Question 1

Find the open interval(s) on which   is increasing or decreasing.&#10;A) increasing on   decreasing on  &#10;B) increasing on   decreasing on  &#10;C) increasing on   decreasing on  &#10;D) increasing on   decreasing on  &#10;E) increasing on   decreasing on

Accepted Answer

increasing on   decreasing on

Question 2

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . Determine the value of after one second. Round numerical values in your answer to one decimal place.

A)

<strong>A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . Determine the value of after one second. Round numerical values in your answer to one decimal place.</strong> A) B) C) D) E) <div style=padding-top: 35px>

B)

C)

D)

E)

Accepted Answer

11eaa595_65d8_abf9_a696_617e8227b758_TB4584_11_TB4584_11

Question 3

Find the relative extremum of   by applying the First Derivative Test.&#10;A) relative minimum:  &#10;B) relative maximum:  &#10;C) relative minimum:  &#10;D) relative minimum:  &#10;E) relative maximum:

Accepted Answer

relative maximum:

Question 4

Find the open interval(s) on which the function   is increasing in the interval   Round numerical values in your answer to three decimal places.&#10;A) increasing on:  &#10;B) increasing on:  &#10;C) increasing on:  &#10;D) increasing on:  &#10;E) increasing on:

Accepted Answer

The answer of Find the open interval(s) on which the...

Question 5

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find the critical number of the function. Round numerical values in your answer to the nearest whole number.

A)

<strong>The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find the critical number of the function. Round numerical values in your answer to the nearest whole number.</strong> A) B) C) D) E) <div style=padding-top: 35px>

B)

C)

D)

E)

Accepted Answer

To find the temperature coefficient of resistance (TCR), we need to differentiate the given resistance expression with respect to temperature T and then multiply the result by T/R. Using a computer algebra system, we get:

dR/dT = 11eaa595_65db_1d09_a696_ffe2c5b66c01_TB4584_11_TB4584_11
TCR = T/R * dR/dT = T/11eaa595_65da_80c4_a696_194e3560321c_TB4584_11_TB4584_11 * 11eaa595_65db_1d09_a696_ffe2c5b66c01_TB4584_11_TB4584_11

Simplifying this expression, we get:

TCR = 11eaa595_65db_441a_a696_173cdd956469_TB4584_11_TB4584_11

Therefore, the best choice is E.

Question 6

Identify the open intervals where the function   is increasing or decreasing.&#10;A) decreasing:   ; increasing:  &#10;B) decreasing on  &#10;C) increasing:   ; decreasing:  &#10;D) increasing:   ; decreasing:  &#10;E) increasing:   ; decreasing:

Accepted Answer

The function is increasing on the interval 11eaa595_65c9_422e_a696_0fe3567e7993_TB4584_11_TB4584_11 to 11eaa595_65c9_422f_a696_1f8a7f94ed46_TB4584_11_TB4584_11 and decreasing on the interval immediately after, which is 11eaa595_65c9_422f_a696_1f8a7f94ed46_TB4584_11_TB4584_11 to infinity. Therefore, choice D is the best option.

Question 7

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . Determine the speed of the ball bearing after t seconds.

A) speed:

meters per second
B) speed:

meters per second
C) speed:

meters per second
D) speed:

meters per second
E) speed:

meters per second

Accepted Answer

Given, distance covered by the ball bearing in t seconds =  $16t^2$. 
Let's evaluate the component of gravitational force acting along the inclined plane, F. This can be done by resolving the gravitational force into two components - one perpendicular to the plane and the other parallel to the plane. The component of gravitational force perpendicular to the plane does not contribute to the motion of the ball bearing along the plane, while the component parallel does. Hence, we evaluate F.

F = mg sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)
where m is the mass of the ball bearing and g is the acceleration due to gravity.

Let's evaluate the acceleration of the ball bearing along the inclined plane, a. 
a = F/m = gsin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)

Using the kinematic equations, we can relate the acceleration, distance, and time as follows: 
distance = ut +  1/2 at^2 
where u is the initial velocity of the ball bearing. Since the ball bearing starts from rest, u = 0. Solving for u, we get:

u = distance/(1/2 at^2) =  2 * $16t^2$/[g sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)]

Using the known values, we get u = 32t^2/[9.8sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)]

Now, using the formula for speed, speed = u + at, we can find the speed of the ball bearing after t seconds:

speed = 32t^2/[9.8sin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)] + [gsin(11eaa595_65d7_736f_a696_cfb7dcaa40e4_TB4584_11_TB4584_11)]t

Substituting the known values, we get: 
speed = 47.8t

Hence, the speed of the ball bearing after t seconds is 11eaa595_65d7_e8a3_a696_dbff6dd1194b_TB4584_11_TB4584_11 meters per second, which is option C.

Question 8

Find the relative minima of   on the interval   by applying the First Derivative Test. Round numerical values in your answer to three decimal places.&#10;A) relative minima:  &#10;B) relative minima:  &#10;C) relative minima:  &#10;D) relative minima:  &#10;E) relative minima:

Accepted Answer

The answer of Find the relative minima of  ...

Question 9

Find the relative maxima of   on the interval   by applying the First Derivative Test. Round numerical values in your answer to three decimal places.&#10;A) relative maxima:  &#10;B) relative maxima:  &#10;C) relative maxima:  &#10;D) relative maxima:  &#10;E) relative maxima:

Accepted Answer

The answer of Find the relative maxima of  ...

Question 10

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find the critical number of the function. Round numerical values in your answer to the nearest whole number.

A)

B)

C)

D)

E)

Accepted Answer

To find the critical number of the function, we need to find the derivative of the function with respect to temperature and set it equal to zero.

Using a computer algebra system to differentiate the function, we get:

R'(T) = 11ea8e7c_65d9_fab7_a696_5225e2a31c20_TB4584_11_TB4584_11 * e^(0.000044*T)

Setting R'(T) = 0 and solving for T, we get:

T = -2474 degrees Celsius

Rounding this to the nearest whole number, we get:

T ≈ -2474°C

Therefore, the answer is choice D, 11eaa595_65da_0b92_a696_7bd3c5e4dc5e_TB4584_11_TB4584_11.

Question 11

Find the open interval(s) on which the function   is increasing in the interval   Round numerical values in your answer to three decimal places.&#10;A) increasing on:  &#10;B) increasing on:  &#10;C) increasing on:  &#10;D) increasing on:  &#10;E) increasing on:

Accepted Answer

The answer of Find the open interval(s) on which the...

Question 12

Identify the open intervals on which the function   is increasing or decreasing.&#10;A) increasing on   decreasing on  &#10;B) increasing on   decreasing on  &#10;C) increasing on   decreasing on  &#10;D) increasing on   decreasing on  &#10;E) increasing on   decreasing on

Accepted Answer

The answer of Identify the open intervals on which the...

Question 13

For the function : (a) Find the critical numbers of f (if any);
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema.
Use a graphing utility to confirm your results.

A) (a)

<strong>For the function : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Use a graphing utility to confirm your results.</strong> A) (a) (b) increasing: ; decreasing: (c) relative max: B) (a) (b) increasing: ; decreasing: (c) relative max: C) (a) (b) decreasing: ; increasing: (c) relative min: D) (a) (b) decreasing: ; increasing: (c) relative min: ; relative max: E) (a) (b) decreasing: ; increasing: (c) relative min:

(b) increasing:

; decreasing:

(c) relative max:

B) (a)

(b) increasing:

; decreasing:

(c) relative max:

C) (a)

(b) decreasing:

; increasing:

(c) relative min:

D) (a)

(b) decreasing:

; increasing:

(c) relative min:

; relative max:

E) (a)

(b) decreasing:

; increasing:

(c) relative min:

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Accepted Answer

The answer of For the function   : (a)...

Question 14

Identify the open intervals where the function   is increasing or decreasing.&#10;A) decreasing on  &#10;B) increasing on  &#10;C)  &#10;D) decreasing on   ; increasing on  &#10;E)

Accepted Answer

The answer of Identify the open intervals where the function...

Question 15

Use the graph of the function   given below to estimate the open intervals on which the function is increasing or decreasing.  &#10;A) increasing on   ; decreasing on  &#10;B) increasing on   ; decreasing on  &#10;C) increasing on   ; decreasing on  &#10;D) increasing on   ; decreasing on  &#10;E) increasing on   ; decreasing on

Accepted Answer

The answer of Use the graph of the function ...

Question 16

Find the critical number of the function   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the critical number of the function...

Question 17

For the function : (a) Find the critical numbers of f (if any);
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema.
Then use a graphing utility to confirm your results.

A) (a) x = 0 ,

(b) increasing:

; decreasing:

(c) relative max:

; relative min:

B) (a) x = 0 ,

(b) decreasing:

; increasing:

(c) relative min:

; relative max:

C) (a) x = 0 ,

(b) increasing:

; decreasing:

(c) relative max:

; no relative min.
D) (a) x = 0 ,

(b) increasing:

; decreasing:

(c) relative max:

; relative min:

E) (a) x = 0 ,

(b) decreasing:

; increasing:

(c) relative min:

; relative max:

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The answer of For the function   : (a)...

Deck 4: Section 3: Applications of Differentiation