Question 1

Determine whether Rolle's Theorem can be applied to  on the closed interval  If Rolle's Theorem can be applied, find all values of c in the open interval  such that

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Question 2

The height of an object t seconds after it is dropped from a height of 550 meters is  Find the average velocity of the object during the first 7 seconds.

Accepted Answer

To find the average velocity of the object during the first 7 seconds, we need to find the displacement of the object during that time interval.

The initial height of the object is 550m and after 7 seconds, its height can be found by:

h(7) = -4.9(7)^2 + 0(7) + 550 ≈ 222.7m

Therefore, the displacement of the object during the first 7 seconds is:

Δy = 222.7 - 550 = -327.3m

The negative sign indicates that the object is moving downwards.

The average velocity during the first 7 seconds is then:

v = Δy/Δt = -327.3/7 ≈ -34.30m/sec

So, the answer is B. The velocity of the object at any time t is given by the derivative of its height function. In this case, the height function is h(t) = 550 - 4.9t^2 (where g=9.8 m/s^2 is the acceleration due to gravity and the negative sign indicates that the object is falling). 
The velocity function is v(t) = h'(t) = -9.8t. 
The average velocity of the object during the first 7 seconds is then:
v_avg = (1/7) integral from 0 to 7 of v(t) dt 
v_avg = (1/7) integral from 0 to 7 of -9.8t dt  
v_avg = (1/7) [-4.9t^2] from 0 to 7 
v_avg = (1/7) [-4.9(7^2 - 0^2)] 
v_avg = -34.3 m/s (rounded to two decimal places) 
Therefore, the best choice is B.

Question 3

The ordering and transportation cost C for components used in a manufacturing process is approximated by  where C is measured in thousands of dollars and x  is the order size in hundreds.According to Rolle's Theorem, the rate of change of the cost must be 0
For some order size in the interval  Find this order size.Round your answer to three decimal
Places.

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The answer of The ordering and transportation cost C for...

Question 4

A company introduces a new product for which the number of units sold S is  where t is the time in months since the product was introduced.Find
The average value of  during the first year.

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The answer of A company introduces a new product for...

Question 5

Determine whether the Mean Value Theorem can be applied to the function  on the closed interval [3,9].If the Mean Value Theorem can be applied, find all numbers c in the open interval (3,9) such that

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The answer of Determine whether the Mean Value Theorem can...

Question 6

Determine whether the Mean Value Theorem can be applied to the function  on the closed interval [0,16].If the Mean Value Theorem can be applied, find all
Numbers c in the open interval (0,16) such that

Accepted Answer

The answer of Determine whether the Mean Value Theorem can...

Question 7

Determine whether Rolle's Theorem can be applied to the function  on the closed interval  If Rolle's Theorem can be applied, find all numbers c in the open
Interval  such that

Accepted Answer

Rolle's Theorem can be applied if the function is continuous on the closed interval and differentiable on the open interval, and if the function values at the endpoints of the closed interval are equal. In this case, the function is continuous and differentiable on the given intervals, and f(-1) = f(1) = 0, so Rolle's Theorem applies. The only number c in the open interval (-1,1) where f'(c) = 0 is c = 0. Rolle's Theorem states that if a function $f(x)$ is continuous on a closed interval $[a,b]$, differentiable on the open interval $(a,b)$, and $f(a) = f(b)$, then there exists at least one number $c$ in $(a,b)$ such that $f'(c) = 0$.

In this case, the given function $f(x) = x^2 - 2x$ is continuous on $[0,2]$ and differentiable on $(0,2)$, and $f(0) = f(2) = 0$. Therefore, we can apply Rolle's Theorem to find a number $c$ in $(0,2)$ such that $f'(c) = 0$.

To find this number $c$, we need to find where the derivative $f'(x) = 2x - 2$ is equal to $0$ on the interval $(0,2)$. Setting $f'(x) = 0$, we get $2x - 2 = 0$, or $x = 1$. Therefore, the only number $c$ in $(0,2)$ where $f'(c) = 0$ is $c = 1$.

Thus, the answer is A, Rolle's Theorem applies; $c=0$.

Question 8

Determine whether Rolle's Theorem can be applied to the function  on the closed interval  If Rolle's Theorem can be applied, find all numbers c in the open interval  such that

Accepted Answer

The answer of Determine whether Rolle's Theorem can be applied...

Question 9

A plane begins its takeoff at 2:00 P.M.on a 2200-mile flight.After 12.5 hours, the plane arrives at its destination.Explain why there are at least two times during the flight when the
Speed of the plane is 100 miles per hour.

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The answer of A plane begins its takeoff at 2:00...

Question 10

Which of the following functions passes through the point  and satisfies

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The answer of Which of the following functions passes through...

Question 11

Determine whether Rolle's Theorem can be applied to the function  on the closed interval  If Rolle's Theorem can be applied, find all numbers c in the open interval  such that

Accepted Answer

The answer of Determine whether Rolle's Theorem can be applied...

Question 12

The height of an object t seconds after it is dropped from a height of 250 meters is  Find the time during the first 8 seconds of fall at which the
Instantaneous velocity equals the average velocity.

Accepted Answer

The answer of The height of an object t seconds...

Question 13

Find a function f that has derivative  and with graph passing through the point (5,6).

Accepted Answer

The answer of Find a function f that has derivative...

Question 14

A company introduces a new product for which the number of units sold S is  where t is the time in months since the product was introduced.During what month does  equal the average value of  during the first year?

Accepted Answer

The average value of the number of units sold during the first year can be calculated by finding the total number of units sold in the first 12 months and dividing by 12. Since the product was introduced at t = 0, we can find the total number of units sold in the first 12 months by evaluating S from t = 0 to t = 12:

S = 11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 |_[t=0 to t=12]
  = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 - 11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11
  = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 - 0

The average value of S during the first year is then:

S_avg = S/12
      = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11/12

We want to find the month where S equals S_avg. In other words, we want to find the value of t that satisfies the equation:

S = S_avg

Substituting the expressions for S and S_avg, we get:

11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 + (t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11/12

Multiplying both sides by 12 and simplifying:

11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 + (t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11
11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 + (t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 - 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = 0
(t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = -11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11
t-12 = -(11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11/1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11)
t = 12 - 11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11/1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11

We can evaluate this expression using a calculator to get:

t ≈ 4.4

Therefore, the answer is D) April.

Quiz 3: Applications of Differentiation

Determine whether Rolle's Theorem can be applied to on the closed interval If Rolle's Theorem can be applied, find all values of c in the open interval such that

The height of an object t seconds after it is dropped from a height of 550 meters is Find the average velocity of the object during the first 7 seconds.

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced.Find The average value of during the first year.

Determine whether the Mean Value Theorem can be applied to the function on the closed interval [3,9].If the Mean Value Theorem can be applied, find all numbers c in the open interval (3,9) such that

Determine whether the Mean Value Theorem can be applied to the function on the closed interval [0,16].If the Mean Value Theorem can be applied, find all Numbers c in the open interval (0,16) such that

Determine whether Rolle's Theorem can be applied to the function on the closed interval If Rolle's Theorem can be applied, find all numbers c in the open Interval such that

Determine whether Rolle's Theorem can be applied to the function on the closed interval If Rolle's Theorem can be applied, find all numbers c in the open interval such that

A plane begins its takeoff at 2:00 P.M.on a 2200-mile flight.After 12.5 hours, the plane arrives at its destination.Explain why there are at least two times during the flight when the Speed of the plane is 100 miles per hour.

Which of the following functions passes through the point and satisfies

Determine whether Rolle's Theorem can be applied to the function on the closed interval If Rolle's Theorem can be applied, find all numbers c in the open interval such that

The height of an object t seconds after it is dropped from a height of 250 meters is Find the time during the first 8 seconds of fall at which the Instantaneous velocity equals the average velocity.

Find a function f that has derivative and with graph passing through the point (5,6) .

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced.During what month does equal the average value of during the first year?