Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
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
(ey + 1) cos x dx + ey sin x dy = 0
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
(y2 − 2xy) dx = (x2 − 2xy) dy
A population grows at the rate of 5% per year. How long does it take for the population to double?
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.
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Solve the following differential equation.
`dy/dx + y = e ^-x`
Choose the correct alternative.
Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in
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`
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation
`y (dy)/(dx) + x` = 0
