Advertisements
Advertisements
प्रश्न
For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then how fast is the slope of curve changing when x = 3?
Advertisements
उत्तर
Slope of curve = `"dy"/'dx"` = 5 – 6x2
⇒ `"d"/"dt" ("dy"/"dx") = -12x * "dx"/"dt"`
= –12 . (3) . (2)
= –72 units/sec.
Thus, slope of curve is decreasing at the rate of 72 units/sec when x is increasing at the rate of 2 units/sec.
APPEARS IN
संबंधित प्रश्न
A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.
Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm.
A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
The radius of an air bubble is increasing at the rate `1/2` cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?
Find the rate of change of the volume of a cone with respect to the radius of its base ?
Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 cm?
The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate ?
An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long?
The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square.
The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?
A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.
Find an angle θ which increases twice as fast as its cosine ?
The surface area of a spherical bubble is increasing at the rate of 2 cm2/s. When the radius of the bubble is 6 cm, at what rate is the volume of the bubble increasing?
The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.
Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate ?
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of the area of the rectangle.
The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. How far is the area increasing when the side is 10 cms?
If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere ?
A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10 cm/sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall ?
The altitude of a cone is 20 cm and its semi-vertical angle is 30°. If the semi-vertical angle is increasing at the rate of 2° per second, then the radius of the base is increasing at the rate of
The coordinates of the point on the ellipse 16x2 + 9y2 = 400 where the ordinate decreases at the same rate at which the abscissa increases, are
The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. The rate of change of lateral surface when the radius = 7 cm and altitude 24 cm is
The distance moved by a particle travelling in straight line in t seconds is given by s = 45t + 11t2 − t3. The time taken by the particle to come to rest is
The radius of a circular plate is increasing at the rate of 0.01 cm/sec. The rate of increase of its area when the radius is 12 cm, is
In a sphere the rate of change of surface area is
A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.
If the rate of change of volume of a sphere is equal to the rate of change of its radius then the surface area of a sphere is ____________.
Let y = f(x) be a function. If the change in one quantity 'y’ varies with another quantity x, then which of the following denote the rate of change of y with respect to x.
What is the rate of change of the area of a circle with respect to its radius when, r = 3 cm
If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?
An edge of a variable cube is increasing at the rate of 10 cm/sec. How fast will the volume of the cube increase if the edge is 5 cm long?
