### Calculus

Derivatives in calculus.

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A ball is thrown vertically upwards with an initial velocity

A ball is thrown vertically upwards with an initial velocity of 60m/s. After time t seconds, its height above the ground is given as s=60t-16t^{2}. Find its instantaneous velocity after t seconds. Discuss the specialty about time t=15/8

A ball is thrown vertically upwards with an initial velocity of 60m/s. After time t seconds, its height above the ground is given as s=60t-16t

Let f(t) be the weight (in grams) of a solid

Let f(t) be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the wieght of the solid at any time t can be be determined from the weight using the formula:

f'(t)=-5(t)(2+f(t))

If there is 3 grams of solid at time t=2...

Let f(t) be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the wieght of the solid at any time t can be be determined from the weight using the formula:

f'(t)=-5(t)(2+f(t))

If there is 3 grams of solid at time t=2...

A one-product firm estimates

(cost and profit) a one-product firm estimates that its daily total cost function (in suitable units) is

C(x)=x^{3}-6x^{2}+13x+15

and its total revenue function is

R(x) = 28...

(cost and profit) a one-product firm estimates that its daily total cost function (in suitable units) is

C(x)=x

and its total revenue function is

R(x) = 28...

A water trough is 10 meter long

A water trough is 10 meter long and a cross-section has the shape of an isosceles trapezoid that is 30 centimeter wide at the bottom, 80 centimeter at the top, and has height 50 centimeter. If the trough is being filled with water the rate of 0.2 meter cube per ...

A water trough is 10 meter long and a cross-section has the shape of an isosceles trapezoid that is 30 centimeter wide at the bottom, 80 centimeter at the top, and has height 50 centimeter. If the trough is being filled with water the rate of 0.2 meter cube per ...

On her way down from Mount Everest

On her way down from Mount Everest, Kim notices for every three meters she travels northwest she climbs a 1/2 meter, and that for every two meters she travels northeast she descends 1/4 meter. In what direction should she start for fastest ...

On her way down from Mount Everest, Kim notices for every three meters she travels northwest she climbs a 1/2 meter, and that for every two meters she travels northeast she descends 1/4 meter. In what direction should she start for fastest ...

To estimate a height of a building, a stone is dropped

Use the position function s(t)= -4.9t^{2} + v_{0}t + s_{0} for free-falling object.

To estimate a height of a building, a stone is dropped from the top a building into a pool of water at ground level. How high is the building if the splash is seen 6...

Use the position function s(t)= -4.9t

To estimate a height of a building, a stone is dropped from the top a building into a pool of water at ground level. How high is the building if the splash is seen 6...

The north-east section of a municipality

The north-east section of a municipality is a rectangle whose 17 km west boundary is Highway 7 and whose 20 km south boundary is Highway 15. A by-pass road is planned to alleviate traffic ...

The north-east section of a municipality is a rectangle whose 17 km west boundary is Highway 7 and whose 20 km south boundary is Highway 15. A by-pass road is planned to alleviate traffic ...

Derivatives

the height of an object dropped from an initial height of 450 feet is given by h=450-16t^2, where t is in seconds and h is in ...

the height of an object dropped from an initial height of 450 feet is given by h=450-16t^2, where t is in seconds and h is in ...

Derivatives

Mila Rockwell invests in classic cars. He recently bought a 1978 convertible valued at $20,000. The value of the car is predicted to appreciate at a rate of 3.5% per year. Find the value of the car after 10, 20, and 40 ...

Mila Rockwell invests in classic cars. He recently bought a 1978 convertible valued at $20,000. The value of the car is predicted to appreciate at a rate of 3.5% per year. Find the value of the car after 10, 20, and 40 ...

Derivatives

A recreational swimming lake is treated periodically to control harmful bacteria growth. Sppose t days after treatment,the concntration of bateria per cubi centimeter is given by:

C(t)=30t^2-240t+500 0how many days aftertreatment willth concentration be ...

A recreational swimming lake is treated periodically to control harmful bacteria growth. Sppose t days after treatment,the concntration of bateria per cubi centimeter is given by:

C(t)=30t^2-240t+500 0

Derivatives

If exactly 208 people sign up for a charter flight, Leisure World Travel Agency charges $296/person. However, if more than 208 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Hint: Let x denote the number of passengers above 208...

If exactly 208 people sign up for a charter flight, Leisure World Travel Agency charges $296/person. However, if more than 208 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Hint: Let x denote the number of passengers above 208...

Derivatives

A plane flying horizontal at an altitude of 1 mi/h and a speed of 500mi/h passes directly over a radar ...

A plane flying horizontal at an altitude of 1 mi/h and a speed of 500mi/h passes directly over a radar ...

Derivatives

A 5 foot ladder is moving away from the wall at the rate 2 m/s. If the ladder is 3 meters away from the wall, find the rate at which the ladder is moving down the wall.

How to solve: well we know the ladder is 5 meters and it's leaning against a wall!! what is this shape?! a right triangle!!! we do the pythagorean theorem!

x^2+y^2 = 5

differentiate this with respect to time

2x dx/dt + 2y dy/dt = 0

now we need to find the value of x, y, and dx/dt. it doesn't matter if x or y get switched. so we could say taht x could be the distance of the base of ladder to the wall. We know this is 3! so x = 3! how do we find out how high the ladder is from the ground!? we use the Pythagorean theorem 3squared + y squared = 5! y is equal to 4! what about dx/dt?! we know that dx/dt rate of change of how fast the ladder is moving away from the wall! we know this is 2 m/s! now that we have these values we substitute

2x dx/dt + 2y dy/dt = 0

2(6)(2) + 2(8)(dy/dt) = 0

dy/dt = -1.5 m/s this means that the rate at which the distance from the top of the ladder to the base is changing at the rate -1.5 m/s! this means it's decreasing! dy/dt is the rate at which the height is changing!

A 5 foot ladder is moving away from the wall at the rate 2 m/s. If the ladder is 3 meters away from the wall, find the rate at which the ladder is moving down the wall.

How to solve: well we know the ladder is 5 meters and it's leaning against a wall!! what is this shape?! a right triangle!!! we do the pythagorean theorem!

x^2+y^2 = 5

differentiate this with respect to time

2x dx/dt + 2y dy/dt = 0

now we need to find the value of x, y, and dx/dt. it doesn't matter if x or y get switched. so we could say taht x could be the distance of the base of ladder to the wall. We know this is 3! so x = 3! how do we find out how high the ladder is from the ground!? we use the Pythagorean theorem 3squared + y squared = 5! y is equal to 4! what about dx/dt?! we know that dx/dt rate of change of how fast the ladder is moving away from the wall! we know this is 2 m/s! now that we have these values we substitute

2x dx/dt + 2y dy/dt = 0

2(6)(2) + 2(8)(dy/dt) = 0

dy/dt = -1.5 m/s this means that the rate at which the distance from the top of the ladder to the base is changing at the rate -1.5 m/s! this means it's decreasing! dy/dt is the rate at which the height is changing!

Derivatives

at 1:00pm ship A is 25miles due south of sip b. if ship A is sailing west at a rate of 1miles/hour and ship B is sailing south at a rate of 20 miles/hur, find the rat at which the distance between the ships is changing at 1:30pm

at 1:00pm ship A is 25miles due south of sip b. if ship A is sailing west at a rate of 1miles/hour and ship B is sailing south at a rate of 20 miles/hur, find the rat at which the distance between the ships is changing at 1:30pm

Derivatives

A window is in the form of a ractangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is tinted only trnasmitting 1/4 the light. The total permimeter is ...

A window is in the form of a ractangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is tinted only trnasmitting 1/4 the light. The total permimeter is ...

Derivatives

The revenue (in dollars) from the sale of x plastic planter boxes is given by

R(x) = 30x − 0.025x2, where 0 x 1, 200.

(a) (4 pts) Find the respective revenue from the sale of 500 plastic planter boxes and

500 + h plastic planter boxes for h = 0.

(b) (3 pts) Find the average rate of change of revenue if the sale is changed from

500 plastic planter boxes and 500 + h plastic planter boxes for h = 0.

(c) (2 pts) Use (b) and limit properties to find the instantaneous rate of change of

revenue at a production level of 500 plastic planter boxes ...

The revenue (in dollars) from the sale of x plastic planter boxes is given by

R(x) = 30x − 0.025x2, where 0 x 1, 200.

(a) (4 pts) Find the respective revenue from the sale of 500 plastic planter boxes and

500 + h plastic planter boxes for h = 0.

(b) (3 pts) Find the average rate of change of revenue if the sale is changed from

500 plastic planter boxes and 500 + h plastic planter boxes for h = 0.

(c) (2 pts) Use (b) and limit properties to find the instantaneous rate of change of

revenue at a production level of 500 plastic planter boxes ...

Derivatives

The line through the origin with slope -2 is tangent to the curve at point P. Find the x and y coordinates of Point P when the tangent is 2x - 3xy/x^2 + y^2 +1

The line through the origin with slope -2 is tangent to the curve at point P. Find the x and y coordinates of Point P when the tangent is 2x - 3xy/x^2 + y^2 +1

Derivatives

two ships deport the same spot at the same time. Ship A travels north at a speed of 16 mi/hr and ship B travels east at a speed of 20 mi/...

two ships deport the same spot at the same time. Ship A travels north at a speed of 16 mi/hr and ship B travels east at a speed of 20 mi/...

Derivatives

A rectangular study area is to be enclosed by a fence and divided into two equal parts with a fence running along the division parallel to one of the sides. if the total area is 384 ft^2, find the dimensions of the study area that will minimize the total length of the ...

A rectangular study area is to be enclosed by a fence and divided into two equal parts with a fence running along the division parallel to one of the sides. if the total area is 384 ft^2, find the dimensions of the study area that will minimize the total length of the ...

Derivatives

the volume of a shere is increasing at a rate of 7cm3/s. find the rate of change of its surface are when its volume is 4/3 cm3

the volume of a shere is increasing at a rate of 7cm3/s. find the rate of change of its surface are when its volume is 4/3 cm3

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