.197

B. 1.373

C. 0.201

D. 1.369

E. -3.381

tan^-1(5/1) = 1.373

For all [theta] defined, sin^2[theta] - cos^2[theta] =

A. -1

B. (sin[theta] - cos[theta])^2

C. -cos2[theta]

D. 1 + cos2[theta]

E. sin2[theta]

sin^2[theta] - cos^2[theta] = -cos2[theta]

If f(y) = (2y - 3)/y, then for which value of y does f(y) = 1?

A. 1/3

B. 3

C. -3

D. -1

E. 1

f(y) = (2y - 3)/y = 2 - 3/y = 1

2 - 2 - 3/y = 1 - 2

-3/y = -1

3/y = 1

y/3 = 1

y = 3

**1.**The x-coordinate of the point of intersection of the graphs of x + 2y = 3 and -3x + y = -2

A. 2

B. 0

C. -2

D. -1

E. 1

**My Answer:**E**Justification:**x + 2y = 3

x - x + 2y = 3 - x

2y = 3 - x

1/2 (2y) = 1/2(3-x)

y = 1/2(3 - x)

-3x + y = -2

3x - 3x + y = -2 + 3x

y = -2 + 3x

-2 + 3x = 1/2(3 - x)

-4 + 6x = 3 - x

-4 + 6x + x = 3 - x + x

-4 + 7x = 3

-4 + 4 + 7x = 3 + 4

7x = 7

(1/7)7x = (1/7)7

x = 1

**2.**If f is a function whose graph is the parabola sketched below, then f(x) > 0 whenever

[PLAIN]http://img703.imageshack.us/img703/1189/capturezd.jpg

A. -1 < x < 3

B. x< -1 or x > 3

C. x > -1

D. x < 3

E. x > 3

**My Answer**: A**Justification:**The answer is not E. because by inspection of the graph provided of the function x values greater than 3 are negative, for example f(6) is clearly negative, hence the answer choice doesn't agree with f(x) > 0.

The answer is not D. because by inspection of the graph provided of the function x values less than 3 are not always positive, for example f(-4) is clearly negative, hence the answer choice doesn't agree with f(x) > 0.

The answer is not C. because by inspection of the graph provided of the function x values greater than -1 are not always positive, for example f(6) is clearly negative, hence the answer choice doesn't agree with f(x) > 0.

The answer is not B. because by inspection of the graph provided of the function x values less than negative one or greater than 3 are negative, for example f(6) is clearly negative as well as f(-4) , hence the answer choice doesn't agree with f(x) > 0.

The only answer choice left is A.

**3.**If log base 3 of (2x + 1) = 2, then x =

A. 10

B. 9

C. 4

D. 2/log(3) - 2

E. 5

**My Answer:**C.**Justification**log base 3 of (2x + 1) = 2

3^2 = 2x + 1

9 = 2x + 1

9 -1 = 2x + 1 - 1

8 = 2x

1/2(8) = 1/2(2x)

4 = x

**4.**x^2 - 2x can be represented as (x + a)^2 + b where

A. {a = -1, b = 1}

B. {a = -1, b = -1}

C. {a = 1, b = -1}

D. {a = -2, b = 2}

E. {a = 2, b = 2}

**My Answer**B.**Justification**(x - 1)^2 - 1

(x -1)(x -1) - 1

x^2 - x - x + 1 - 1

X^2 -2x

**5.**A certain cougar population increases by a factor of 1.2 every year. (for example, if there are 100 cougars now, a year from now there will be 120.) Of the following, which best approximates the factor by which the population will increase over a 15-year period?

A. 18.000

B. 15.000

C. 7.400

D. 15.407

E. 26.986

**My Answer**A.**Justification**1.2 = 6/5

15(6/5)

3(6)

18

**6.**If a^2 is approximately equal to 3000, then, of the following, which best approximates a^4?

A. 9,000

B. 900,000

C. 9,000,000

D. 3000^4

E. 6,000

**My Answer**C.**Justification**3000^2 = 9,000,000

**7.**The algebraic quantity x - y is a factor of how many of the following?

{x^2 - y^2} {x^2 + y^2} {x^3 - y^3} {x^3 + y^3}

A. none

B. one only

C. two only

D. three only

E. all four

**My Answer**C.**Justification**x^2 - y^2 = (x - y)(x + y)

x^2 + y^2, cannot be factored

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

x^3 + y^3 = (x + y)(x^2 - xy + y^2), cannot be factored further

hence there are only two that have x - y as a factor

**8.**Simplify the expression SQRT(9u^6 + 36v^4), if u > 0 and v > 0.

A. 3u^3 + 6v^2

B. 3SQRT(u^6 + 2v^2)

C. 18u^3 v^2

D. Cannot be simplified more.

E. None of these

**My Answer:**E.**Justification**SQRT(9u^6 + 36v^4)

SQRT(9(u^6 + 4v^4))

3SQRT(u^6 + 4v^4)

I chose E. because this cannot be simplified further and is not one of the other options

**9.**A child 4 feet high casts a shadow 6 feet long. At the same time, the length of the shadow cast by a pole is 12 feet. what is the height of the pole in feet?

A. 18

B. 16

C. 8

D. 6

E. 14

**My Answer**C.**Justification**tan^-1 (4/6) = Ans

tan(Ans) = x/12

12 tan(Ans) = x

8 = x

**10.**In the figure shown below, what is the distance between (-2, 4) and (2, 7)?

[PLAIN]http://img69.imageshack.us/img69/8663/capturellc.jpg

A. 9

B. 8

C. 7

D. 6

E. 5

**My Answer:**E.**Justification:**SQRT( (7 - 4)^2 + (2 + 2)^2 )

SQRT( 3^2 + 4^2 )

SQRT( 9 + 16 )

SQRT( 25 )

5

**11.**If P is proportional to T and P = 4 when T = 1/3, then when P = 1, what is T equal to?

A. 1/12

B. 1/3

C. 3

D. 12

E. None of these.

**My Answer:**E**Justification:**PT = (1/3) 4 = 4/3

1T = 4/3

1/1 (1T) = 1/1 (4/3)

T = 4/3

**12.**If ( (3x - 3)(x + 2) )/(x - 2) = 0, then x =

A. 2 or -1

B. -2 or 1

C. 2 or 1

D. -2, 1, or 2

E. -1, 2, or -2

**My Answer:**B**Justification:**3x - 3 = 0

3x - 3 + 3 = 3

3x = 3

1/3 (3x) = 1/3 (3)

x = 1

x + 2 = 0

x = -2

**13.**In the standard coordinate system, the graph of a line with slope 2/3 is

A. a vertical line

B. a horizontal line

C. a line through the origin

D. a line falling to the right

E. a line rising to the right

**My Answer:**E.**Justification:**A line with a first derivative that is positive is "rising to the right" I guess... I thought the term was funny

**!4.**What is the area of the rectangle shown below (not drawn to scale)?

[PLAIN]http://img225.imageshack.us/img225/97/capturewl.jpg

A. 1.320

B. 1705

C. 4.774

D. 1.023

E. 0.990

**My Answer:**D.**Justification:**1.1 = 11/10

(11/10)^2 + 2(11/10)

121/100 + 22/10

121/100 + 220/100

341/100

341/100(1.4 - 1.1)

341/100(.3)

341/100(3/10)

1023/1000

should be slightly greater than 1 so I chose D

**15.**The width of a certain rectangle is 1 inch more than its length. What is the length of the rectangle if the perimeter of the rectangle is 26 inches?

A. 5 inches

B. 8 inches

C. 5.5 inches

D. 6.5 inches

E. 6 inches

**My Answer:**E.**Justification:**26 = 2(L + W)

W = L + 1

26 = 2(L + L + 1)

26 = 2(2L + 1)

13 = 2L + 1

13 - 1 = 2L + 1 - 1

12 = 2L

1/2(12) = 1/2(2L)

6 = L

**16.**Definition: A function f has a maximum value at c if f(c) => f(x) for every x if in the domain of f. The domain of the function whose graph is shown below is [-1, 2]. At which of the following numbers does the function appear to have a maximum value?

[PLAIN]http://img189.imageshack.us/img189/8938/capturejg.jpg

A. -1

B. 0

C. 1

D. 1.5

E. 2

**My Answer:**B.**Justification:**I don't like how the defined maximum value of a function but either way based off of observation of the graph the function has a maximum value at would appear to be a number close to 0.

**17.**The inequality |x - 6| <= 9 is equivalent to:

A. -6 <= x <= 9

B. -9 <= x <= 6

C. x <= -3 and x => 15

D. -3 <= x <= 6

E. -3 <= x <= 15

**My Answer:**E.**Justification:**x - 6 <= 9

x -6 + 6 <= 9 + 6

x <= 15

6 - x <= 9

6 - 6 - x <= 9 - 6

- x <= 3

1/-1 (- x) => 1/-1 (3)

x => -3

-3 <= x <= 15

**18.**If f(x) = x^2 for all x, then f(x+h) - f(x) =

A. h^2

B. f(h)

C. 2hx

D. 2hx + h^2

E. x^2

**My Answer:**D**Justification:**f(x+h) - f(x) = (x + h)^2 - x^2

= (x + h)(x + h) - x^2

= x^2 + 2xh + h^2 - x^2

= 2xh + h^2

**19.**How many distinct numbers in the interval 1 <= x <= 3 are solutions to the equation x(x^2 + 1)sinx = 0?

A. 0

B. 1

C. 2

D. 3

E. 4

**My Answer:**A.**Justification:**x(x^2 + 1)sinx = 0 has only one real solution at x = 0, which isn't in the domain the question provided.

**20.**Simplify: ( x^2/( 2 x^(1/3) ) )^-3

A. x^-3/8

B. 8x^-5

C. 8x^3

D. X^3/8

E. None of these

**My Answer:**E.**Justification:**( x^2/( 2 x^(1/3) ) )^-3 = ( 8x )/ x^6 = 8/x^5

Most would consider answer B unsimplified because of the negative exponent.

**21.**The equation of the line passing through points (1, 5/2) and (4, 10) is y = mx + b, what is b equal to?

A. 2/5

B. -5/2

C. 5/2

D. 23/10

E. 0

**My Answer:**E.first derivative = (10 - 5/2)/(4 - 1) = (20/2 - 5/2)/3 = (15/2)/3 = 15/6 = 5/2

y = 5/2 x + b

5/2 = 5/2 (1) + b

5/2 - 5/2 = 5/2 - 5/2 + b

0 = b

**22.**Of the following, which best approximates the radian measure of the angle [theta] shown? (Note: the picture is not drawn to scale."

[PLAIN]http://img846.imageshack.us/img846/6394/capturecvv.jpg

A. 0.197

B. 1.373

C. 0.201

D. 1.369

E. -3.381

**My Answer:**B.**Justification:**tan^-1(5/1) = 1.373

**23.**For all [theta] defined, sin^2[theta] - cos^2[theta] =

A. -1

B. (sin[theta] - cos[theta])^2

C. -cos2[theta]

D. 1 + cos2[theta]

E. sin2[theta]

**My Answer:**C.**Justification:**sin^2[theta] - cos^2[theta] = -cos2[theta]

**24.**If f(y) = (2y - 3)/y, then for which value of y does f(y) = 1?

A. 1/3

B. 3

C. -3

D. -1

E. 1

**My Answer:**B.**Justification:**f(y) = (2y - 3)/y = 2 - 3/y = 1

2 - 2 - 3/y = 1 - 2

-3/y = -1

3/y = 1

y/3 = 1

y = 3

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