II PUC CHAPTER 6 APPLICATION OF DERIVATIES Total marks 10

Transcription

1 II PUC CHAPTER 6 APPLICATION OF DERIVATIES Total marks 10 1 mark 2 marks 3 marks 4 marks 5 marks 6 Marks TOTAL MARKS TWO MARK QUESTIONS 1. Find the approximate change in the volume V of a cube of a side x meters caused by increasing by 3%. (March 2017) 2. Find the point on the curve at which the tangents are parallel to x- axis. (July 2017) 3. Find the approximate of ( ) using differentials. (March 2016) 4. Find the intervals in which the function given by ( ) is strictly increasing. (July 2016) 5. The slope of the tangent to the curve and at the point whose x- coordinate is 10. (March 2015) 6. Find the approximate of using differentials. (July 2015) 7. Find the approximate change in the volume V of a cube of a side x meters caused by increasing by 2%. (March 2014) 8. Find approximate of by using differential. (July 2014) 9. Find approximate of ( ) by using differentials. 10. Find the approximate of using differentials. 11. Find the approximate value of f(3.02), where ( ) If the radius of the sphere is measured as 9 cm with an error, 0.03cm, then find the approximate error in calculating its volume. 13. If the radius of the sphere is measured as 7 cm with an error, 0.02cm, then find the approximate error in calculating its volume. 14. Find the slope of tangent to the curve 15. Find the slope of tangent to the curve at x= Find the slope of normal to the curve given by at a point. 1

2 17. The slope of the tangent to the curve at the point whose x- coordinate is Find the equation of normal to the curve at the point (1, 1). 19. Show that the equation of tangent to the parabola at ( ). 20. Find the points on the curve at which the tangent is parallel to the. 21. Find a point on the curve at which the tangent is. 22. Find the intervals in which the function given by ( ) is strictly increasing. 23. Find the intervals in which the function given by ( ) is strictly increasing. 24. Prove that the logarithmic function is strictly increasing on (0, ). 25. Find the local maximum value of the function ( ). 26. Find the maximum and minimum values of the function ( ). THREE MARK QUESTIONS 1. Find two positive numbers x and y so that and x is a maximum. (March 2017) (July 2014) 2. Find two numbers whose sum is 15 and the sum of whose squares is minimum. (July 2017) 3. Find two numbers whose product is 100 and whose sum is minimum. (March 2016) 4. Find the approximate of ( ) using differentials. (July 2016) 5. Find two positive numbers whose sum is 16 and the sum of whose squres is minimum. (March 2015) 6. Find two numbers whose sum is 24 and whose product is as large as possible. (July 2015) 7. Find the intervals in which the function given by ( ) is i) strictly increasing ii) strictly decreasing. (March 2014) 8. Find the intervals in which the function given by ( ) is i) strictly increasing ii) strictly decreasing. 9. Find the intervals in which the function given by ( ) is 2

3 i) strictly increasing ii) strictly decreasing. 10. Find the intervals in which the function given by ( ) is i) strictly increasing ii) strictly decreasing. 11. Find the least value of a such that the function ( ) is strictly increasing on (1, 2). 12. Find the point at which the tangent to the curve has its slope. 13. Find the equation of tangent to the curve given by at a point. 14. Find the equation of tangent and normal to the curve ( ). 15. Find the equation of tangent and normal to the curve ( ). 16. Find the equation of tangent to the curve which is parallel to the line. 17. Prove that the curves and cut at right angles if. 18. Find the absolute maximum value and absolute minimum value of the function ( ) [ ]. 19. Find the maximum value of the function ( ) in [1, 5]. 20. Find the local maximum and local minimum of the function ( ). 21. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum. 22. Find two positive numbers x and y so that and is a maximum. 23. Show that all rectangles inscribed in a given fixed circle, the square has the maximum area. 24. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base. 25. A square piece of tin of side 18 cm is to be made in to a box without top by cutting off square from each corner and folding up the flaps of the box. What should be the side of the square to be cut off so that the volume of the box is maximum possible? 3

4 FIVE MARK QUESTIONS 1. The length x of a rectangle is decreasing at the rate of 5cm/m and the width y increasing at the rate of 4cm/m. When x = 8cm and y = 6cm, find the rates of changes of a) the perimeter b) the area of the rectangle. (July 2014) (March 2017) 2. Sand is pouring from a pipe at the rate of 12 cm 3 /s the falling sand forms a cone on the ground in such a way that the height of cone is always 1/6 th of radius of the base. How fast height of the sand cone increasing when the height is 4 cm? (July 2015) (July 2017) 3. The length of rectangle is decreasing at the rate of 3cm/min and the width y is increasing at the rate of 2 cm/min. when x=10 cm and y=6 cm find the rate of change of a) the perimeter b) the area of rectangle. (March 2015) (March 2016) 4. A ladder 5m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 2cm/s. How fast is its height of the ladder decreasing when the foot of the ladder is 4m away from the wall? (July 2016) 5. A ladder 24 ft long is leans against a vertical wall. The lower end is moving away at the rate of 3 ft/sec. Find the rate at which the top of the ladder is moving downwards, if its foot is 8 ft from the wall. (March 2014) 6. The volume of a cube is increasing at the rate of 8 c.c/s. How fast is the surface area increasing when the length of an edge is 12cm? 7. A stone is dropped into a quiet lake and waves in circles at the speed of 5cm/s. At the instant when the radius of circular wave is 8 cm, how fast is the enclosed area is increasing? 8. A balloon, which always remains spherical, has a variable diameter ( ). Find the rate of change of its volume w.r.t. x 9. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm. 10. A water tank has the slope of an inverted right circular cone with its axis vertical and lower most. Its semi-vertical angle is tan -1 (0.5). Water is poured into it at a constant rate of 5 cubicmeter per hour. Find the rate at which the level of water is rising at the instant when the depth of water in the tank is 4m. 4

5 11. A stone is dropped into a pond, waves in the form of circles are generated and the radius of the outer most ripple increases at the rate of 2 inches/sec. How fast is the area increasing when the radius is 5 inches? 12. Water is being poured in to a right circular cone at the rate of 24cc/s. the radius of the circular base is 15 cm and the height of the cone is 40 cm. Find the rate at which the depth of water and the radius of water cone increases when the depth of the water is 16 cm. 13. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm. 14. The volume of a cube is increasing at the rate of 9 c.c/s. How fast is the surface area increasing when the length of an edge is 10 cm? 15. The radius of an air bubble is increasing at the rate of ½ cm/s. at what rate is the volume of the bubble increasing when the radius is 1 cm. 5

6 6