# Ch05. Continuity and Differentiability Class 12 Maths (Part -1)

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The whole chapter Continuity and Differentiability is divided into three courses. This course is the part one. In this online course, you will learn continuity, derivative of composite functions, chain rule, derivative of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. This course also has NCERT Exercise solutions, some important NCERT Examples and assignment based on NCERT Exemplar, R.D. Sharma, board’s question bank etc. Please note, all three courses based on Continuity and Differentiability do not have Rolle’s Theorem, Mean Value Theorem, NCERT Exercise 5.8 and questions based on them in Miscellaneous exercise.

## Course Content

Continuous Functions
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Differentiation
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The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

Discuss the continuity of following functions:

1. {\displaystyle f(x)= \begin{cases} \begin{aligned} 2x+3,\,\,\,\text{if}\,\,x\le 2 \\    2x-3,\,\,\,\text{if}\,\,x>2  \end{aligned} \end{cases} }
2. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{|x|}{x},\text{if}\,\,x\ne 0 \\    0,\text{if}\,\,x=0  \end{aligned} \end{cases} }
3. {\displaystyle f(x)= \begin{cases} \begin{aligned} x+1,\text{if}\,\,x\ge 1 \\    {{x}^{2}}+1,\text{if}\,\,x<1  \end{aligned} \end{cases} }
4. {\displaystyle f(x)= \begin{cases} \begin{aligned} {{x}^{10}}-1,\,\,\text{if}\,\,x\le 1 \\    {{x}^{2}},\,\,\text{if}\,\,x>1  \end{aligned} \end{cases} }
5. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{x}{|x|},\text{if}\,\,x<0 \\    -1,\text{if}\,\,x\ge 0  \end{aligned} \end{cases} }
6. {\displaystyle f(x)= \begin{cases} \begin{aligned} {{x}^{3}}-3,\text{if}\,\,x\le 2 \\    {{x}^{2}}+1,\text{if}\,\,x>2  \end{aligned} \end{cases} }
7. {\displaystyle f(x)= \begin{cases} \begin{aligned} x+5,\text{if}\,\,x\le 1 \\    x-5,\text{if}\,\,x>1  \end{aligned} \end{cases} }
8. {\displaystyle f(x)= \begin{cases} \begin{aligned} |x|+3,\,\,\text{if}\,\,x\le -3 \\   -2x,\,\,\text{if}\,\,-3<x<3 \\   6x+2,\,\,\text{if}\,\,x\ge 3  \end{aligned} \end{cases} }
9. {\displaystyle f(x)= \begin{cases} \begin{aligned} 3,\,\,\text{if}\,\,0\le x\le 1 \\    4,\,\,\text{if}\,\,1<x<3 \\    5,\,\,\text{if}\,\,3\le x\le 10  \end{aligned} \end{cases} }
10. {\displaystyle f(x)= \begin{cases} \begin{aligned} 2x,\text{if}\,\,x<0 \\    0,\text{if}\,\,0\le x\le 1 \\    4x,\text{if}\,\,x>1  \end{aligned} \end{cases} }
11. {\displaystyle f(x)= \begin{cases} \begin{aligned} -2,\text{if}\,\,x\le -1 \\    2x,\text{if}\,\,-1<x\le 1 \\    2,\text{if}\,\,x>1  \end{aligned} \end{cases} }
12. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{{{e}^{x}}-1}{\log (1+2x)},x\ne 0 \\   7,x=0  \end{aligned} \end{cases} }
13. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{\sin x}{x},\text{if}\,\,x<0 \\   x+1,\text{if}\,\,x\ge 0  \end{aligned} \end{cases} }
14. {\displaystyle f(x)= \begin{cases} \begin{aligned} {{x}^{2}}\sin \frac{1}{x},\text{if}\,\,x\ne 0 \\   0,\text{if}\,\,x=0  \end{aligned} \end{cases} }
15. {\displaystyle f(x)= \begin{cases} \begin{aligned} \sin x-\cos x,\text{if}\,\,x\ne 0 \\    -1,\text{if}\,\,x=0  \end{aligned} \end{cases} }
16. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{\sin 3x}{\tan 2x},x<0 \\    \frac{3}{2},x=0 \\    \frac{\log (1+3x)}{{{e}^{2x}}-1},x>0  \end{aligned} \end{cases} }

### Assignment – 2

Find the value k, a, b, c, or any other unknown, if following functions are continuous:

1. {\displaystyle f(x)= \begin{cases} \begin{aligned} k{{x}^{2}},\text{if}\,x\le 2 \\    3,\text{if}\,x>2  \end{aligned} \end{cases} }
2. {\displaystyle f(x)= \begin{cases} \begin{aligned} kx+1,\text{if}\,\,x\le \pi  \\    \cos x,\text{if}\,\,x>\pi   \end{aligned} \end{cases} }
3. {\displaystyle f(x)= \begin{cases} \begin{aligned} kx+1,\text{if}\,\,x\le 5 \\    3x-5,\text{if}\,\,x>5  \end{aligned} \end{cases} }
4. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{{{x}^{2}}-2x-3}{x+1},x\ne -1 \\   k,x=-1  \end{aligned} \end{cases} }
5. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{k\cos x}{\pi -2x},\text{if}\,\,x\ne \frac{\pi }{2} \\    3,\text{if}\,\,x=\frac{\pi }{2}  \end{aligned} \end{cases} }
6. {\displaystyle f(x)= \begin{cases} \begin{aligned} 5,\text{if}\,\,x\le 2 \\   ax+b,\text{if}\,\,2<x<10 \\   21,\text{if}\,\,x\ge 10  \end{aligned} \end{cases} }
7. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{1-\cos 2x}{2{{x}^{2}}},x\ne 0 \\   k,x=0  \end{aligned} \end{cases} }
8. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{\log (1+ax)-\log (1-bx)}{x},x\ne 0 \\   k,x=0  \end{aligned} \end{cases} }
9. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{{{\sin }^{2}}kx}{{{x}^{2}}},x\ne 0 \\   1,x=0  \end{aligned} \end{cases} }
10. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{1-\cos 4x}{{{x}^{2}}},x<0 \\   k,x=0 \\   \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4},x>0  \end{aligned} \end{cases} }
11. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{x}{|x|+2{{x}^{2}}},x\ne 0 \\   k,x=0  \end{aligned} \end{cases} }
12. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{\sin (a+1)x+\sin x}{x},x<0 \\    c,x=0 \\    \frac{\sqrt{x+b{{x}^{2}}}-\sqrt{x}}{b{{x}^{\frac{3}{2}}}},x>0  \end{aligned} \end{cases} }
13. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{1-\cos kx}{x\sin x},x\ne 0 \\   \frac{1}{2},x=0  \end{aligned} \end{cases} }
14. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{x-4}{|x-4|}+a,x<4 \\    a+b,x=4 \\   \frac{x-4}{|x-4|}+b,x>4  \end{aligned} \end{cases} }
15. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{1-{{\sin }^{3}}x}{3{{\cos }^{2}}x},x<\frac{\pi }{2} \\   a,x=\frac{\pi }{2} \\   \frac{b\,(1-\sin x)}{{{(\pi -2x)}^{2}}},x>\frac{\pi }{2}  \end{aligned} \end{cases} }
16. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{\sqrt{1+px}-\sqrt{1-px}}{x},-1\le x<0 \\   \frac{2x+1}{x-2},0\le x\le 1  \end{aligned} \end{cases} }
17. {\displaystyle f(x)= \begin{cases} \begin{aligned} a\,\,\sin \frac{\pi }{2}(x+1),x\le 0 \\   \frac{\tan x-\sin x}{{{x}^{3}}},x>0  \end{aligned} \end{cases} }
18. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{\log (1+3x)-\log (1-2x)}{x},x\ne 0 \\   k,x=0  \end{aligned} \end{cases} }
19. {\displaystyle f(x)= \begin{cases} \begin{aligned} \frac{1-\cos 10x}{{{x}^{2}}},x<0 \\   a,x=0 \\   \frac{\sqrt{x}}{\sqrt{625+\sqrt{x}}-25},x>0  \end{aligned} \end{cases} }
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