Question 1

A rectangular page is to contain 36 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. ​

Accepted Answer

The least amount of paper will be used when the page is a square, as this shape has the smallest perimeter for a given area. Since the print area is 36 square inches and the margins are 1 inch on each side, the total page size must be larger to accommodate the margins. For a square print area, each side of the print area is √36 = 6 inches. Adding 2 inches to each dimension for the margins (1 inch on each side) gives a total page size of 8 inches by 8 inches.

Question 2

The graph shows the temperature T, in degrees Fahrenheit, of molten glass t seconds after it is removed from a kiln. ​  ​
Find  .
​

Accepted Answer

B

Question 3

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

Accepted Answer

C

Question 4

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 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 400 miles per hour. ​

Accepted Answer

The average speed of the plane is calculated by dividing the total distance by the total time of the flight. For a 3400-mile flight that takes 7.5 hours, the average speed is $ \frac{3400 \text{ miles}}{7.5 \text{ hours}} = 453.33 \text{ miles per hour} $. The Mean Value Theorem suggests that at some point(s) during the flight, the plane must travel at its average speed. However, the explanation that there must be at least two times the plane's speed is exactly 400 miles per hour is not directly related to the Mean Value Theorem but rather to the practical aspects of a plane's speed changes (acceleration and deceleration) around this average speed.

Question 5

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

Accepted Answer

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

Question 6

A sector with central angle  is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of  such that the volume of the cone is a maximum. ​

Accepted Answer

The answer of A sector with central angle  is...

Question 7

Find all points of inflection on the graph of the function  . ​

Accepted Answer

The answer of Find all points of inflection on the...

Question 8

Find the differential dy of the function  . ​

Accepted Answer

The answer of Find the differential dy of the function...

Question 9

Find two positive numbers such that the sum of the first and twice the second is 88 and whose product is a maximum. ​

Accepted Answer

The answer of Find two positive numbers such that the...

Question 10

On a given day, the flow rate F (cars per hour) on a congested roadway is given by  , where v is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour. ​

Accepted Answer

The answer of On a given day, the flow rate...

Question 11

Use a graphing utility to graph the function  and locate the absolute extrema of the function on the interval  . ​

Accepted Answer

The answer of Use a graphing utility to graph the...

Question 12

The radius of a spherical balloon is measured to be 14 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the surface area of the sphere. Round your answer to two decimal places. ​

Accepted Answer

The answer of The radius of a spherical balloon is...

Question 13

Find all points of inflection on the graph of the function  . ​

Accepted Answer

The answer of Find all points of inflection on the...

Question 14

Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 27%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.) ​

Accepted Answer

The answer of Assume that the amount of money deposited...

Question 15

Find the limit. ​  ​

Accepted Answer

The answer of Find the limit. ​  ​
...

Question 16

A business has a cost of  for producing x units. The average cost per unit is  Find the limit of  as x approaches infinity. ​

Accepted Answer

The answer of A business has a cost of ...

Question 17

Identify the open intervals where the function  is increasing or decreasing. ​

Accepted Answer

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

Question 18

The measurement of the side of a square floor tile is 14 inches, with a possible error of  inch. Use differentials to approximate the possible propagated error in computing the area of the square. ​

Accepted Answer

The answer of The measurement of the side of a...

Question 19

The graph of a function f is shown below. Sketch the graph of the derivative  . ​  ​

Accepted Answer

The answer of The graph of a function f is...

Question 20

Sketch the graph of the function  using any extrema, intercepts, symmetry, and asymptotes. ​

Accepted Answer

The answer of Sketch the graph of the function ...

Quiz 4: Applications of Differentiation

A rectangular page is to contain 36 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

The graph shows the temperature T, in degrees Fahrenheit, of molten glass t seconds after it is removed from a kiln. Find .

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

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 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 400 miles per hour.

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

A sector with central angle is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of the cone is a maximum.

Find all points of inflection on the graph of the function .

Find the differential dy of the function .

Find two positive numbers such that the sum of the first and twice the second is 88 and whose product is a maximum.

On a given day, the flow rate F (cars per hour) on a congested roadway is given by , where v is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour.

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval .

The radius of a spherical balloon is measured to be 14 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the surface area of the sphere. Round your answer to two decimal places.

Find all points of inflection on the graph of the function .

Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 27%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)

Find the limit.

A business has a cost of for producing x units. The average cost per unit is Find the limit of as x approaches infinity.

Identify the open intervals where the function is increasing or decreasing.

The measurement of the side of a square floor tile is 14 inches, with a possible error of inch. Use differentials to approximate the possible propagated error in computing the area of the square.

The graph of a function f is shown below. Sketch the graph of the derivative .

Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes.

Quiz 4: Applications of Differentiation

A rectangular page is to contain 36 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. ​

The graph shows the temperature T, in degrees Fahrenheit, of molten glass t seconds after it is removed from a kiln. ​ ​ Find . ​

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

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 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 400 miles per hour. ​

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

A sector with central angle is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of the cone is a maximum. ​

Find all points of inflection on the graph of the function . ​

Find the differential dy of the function . ​

Find two positive numbers such that the sum of the first and twice the second is 88 and whose product is a maximum. ​

On a given day, the flow rate F (cars per hour) on a congested roadway is given by , where v is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour. ​

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval . ​

The radius of a spherical balloon is measured to be 14 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the surface area of the sphere. Round your answer to two decimal places. ​

Find all points of inflection on the graph of the function . ​

Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 27%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.) ​

Find the limit. ​ ​

A business has a cost of for producing x units. The average cost per unit is Find the limit of as x approaches infinity. ​

Identify the open intervals where the function is increasing or decreasing. ​

The measurement of the side of a square floor tile is 14 inches, with a possible error of inch. Use differentials to approximate the possible propagated error in computing the area of the square. ​

The graph of a function f is shown below. Sketch the graph of the derivative . ​ ​

Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes. ​

A rectangular page is to contain 36 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

The graph shows the temperature T, in degrees Fahrenheit, of molten glass t seconds after it is removed from a kiln. Find .

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

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 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 400 miles per hour.

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

A sector with central angle is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of the cone is a maximum.

Find all points of inflection on the graph of the function .

Find the differential dy of the function .

Find two positive numbers such that the sum of the first and twice the second is 88 and whose product is a maximum.

On a given day, the flow rate F (cars per hour) on a congested roadway is given by , where v is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour.

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval .

The radius of a spherical balloon is measured to be 14 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the surface area of the sphere. Round your answer to two decimal places.

Find all points of inflection on the graph of the function .

Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 27%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)

Find the limit.

A business has a cost of for producing x units. The average cost per unit is Find the limit of as x approaches infinity.

Identify the open intervals where the function is increasing or decreasing.

The measurement of the side of a square floor tile is 14 inches, with a possible error of inch. Use differentials to approximate the possible propagated error in computing the area of the square.

The graph of a function f is shown below. Sketch the graph of the derivative .

Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes.