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Chapter: 12th Maths : UNIT 7 : Applications of Differential Calculus

Exercise 7.1 : Meaning of Derivatives

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EXERCISE 7.1

1. A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres.

(i) Find the average velocity between t = 3 and t = 6 seconds.

(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.


2. A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t2 in t seconds.

(i) How long does the camera fall before it hits the ground?

(ii) What is the average velocity with which the camera falls during the last 2 seconds?

(iii) What is the instantaneous velocity of the camera when it hits the ground?


3. A particle moves along a line according to the law s (t= 2t3  9t2 + 12t  4 , where t  0 .

(i) At what times the particle changes direction?

(ii) Find the total distance travelled by the particle in the first 4 seconds.

(iii) Find the particle’s acceleration each time the velocity is zero.


4. If the volume of a cube of side length x is v = x3 . Find the rate of change of the volume with respect to x when x = 5 units.


5. If the mass m ( x) (in kilograms) of a thin rod of length x (in metres) is given by, m ( x) = √3x then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres.


6. A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?


7. A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shore line. How fast is the beam moving along the shore line when it makes an angle of 45°with the shore?


8. A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?


9. A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall,

(i) how fast is the top of the ladder moving down the wall?

(ii) at what rate, the area of the triangle formed by the ladder, wall, and the floor, is changing?


10. A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car?


 

Answers:

(1) (i) 21m/s (ii) 15 m/s and 27 m/s

(2) (i) 5sec (ii) 128 ft/s (iii)160 ft/s

(3) (i) 1,2 sec (ii) 34 m (iii) –6 m / s2 , 6 m / s2

(4) 75 units

(5) 1/2  kg/m, 1/6 kg/m

(6) 20π sq.cm/s

(7) 2π km/s

(8) 9/10π  m/min

(9) (i) −8/3 m/s (ii) 26.83 sq.m/sec

(10) 70 km/hr.

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12th Maths : UNIT 7 : Applications of Differential Calculus : Exercise 7.1 : Meaning of Derivatives | Problem Questions with Answer, Solution

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12th Maths : UNIT 7 : Applications of Differential Calculus


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