Compute the value of:
For the integrand
This gives a new lower bound and upper bound Now, our integral becomes:
Since the antiderivative of cos(u) is sin (u) , applying the fundamental theorem of calculus, we get:
Which of the following statements is true?
is a continuously increasing function, and for a continuously increasing function f(x)
But in question, summation of L.H.S. above, a = 3 and in R.H.S, a = 2, so we don't know whether S > T. So we compute some initial values :
and since we already know that
Use Integration by Parts
ln(x) dx
set
u = ln(x), dv = dx
then we find
du = (1/x) dx, v = x
substitute
substitute u=ln(x), v=x, and du=(1/x)dx
(a) Find the points of local maxima and minima, if any, of the following function defined in 0 ≤ x ≤ 6.
x^{3}  6x^{2} + 9x + 15
(b) Integrate
f''(1) < 0, so x = 1 is point of local maxima, f''(3) > 0, so x = 3 is point of local minima.
Also the end points 0 and 6 are critical points. 0 is point of local minima, because it is to the left of x = 1 (which is point of maxima). Similarly x = 6 is point of local maxima.
(b) Since xcosx is an odd function, by the properties of definite integration, answer is 0.
then the value of k is equal to ______.
There is a mod term in the given integral. So, first we have to remove that. We know that x is always positive here and sin x is positive from 0 to π. From π to 2π, x is positive while sin x changes sign. So, we can write
Take the series as 1/x
and use Riemann integral to evaluate the series
integrate 1/ x dx from x=1001 to 3001 then we get 1.0979 which is in between 1 and 3/2
So C is correct option. 1<X<3/2
The value of the integral given below is
Integral of a multiplied by b equals
a multiplied by integral of b
minus
integral of derivative of a multiplied by integral of b
The value of
Since substituting x=1 we get 0/0 which is indeterminate
after applying L'H we get ((7x^{6})(10x^{4}))/((3x^{2})(6x))
now substituting x=1 we get 3/3
=1
hence answer is 1
What is the value of the following limit?
Since we have a 0/0 form, we can apply the L'Hôpital's rule.
Hence, option c is correct.
=____.
substitute h=x4..so it becomes lim_{h>0 (sinh)/h ...} which is a standard limit.. Ans would be 1.
What is the value of
i will solve by two methods
method 1
y=lim_{n>inf} (11/n)^{2n}
taking log
log y =lim_{n>inf} 2n log(11/n) =lim_{n>inf} log (11/n)/(1/2n)(converted this so as to have form 0/0)
apply l hospital rule
log y=lim_{n>inf} (1/11/n)1/n^{2} / (1/2n^{2}) log y=^{2}
y=e^{2} method two
it takes 1 to power infinity form
lim_{x>inf}f(x)g(x)
=elim_{x>inf}(f(x)1)g(x) (f(x)1)*g(x)=1/n*2n=2 ie 2
constant so we get final ans e^{2}
The limit equals.
In questions 2.1 to 2.10 below, each blank (___) is to be suitably filled in.
Use LH rule:
First Derivative: [x(e^{x}) + (e^{x}1)  2(sinx)]/[xsinx + (1  cosx)]
Second Derivative: [xe^{x} + e^{x} + e^{x}  2cosx]/[{xcosx + sinx + sinx]
Third Derivative: [xe^{x} + e^{x} + e^{x} + e^{x} + 2sinx]/[xsinx + cosx + cosx + cosx]
Put x = 0: [0+1+1+1+0]/[0+1+1+1] = 3/3 = 1.
Apply an exponential of a logarithm to the expression.
Since the exponential function is continuous, we may factor it out of the limit.
The numerator of grows asymptotically slower than its denominator as x approaches ∞.
Since grows asymptotically slower than e^{z} as x approaches
e^{0}
now to calculate values we use Squeezing Theorem.
Hence,
The limit of
Now we can see that after the 10/10 term, all subsequent terms are < 1, and keep decreasing. As we increase the value of n it the product will get close to 0.
Apply an exponential of a logarithm to the expression.
Since the exponential function is continuous, we may factor it out of the limit.
Logarithmic functions grow asymptotically slower than polynomials.
since log(x) grows asymptotically slower than the polynomial x as x approaches ∞,
A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve is 3x^{4} + 24x^{2} + 37 is
at it means that x = 0 is local minima.
but at so we can't apply second derivative test. So, we can apply first derivative test.
is not changing sign on either side of 2. So,x = 2 is neither maxima nor minima.
So, only one extremum i.e. x=0.
Find the minimum value of . 3  4x + 2x^{2}.
Determine the number of positive integers (≤ 720) which are not divisible by any of 2, 3 or 5.
(x) = 34x+2x^{2}
f'(x) = 4 + 4x = 0 => x=1
f''(x) = 4
f''(1) = 4>0, therefore at x=1 we will get mimimum value, which is : 3  4(1) + 2(1)^{2} = 1
The maximum possible value of xy^{2}z_{3} subjected to condition x,y,z>0and x+y+z=3 is
Given , x + y + z = 3
==> x + (y/2) + (y/2) + (z/3) + (z/3) + (z/3) = 3
Now using A.M. G.M inequality , we have :
[x + (y/2) + (y/2) + (z/3) + (z/3) + (z/3) ] / 6 >= (x . (y/2) .(y/2) . (z/3) . (z/3) .(z/3))^(1/6)
==> (x . (y/2) .(y/2) . (z/3) . (z/3) .(z/3))^(1/6) <= 1/2
==> (x . (y/2) .(y/2) . (z/3) . (z/3) .(z/3)) <= 1 / 64
==> x . y^{2} . z^{3} <= 108 / 64 = 27 / 16
If f(x) is defined as follows, what is the minimum value of f(x) for x ∊ (0, 2] ?
What is the maximum value of the function f(x) = 2x^{2}  2x + 6 in the interval [0,2] ?
is an extremum (either maximum or minimum).
So, the maximum value is at x = 2 which is 10 as there are no other extremum for the given function.
Consider the problem of maximizing x^{2}  2x + 5 such that 0 < x , 2. The value of x at which the maximum is achieved is:
Since a polynomial is defined and continuous everywhere, we only need to check the critical point and the boundaries.
Critical point: which is the minimum.
Boundaries:
Since p(x) increases as x goes farther away from the 1, but
p(x) is defined on an open interval,
p(x) never attains a maximum!
Hence, e. None of the above is the correct answer.
The function attains a minimum at x = ?
Digvijay is right, f(x) will be minimum at x = 2,
Here is another approach.
Since log and exponent are monotonically increasing functions, the problem of minimizing f(x) can be reduced to just minimizing the quadratic expression
x^{2}  4x + 5,
this quadratic expression can be written as (x^{2}  4x + 4) + 1 which is equal to
(x  2)^{2} + 1.
now since (x  2)^{^}2 can not be less than 0, so (x  2)^{^}2 + 1 can not be less than 1.
Also (x  2)^{2} + 1 will be at its minimum value (= 1), when x = 2.
so value of f(x) will be minimum at x = 2.
Suppose that f(x) is a continuous function such that 0.4 ≤ f(x) ≤ 0.6 for 0 ≤ x ≤ 1. for Which of the following is always true?
This is a repeating question on continuity. Let me solve it a nonstandard way  which should be useful in GATE.
From the question f is a function mapping the set of real (or rational) numbers between [0,1] to [0.4,0.6]. So, clearly the codomain here is smaller than the domain set. The function is not given as onto and so, there is no requirement that all elements in codomain set be mapped to by the domain set. We are half done now. Lets see the options:
A False, as we can have f(0.5) = 0.4, continuity does not imply anything other than all points being mapped being continuous.
C. Again false, we can have for all .
D. False, same reason as for A.
D. False, same reason as for A.
Only B option left which needs to be proved as correct now since we also have E option. We know that for a function all elements in domain set must have a mapping. All these can map to either 1 or more elements but at least one element must be there in the range set. i.e., f(x) = y is true for some y which is in [0.4,0.6]. In the minimal case this is a single element say c. Now for x = 1/0.8, option B is true. In the other case, say the minimal value of f(x) = a and the maximum value be f(x) = b. Now,
as per Intermediate Value theorem between and are also in the range set as f is continuous. Now, we need to consider x in the range [0.5, 0.75] as then only f(x) can be 0.8x and be in [0.4,0.6]. In our case we have
Lets assume Now, for all other points in must be between a and b and all points between a and b must be mapped by some x.
Moreover,for aand for So, if we plot this line should cross
must be above or equal to the line 0.8x (shown below) and for x = 0.75 it must be belowor equal which means an intersection must be there.
This shows there exist some a stronger case than option B. So, B option is true. Now please try for and see if it is true.
x>=4 and y>=4 , So we can take both x =5 & y = 5
x+y >= 10 => Satisfied , 5+5 = 10
2x + 3y >= 20. Satisfied.
This is infact minimum value.
Other options =>
4,4 => x+y constraint fail
4,5 => x+y fail
6,4 => Still giving 52 as sum which is more than 50 !, This can not be answer.
7,3 => 49+9 > 58 > 50.
The minimum of the function
Minimum value of function occurs at end points or critical points
f'(x)=1+logx
Equate it to 0
x=1/e
f''(x)=1/x
Put x=1/e f''(x)=e so minima at 1/e But 1/e=0.36
But x∈[1/2,infinity)
So min occurs at 1/2
So min value=1/2 log 1/2
So ans is c
Consider the differential equation . Which of its equilibria is unstable?
For unstable equilibrium point, dx/dt >0
At x = 0, dx/dt = (10)(20)(30) = 6 > 0
Hence x = 0 is point of unstable equilibrium.
We can understand equilibrium in terms of radioactive decay.
Let dN/dt = KN ;K>0 its significance is that an element is loosing energy so it is getting stability because we know more energy an element gets,more destability it gains and vice versa
Consider the function f(x) = sin x in the interval x = [π/4 , 7π /4]. The number and location(s) of the local minima of this function are
which lie between given domain in question
it means it is local maxima and at which is local minima and since it at is local maxima so before it graph is strictly increasing so is also local minima
so there is two local minima
Sine function increases till π/2 and so for the considered interval π/4 would be a local minimum. From π/2, value of sine keeps on decresing till 3π/2 and hence 3π/2 would be another local minima. So, (D) is the correct answer here.
The equation 7x^{7} + 14x^{6} + 12 x^{5} + 3x^{4} + 12x^{3}+ 10x^{2} + 5x+ 7 = 0 has
Since the polynomial has highest degree 7. So there are 7 roots possible for it
now suppose if an imaginary number a+bi is also root of this polynomial then abi will also be the root of this polynomial
That means there must be even number of complex root possible becoz they occur in pair.
Now we will solve this question option wise
A) All complex root
This is not possible. The polynomial has 7 roots and as I mention a polynomial should have even number of complex root and 7 is not even. So this option is wrong
B) At least one real root
This is possible. Since polynomial has 7 roots and only even number of complex root is possible, that means this polynomial has max 6 complex roots and Hence minimum one real root. So this option is correct
C) Four pairs of imaginary roots 4 pair means 8 complex root. But this polynomial can have atmost 7 roots. So this option is also wrong
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