Class 12

Math

Calculus

Continuity and Differentiability

Examine the continuity of the function $f(x)=2x_{2}−1$at $x=3$.

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Find the value of $a$ for which $f(x)={x_{2},x∈Qx+a,x∈/Q$ is not continuous at any $x˙$

Discuss the differentiability of $f(x=cos_{−1}(1+x_{2}1−x_{2} )$

If $x=secθ−cosθ$ and $y=sec_{n}θ−cos_{n}θ$ then show that $(x_{2}+4)(dxdy )_{2}=n_{2}(y_{2}+4)$

Let $f:(−∞,∞)0,∞ $ be a continuous function such that $f(x+y)=f(x)+f(y)+f(x)f(y),∀x∈R˙$ Also $f_{′}(0)=1.$ Then $[f(2)]$ equal \displaystyle{\left({\left[{\dot}\right]}\right.} represents the greatest integer function$)$ $5$ b. $6$ c. $7$ d. $8$

If $f$ is an even function such that $(lim)_{h0}hf(h)−f(0) $ has some finite non-zero value, then prove that $f(x)$ is not differentiable at $x=0.$

Find the values of $a$ if $f(x)=(lim)_{n→∞}x_{2n}+a+1ax_{2n}+2 $ is continuous at $x=1.$

A curve in the xy-plane is parametrically given by $x=t+t_{3}andy=t_{2},wheret∈R$ is the parameter. For what value(s) of $t$ is $dxdy =21 ?$ $31 $ b. $2$ c. $3$ d. $1$

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 $g(c_{1})=0.$