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put a-x =t and we get now by writing cot in terms of sin and cos we can just take out the cos as cos(0) = 1, and hence what remains is and we know that the form oflimit x/sin(x) as x->0 is 1 hence...

Ok let me give you an intuitive feeling about this Imagine a differential curve with a local maxima (basically a peak). Now can you imagine that (shown in figure) that the slope goes from positive to...

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Dear Student, d/dx(dy/dx)=d^2y/dx^2 becuase of y in the numerator. Thanks.

here the answer is zero because here 2x/3x here x and x is cancel so after here 2/3 so the answer is 0

from equation we take (dx) 2 why we can’t take that we take only dx 2 why?….….….….….….….….….….….….….….….….…....

My dear friend the answer for your question is 5 it will be come with formulae of differentiatio all the best for your bright future

because w.r.t of x we are differentiating, that why we are not writting d^2x^2,, we are writting dx^2

Around the 1670’s, two great men discovered and developed calculus independently from each other. Sir Isaac Newton of England, and Gottfried Wilhelm Leibniz of Germany, both did quite a lot of...

Let tangent and normal to a parabola at a point P(x1,y1) be extended to meet the axis of the parabola at N and T respectively. PT is termed the length of the tangent. PN is termed the length of the...

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Dear Student, Let P(x,y) be any point on any general curve. PG be a normal to the curve at point P and G be the point of axis axis. Let PN be the lilne perpendicular to x-axis. So the Projection of...

Solution in the figure

you have to solve this by induction make the hypothesis by observation that the kth derivative of f(x) where k<n is of the form and hence we get the nthe derivative as

The points of interest are just the 0 and 1 f(0) = f(1) = 0 we can see that for x = 0 both differentials are equal and hence its a differential point but for x=1 they are not equal and hence the only...

the two curve touches each other and hence the gradient at the point (m,n) is same for both hence differentiating the second curve we get at x =m and y = n, dy/dx = -b/a (gradient of 1 st curve) so...

The limit is of the form so we can write the limit as so we find thus using the expansion of the under root we can easily find the limit to be and hence

We start with substituting x = a + t, and hence as x tends to a, t tends to 0. so the limit now becomes this can be written in form of standard limit multiples as now as per the standard limit rules...

By expanding the tan2x and cos2x terms we get simplifying we get now each of the multiplying term has limit 1 as x tends to 0 thus L = 1/2

the mean value theorem generally gives the boundary values of a functions hence if question involves anything about boundary values of a given function use mean value theorem concept.

For a cubic equation to have three real roots it should have one maxima and one minima and hence to find the conditions find critical points c1 c2 then used f(c1)f(c2)

Write the equation for the tangent to curve at general point(x1,y1) and as shortest distance is along parallel line put slope of this tangent equal to slope of line and by using equation of curve...

Dear Student, If the degree of a polynomialf(x) is even and the leading coefficient is positive, thenf(x) → ∞ asx→ ±∞. Iff(x) is an even degree polynomial with negative...

Probality functions is a function from power set of A to the real numbers or we can say that if no of elements in A is s then probability function is f:2^s--R

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