An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
Let x be the length of a side and V be the volume of the cube. Then,
V = x3.
\begin{align}\therefore \frac{dV}{dt}=3x^2.\frac{dx}{dt}\;\;\;[By\; Chain \;Rule]\end{align}
It is given that,
\begin{align} \frac{dx}{dt}=3 \;cm^2/s\end{align}
\begin{align}\therefore \frac{dV}{dt}=3x^2.(3) = 9x^2\end{align}
Thus, when x = 10 cm,
\begin{align} \frac{dV}{dt}=9 (10)^2=900 \;cm^3/s\end{align}
Hence, the volume of the cube is increasing at the rate of 900 cm3/s when the edge is 10 cm long.
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.
Let f : X → Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f).
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
(A) 10π (B) 12π (C) 8π (D) 11π
y = ex +1 : yn -y' = 0
The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
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.
The order of the differential equation
\begin{align}2x^2\frac{d^2y}{dx^2}\;- \;3\frac{dy}{dx}\;+ y=\;0\end{align}
is (A) 2 (B) 1 (C) 0 (D) not defined
Classify the following as scalar and vector quantities.
(i) time period (ii) distance (iii) force
(iv) velocity (v) work done
why dv/dt=3x^2.dx/dt