\(\int \sqrt{1+x^2}dx\) is equal to
The maximum value of\( [x(x-1)+1]^{\frac{1}{3}},0≤x≤1\) is
Find the maximum value of 2x3−24x+107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find the intervals in which the function f given by f(x)=x3+\(\frac{1}{x^3}\),x≠0 is (i) increasing (ii) decreasing
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \(tan^{-1}\sqrt{2}.\)
Show that the right circular cone of least curved surface and given volume has an altitude equal to\(\sqrt{2}\) time the radius of the base.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i)f(x)=x3,x∈[-2,2] (ii) f(x)=sin x+cos x,x∈[0,π] (iii) f(x)=4x-1/2x2,x∈[-2,\(\frac{9}{2}\)] (iv) f(x)=(x-1)2+3,x∈[-3,1]
Prove that the following functions do not have maxima or minima: (i) f(x) = ex (ii) g(x) = logx (iii) h(x) = x3 + x2+x+1
Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = (2x − 1)2 + 3 (ii) f(x) = 9x2+12x+2 (iii) f(x) = −(x − 1)2+ 10 (iv) g(x) = x3 +1
Find the approximate change in volume V of a cube of side x meters caused by increasing side by 1%.
Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15
Find the approximate value of f (2.01), where f (x) = 4x2+5x + 2
The anti derivative of \(\bigg(\sqrt x+\frac{1}{\sqrt x}\bigg)\) equals
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.