Question 1

A plane begins its takeoff at 2:00 P.M.on a 2100-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. ​

Accepted Answer

The average speed of the plane is 168 mi/hr (2100 miles / 12.5 hours). By the Mean Value Theorem, there must be at least one time when the speed of the plane is exactly 168 mi/hr. Since the plane accelerates from 0 to 168 mi/hr and decelerates back to 0, it must pass through 100 mi/hr at least twice.

Question 2

Locate the absolute extrema of the function  on the closed interval  . ​

Accepted Answer

The answer of Locate the absolute extrema of the function...

Question 3

Find the value of the derivative (if it exists)of the function  at the extremum point  . ​  ​

Accepted Answer

The answer of Find the value of the derivative (if...

Question 4

Determine whether the Mean Value Theorem can be applied to the function  on the closed interval  .If the Mean Value Theorem can be applied,find all numbers c in the open interval  such that  . ​

Accepted Answer

The function is continuous on the closed interval and differentiable on the open interval. Therefore, the Mean Value Theorem can be applied. By the Mean Value Theorem, there exists at least one number c in the open interval (0,1) such that f'(c)=(f(1)-f(0))/(1-0). In this case, we have f(x)=x^3-2x+1, f(1)=0 and f(0)=1, so (f(1)-f(0))/(1-0)=-1. We need to find a number c in the open interval (0,1) such that f'(c)=-1. We have f'(x)=3x^2-2, so we need to solve the equation 3c^2-2=-1, which gives c=1/sqrt(3). Therefore, the answer is A.

Question 5

Find any critical numbers of the function  . ​

Accepted Answer

The answer of Find any critical numbers of the function...

Question 6

Locate the absolute extrema of the function  on the closed interval  . ​

Accepted Answer

The answer of Locate the absolute extrema of the function...

Question 7

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. ​

Accepted Answer

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

Question 8

Determine whether the Mean Value Theorem can be applied to the function  on the closed interval  .If the Mean Value Theorem can be applied,find all numbers c in the open interval  such that  . ​

Accepted Answer

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

Question 9

Find all critical numbers of the function  . ​

Accepted Answer

The answer of Find all critical numbers of the function...

Question 10

Find all critical numbers of the function  ,  .​ ​

Accepted Answer

The answer of Find all critical numbers of the function...

Question 11

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

Question 12

Find a function f that has derivative  and with graph passing through the point  . ​

Accepted Answer

Since the given derivative is a constant function and the point (0,1) lies on the graph, the function must be of the form f(x) = 11x + 1. Plugging in x = 0 gives f(0) = 1 which matches the given point. Therefore, the correct answer is D.

Question 13

Locate the absolute extrema of the given function on the closed interval  . ​  ​

Accepted Answer

The answer of Locate the absolute extrema of the given...

Question 14

Locate the absolute extrema of the function  on the closed interval  . ​

Accepted Answer

The answer of Locate the absolute extrema of the function...

Question 15

Locate the absolute extrema of the function  on the closed interval  . ​

Accepted Answer

The answer of Locate the absolute extrema of the function...

Question 16

Find the value of the derivative (if it exists)of  at the indicated extremum. ​  ​

Accepted Answer

The answer of Find the value of the derivative (if...

Question 17

Find the value o the derivative (if it exists)of the function  at the extremum point  . ​
​  ​

Accepted Answer

At the extremum point, the derivative of the function is equal to 0. This is because at the extremum point, the function changes from increasing to decreasing or vice versa, and the slope (derivative) is equal to 0 at that point. Therefore, the answer is D, 0.

Question 18

Determine whether the Mean Value Theorem can be applied to the function  on the closed interval  .If the Mean Value Theorem can be applied,find all numbers c in the open interval  such that  . ​

Accepted Answer

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

Question 19

Which of the following functions passes through the point  and satisfies  ? ​
​

Accepted Answer

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

Question 20

Find all critical numbers of of the function  ,  . ​
​

Accepted Answer

The answer of Find all critical numbers of of the...

Quiz 4: Applications of Differentiation

A plane begins its takeoff at 2:00 P.M.on a 2100-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.

Locate the absolute extrema of the function on the closed interval .

Find the value of the derivative (if it exists) of the function at the extremum point .

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

Find any critical numbers of the function .

Locate the absolute extrema of the function on the closed interval .

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 .If the Mean Value Theorem can be applied,find all numbers c in the open interval such that .

Find all critical numbers of the function .

Find all critical numbers of the function , .

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?

Find a function f that has derivative and with graph passing through the point .

Locate the absolute extrema of the given function on the closed interval .

Locate the absolute extrema of the function on the closed interval .

Locate the absolute extrema of the function on the closed interval .

Find the value of the derivative (if it exists) of at the indicated extremum.

Find the value o the derivative (if it exists) of the function at the extremum point .

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

Which of the following functions passes through the point and satisfies ?

Find all critical numbers of of the function , .

Quiz 4: Applications of Differentiation

A plane begins its takeoff at 2:00 P.M.on a 2100-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. ​

Locate the absolute extrema of the function on the closed interval . ​

Find the value of the derivative (if it exists) of the function at the extremum point . ​ ​

Determine whether the Mean Value Theorem can be applied to the function on the closed interval .If the Mean Value Theorem can be applied,find all numbers c in the open interval such that . ​

Find any critical numbers of the function . ​

Locate the absolute extrema of the function on the closed interval . ​

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 .If the Mean Value Theorem can be applied,find all numbers c in the open interval such that . ​

Find all critical numbers of the function . ​

Find all critical numbers of the function , .​ ​

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? ​

Find a function f that has derivative and with graph passing through the point . ​

Locate the absolute extrema of the given function on the closed interval . ​ ​

Locate the absolute extrema of the function on the closed interval . ​

Locate the absolute extrema of the function on the closed interval . ​

Find the value of the derivative (if it exists) of at the indicated extremum. ​ ​

Find the value o the derivative (if it exists) of the function at the extremum point . ​ ​ ​

Determine whether the Mean Value Theorem can be applied to the function on the closed interval .If the Mean Value Theorem can be applied,find all numbers c in the open interval such that . ​

Which of the following functions passes through the point and satisfies ? ​ ​

Find all critical numbers of of the function , . ​ ​

A plane begins its takeoff at 2:00 P.M.on a 2100-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.

Locate the absolute extrema of the function on the closed interval .

Find the value of the derivative (if it exists) of the function at the extremum point .

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

Find any critical numbers of the function .

Locate the absolute extrema of the function on the closed interval .

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 .If the Mean Value Theorem can be applied,find all numbers c in the open interval such that .

Find all critical numbers of the function .

Find all critical numbers of the function , .

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?

Find a function f that has derivative and with graph passing through the point .

Locate the absolute extrema of the given function on the closed interval .

Locate the absolute extrema of the function on the closed interval .

Locate the absolute extrema of the function on the closed interval .

Find the value of the derivative (if it exists) of at the indicated extremum.

Find the value o the derivative (if it exists) of the function at the extremum point .

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

Which of the following functions passes through the point and satisfies ?

Find all critical numbers of of the function , .