Class 12

Math

Calculus

Continuity and Differentiability

If u, v and w are functions of x, then show that $dxd (u.v.w)=dxdu v.w+u.dxdv .w+u.vdxdw $ in two ways - first by repeated application of product rule, second by logarithmic differentiation.

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Let $f(x)=(lim)_{x→∞}r=0∑n−1 (rx+1){(r+1)x+1}x $ . Then (A) $f(x)$ is continuous but not differentiable at $x=0$ (B) $f(x)$ is both continuous but not differentiable at $x=0$ (C) $f(x)$ is neither continuous not differentiable at $x=0$ (D) $f(x)$ is a periodic function.

Suppose that $f(x)$ is differentiable invertible function $f_{prime}(x)=0andh_{prime}(x)=f(x)˙$ Given that $f(1)=f_{prime}(1)=1,h(1)=0$ and $g(x)$ is inverse of $f(x)$ . Let $G(x)=x_{2}g(x)−xh(g(x))∀x∈R˙$ Which of the following is/are correct? $G_{prime}(1)=2$ b. $G_{prime}(1)=3$ c.$G_{1}=2$ d. $G_{1}=3$

f is a continous function in $[a,b]$; g is a continuous function in [b,c]. A function h(x) is defined as $h(x)=f(x)forx∈[a,b),g(x)forx∈(b,c]$ if f(b) =g(b) then

Statement 1: Minimum number of points of discontinuity of the function $f(x)=(g(x)[2x−1]∀x∈(−3,−1)$ , where [.] denotes the greatest integer function and $g(x)=ax_{3}+x_{2}+1$ is zero. Statement 2: $f(x)$ can be continuous at a point of discontinuity, say $x=c_{1}$ of $[2x−1]$if $g(c_{1})=0.$

Draw the graph and find the points of discontinuity $f(x)=[2cosx]$ , $x∈[0,2π]$ . ([.] represents the greatest integer function.)

If $f(x)=n=1∏100 (x−n)_{n(101−n)}$; then $f_{′}(101)f(101) =$

Find the number of integers lying in the interval (0,4) where the function $f(x)=(lim)_{n∞}(2cos(πx) )_{2n}$ is discontinuous

Which of the function is non-differential at $x=0?$ $f(x)=∣∣ x_{3}∣∣ $