The anti derivative of sin 2x is a function of x whose derivative is sin 2x.
It is known that,
\begin{align} \frac {d}{dx} (cos 2x) = 2 sin2x \end{align}
⇒ \begin{align} sin 2x =-\frac {1}{2} \frac {d}{dx}(cos 2x) \end{align}
∴ \begin{align} sin 2x = \frac {d}{dx}\left(-\frac {1}{2}cos 2x\right) \end{align}
Therefore, the anti derivative of sin2x is
\begin{align} sin 2x \;is -\frac {1}{2}cos 2x \end{align}
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}
Represent graphically a displacement of 40 km, 30° east of north.
If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).
If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
In Figure, identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal
Represent graphically a displacement of 40 km, 30° east of north.
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
The total cost C (x) in Rupees associated with the production of x units of an item is given by
C(X) = 0.007 x3 - 0.003x2 + 15x + 4000
Find the marginal cost when 17 units are produced.
Classify the following as scalar and vector quantities.
(i) time period (ii) distance (iii) force
(iv) velocity (v) work done