\begin{align} \int\left(1-x\right).\sqrt {x}.dx\end{align}
\begin{align} =\int \left(\sqrt {x} - x^\frac32\right).dx\end{align}
\begin{align} =\frac{x^{\displaystyle\frac32}}{\displaystyle\frac32} - \frac{x^{\displaystyle\frac52}}{\displaystyle\frac52}+C \end{align}
\begin{align} =\frac23\;x^\frac32\; - \frac25\;x^\frac52\;+C \end{align}
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)
Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Determine order and degree(if defined) of differential equation (ym)2 + (yn)3 + (y')4 + y5 =0
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Represent graphically a displacement of 40 km, 30° east of north.