Advertisements
Advertisements
प्रश्न
For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3
Advertisements
उत्तर
The given curve is y = 5x – 2x3 and `dx/dt = 2` units/sec
y = 5x – 2x3
Differentiating both sides w.r.t x, we get
Slope of the curve = `dy/dx = 5 - 6x^2`
Differentiating both sides w.r.t t, we get
`=> d/dt (dy/dx) = 0 - 12x dx/dt`
`=> d/dt (dy/dx)_(x = 3) = 0 - 12 xx 3 xx2 = -72 "units/sec"`
Thus, the slope of the curve is decreasing at the rate of 72 units/sec when x = 3
APPEARS IN
संबंधित प्रश्न
Which of the following differential equations has y = c1 ex + c2 e–x as the general solution?
(A) `(d^2y)/(dx^2) + y = 0`
(B) `(d^2y)/(dx^2) - y = 0`
(C) `(d^2y)/(dx^2) + 1 = 0`
(D) `(d^2y)/(dx^2) - 1 = 0`
Form the differential equation of the family of curves represented by y2 = (x − c)3.
Find the differential equation of the family of curves, x = A cos nt + B sin nt, where A and B are arbitrary constants.
Form the differential equation of the family of curves represented by the equation (a being the parameter):
(2x − a)2 − y2 = a2
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = a2
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + (y − b)2 = 1
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax3
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x\]
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} - y = \cos 2x\]
Find one-parameter families of solution curves of the following differential equation:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]
Find one-parameter families of solution curves of the following differential equation:-
\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]
Find one-parameter families of solution curves of the following differential equation:-
\[e^{- y} \sec^2 y dy = dx + x dy\]
Form the differential equation representing the family of curves y = mx, where m is an arbitrary constant.
Form the differential equation representing the family of curves y = e2x (a + bx), where 'a' and 'b' are arbitrary constants.
Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.
Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.
Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
Find the differential equation of system of concentric circles with centre (1, 2).
Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P(x, y) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB.
The differential equation of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is ______.
Family y = Ax + A3 of curves will correspond to a differential equation of order ______.
The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is ______.
The differential equation of the family of curves y2 = 4a(x + a) is ______.
The differential equation representing the family of circles x2 + (y – a)2 = a2 will be of order two.
Find the equation of the curve at every point of which the tangent line has a slope of 2x:
