Advertisements
Advertisements
प्रश्न
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
Advertisements
उत्तर
We have,
\[\frac{dC}{dx} = 2 + 0 . 15x\]
\[ \Rightarrow dC = \left( 2 + 0 . 15x \right)dx\]
Integrating both sides with respect to x, we get
\[C = 2x + \frac{0 . 15}{2} x^2 + K . . . . . \left( 1 \right)\]
\[\text{ At }C\left( 0 \right) = 100,\text{ we have }\]
\[100 = 2\left( 0 \right) + \frac{0 . 15}{2} \left( 0 \right)^2 + K\]
\[ \Rightarrow K = 100\]
Putting the value of T in (1), we get
\[C = 2x + \frac{0 . 15}{2} x^2 + 100\]
\[ \Rightarrow C = 0 . 075 x^2 + 2x + 100\]
APPEARS IN
संबंधित प्रश्न
Show that y = AeBx is a solution of the differential equation
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
x2 dy + y (x + y) dx = 0
2xy dx + (x2 + 2y2) dy = 0
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The solution of the differential equation y1 y3 = y22 is
The differential equation satisfied by ax2 + by2 = 1 is
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Solve: `("d"y)/("d"x) + 2/xy` = x2
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve: ydx – xdy = x2ydx.
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation
`x + y dy/dx` = x2 + y2
