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January 25, 2015

January 25, 2015

Posted by **Samantha** on Thursday, December 19, 2013 at 7:18pm.

1. Determine which of the following is true for the function f(x) = x^3 +5x^2 - 8x +3.

I. f(x) has a relative minimum at x = 2/3.

II. f(x) has a relative maximum at x = 2/3.

III. f(x) has a zero at x = 2/3

I only

*II only

III only

I and III only

II and III only

2. Determine the equation of the normal line to y = x^2 + 5 at the point (2, 9)

y=-1/4x + 17/4

*y=1/2x + 8

y=4x+1

y=4x-34

y=-1/4x + 19/2

3. If 12 ft^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

2 ft^3

4 ft^3

5 ft^3

*8.5 ft^3

9 ft^3

4. Find the tangent line approximation, L(x), of f(x)=x^2/3 at x = 8.

L(x) = 4/3x-20/3

L(x) = 2/3x+8

L(x) = 4(x-8)

L(x) = 1/3x + 4/3

*L(x) = 4/3x - 8/3

5. A balloon is rising at a constant speed of 5 ft/sec. A boy is cycling along a straight road at a constant speed of 15 ft/sec. When he passes under the balloon, it is 5 feet above him. Approximately how fast is the distance between the boy and the balloon increasing 3 seconds after he has passed underneath it?

12 ft/sec

16 ft/sec

20 ft/sec

*25 ft/sec

30 ft/sec

6. A factory is manufacturing a rectangular storage container with an open top. The volume of the container is 10 ft^3, and the length of the base is twice the width. The material for the base costs $10 per square foot, and the material for the sides costs $6 per square foot. Find the cheapest cost to make the container, given the conditions.

$27.85

$46.19

*$87.24

$147.85

$163.54

7. The edge of a cube was found to have a length of 50 cm with a possible error in measurement of 0.1 cm. Based on the measurement, you determine that the volume is 125,000 cm^3. Use tangent line approximation to estimate the percentage error in volume.

0.6%

0.9%

*1.2%

1.5%

1.8%

8. An inverted conical tank (with vertex down) is 14 feet across the top and 24 feet deep. If water is flowing in at a rate of 12 ft^3/min, find the rate of change of the depth of the water when the water is 10 feet deep.

0.007 ft/min

0.449 ft/min

0.018 ft/min

*0.051 ft/min

0.065 ft/min

9. For the function f(x)=Inx/x^2, find the approximate location of the critical point in the interval (0, 5).

(0.5, −2.773)

(1, 0)

(1.649, 0.184)

*(2, 0.173)

(0.778, −1.813)

- CALCULUS - Check my answers :) -
**Damon**, Thursday, December 19, 2013 at 7:31pm1. Determine which of the following is true for the function f(x) = x^3 +5x^2 - 8x +3.

I. f(x) has a relative minimum at x = 2/3.

II. f(x) has a relative maximum at x = 2/3.

III. f(x) has a zero at x = 2/3

I only

*II only

III only

I and III only

II and III only

================================

f' = 3 x^2 + 10 x - 8

zero at x = 2/3 and at a = -2

f" = 6 x + 10

curvature is POSITIVE at x = 2/3

the function is headed UP

That is a MINIMUM

- CALCULUS - Check my answers :) -
**Damon**, Thursday, December 19, 2013 at 7:36pm2. Determine the equation of the normal line to y = x^2 + 5 at the point (2, 9)

y=-1/4x + 17/4

*y=1/2x + 8

y=4x+1

y=4x-34

y=-1/4x + 19/2

==================================

dy/dx = 2x

at x = 2, slope = 2*2 =4

so m of normal= -1/slope = -1/4

y = -1/4 x + b

9 = -1/4 *2 + b

9 = -1/2 + b

b = 19/2 so I say the LAST one

- CALCULUS - Check my answers :) -
**Damon**, Thursday, December 19, 2013 at 7:49pm3. If 12 ft^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

2 ft^3

4 ft^3

5 ft^3

*8.5 ft^3

9 ft^3

=================================

side length s and height h

A = 4 s h + s^2 = 12

4 s h = 12 - s^2

h = (3/s) - s/(4)

V = s^2 h

V = 3 s - s^3/4

dV/ds = 3 - (3/4)s^2

at max or min dV/ds = 0

so

3/4 s^2 = 3

s = 2

then find h

h = 3/2 - 2/4

h =1

volume = h s^2 = 1 (4) = 4

- Now you check for a while ! -
**Damon**, Thursday, December 19, 2013 at 7:50pmI am getting bored

- CALCULUS - Check my answers :) -
**mathidislikeperson**, Thursday, December 19, 2013 at 9:02pmWell Damon, calculus isnt supposed to be.. interesting. Haha

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**Samantha**, Thursday, December 19, 2013 at 9:24pmIs number 4 the last one?

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**Samantha**, Thursday, December 19, 2013 at 9:28pmAnd 5 would be 25?

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**Reiny**, Thursday, December 19, 2013 at 10:18pm#4, the way you typed it, ...

y' = 2x/3

when x = 8, y' = 16/3, which is not found in any of your choices as the slope

if you meant , f(x) = x^(2/3)

then y' = (2/3)x^(-1/3) = (2/3)(1/x^(1/3))

when x = 8

y' = (2/3)(1/2) = 1/3

the only one that has a slope of 1/2 is

L(x) = 1/3x + 4/3

- CALCULUS - Check my answers :) -
**Reiny**, Thursday, December 19, 2013 at 10:30pm#5

At a time of t seconds after the boy passed under the balloon,

distance covered by boy = 15t

height of balloon = 5 + 5t

let the distance between them be d ft

d^2 = (5t+5)^2 + (15t)^2

2d dd/dt = 2(5t+5)(5) + 2(15t)(15)

dd/dt =( 5(5t+5) + 15(15t) )/d

when t = 3,

d^2 = 20^2 + 45^2

d = √2425

dd/dt = (5(20) + 15(45))/√2425

= 775/(5√97)

= appr 15.74

which is none of the answers unless it was rounded off to 16 ft/sec

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