Advertisements
Advertisements
प्रश्न
A particle moves along the curve y = x2 + 2x. At what point(s) on the curve are the x and y coordinates of the particle changing at the same rate?
Advertisements
उत्तर
\[\text { Here, } \]
\[y = x^2 + 2x\]
\[ \Rightarrow \frac{dy}{dt} = \left( 2x + 2 \right)\frac{dx}{dt}\]
\[ \Rightarrow 2x + 2 = 1 \left[ \because \frac{dy}{dt} = \frac{dx}{dt} \right]\]
\[ \Rightarrow 2x = - 1\]
\[ \Rightarrow x = \frac{- 1}{2}\]
\[\text { Substituting x }=\frac{- 1}{2}\text { in y }= x^2 +2x, \text { we get }\]
\[y = \frac{- 3}{4}\]
\[\text { Hence, the coordinates of the point are } \left( \frac{- 1}{2}, \frac{- 3}{4} \right) .\]
APPEARS IN
संबंधित प्रश्न
The rate of growth of bacteria is proportional to the number present. If, initially, there were
1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½
hours.
[Take `sqrt2` = 1.414]
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.
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 total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.
Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm ?
The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm.
A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases ?
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 radius of a cylinder is increasing at the rate 2 cm/sec. and its altitude is decreasing at the rate of 3 cm/sec. Find the rate of change of volume when radius is 3 cm and altitude 5 cm.
The volume of a cube is increasing at the rate of 9 cm3/sec. How fast is the surface area increasing when the length of an edge is 10 cm?
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 side of an equilateral triangle is increasing at the rate of \[\frac{1}{3}\] cm/sec. Find the rate of increase of its perimeter ?
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 ?
A cone whose height is always equal to its diameter is increasing in volume at the rate of 40 cm3/sec. At what rate is the radius increasing when its circular base area is 1 m2?
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
Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is
A man 2 metres tall walks away from a lamp post 5 metres height at the rate of 4.8 km/hr. The rate of increase of the length of his shadow is
A man of height 6 ft walks at a uniform speed of 9 ft/sec from a lamp fixed at 15 ft height. The length of his shadow is increasing at the rate of
A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate
If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius
The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side
x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of second square with respect to the area of first square.
A particle is moving along the curve x = at2 + bt + c. If ac = b2, then particle would be moving with uniform ____________.
Total revenue in rupees received from the sale of x units of a product is given by R(x)= 3x2+ 36x + 5. The marginal revenue, when x = 15 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
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 3y = ax3 + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.
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?
The median of an equilateral triangle is increasing at the ratio of `2sqrt(3)` cm/s. Find the rate at which its side is increasing.
When \(x\) and \(y\) both vary with respect to \(t\), which formula gives the rate of change of \(y\) with respect to \(x\)?
For the Chain Rule for Rates \[\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\], which condition is required?
The total cost in Rupees is \[\text{C}(x)=0.005x^3-0.02x^2+30x+5000\]. Which expression is the marginal cost?
For \[\text{C}(x)=0.005x^3-0.02x^2+30x+5000\], what is the marginal cost when \[x=3\]?
What does a negative derivative mean for a quantity?
