Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%
Solution and Explanation
The surface area of a cube (S) of side x is given by S = 6x2
\(\Rightarrow\)\(\frac{ds}{dx}\)=(\(\frac{ds}{dx}\))∇x
\(\Rightarrow\)(12x)∇x
\(\Rightarrow\)(12x)(0.01x) [ as1% of x is 0.01x]
\(\Rightarrow\)0.12x2
Hence, the approximate change in the surface area of the cube is 0.12x2 m2.
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Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
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Top CBSE CLASS XII Applications of Derivatives Questions
- Show that of all the rectangles inscribed in a given fixed circle,the square has the maximum area.
- Find the rate of change of the area of a circle with respect to its radius r when (a) r=3 cm (b) r=4 cm
- The volume of a cube is increasing at the rate of \(8 cm^3 /s\). How fast is the surface area increasing when the length of an edge is \(12 cm\)?
- The radius of a circle is increasing uniformly at the rate of \(3 cm/s\). Find the rate at which the area of the circle is increasing when the radius is \(10 cm\).
- 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?
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
Read More: Application of Derivatives