Continuity and Differentiability Part - 1

Questions Discussed:

Discuss the continuity of following functions:

Question 1. { \displaystyle f(x)= \begin{cases} 2x+3, & \text{ if } x\le 2 \\ 2x-3, & \text{ if } x>2 \end{cases} }

Question 2. { \displaystyle f(x)= \begin{cases} \frac{|x|}{x}, & \text{ if } x\ne 0 \\ 0, & \text{ if } x = 0 \end{cases} }

Question 3. { \displaystyle f(x)= \begin{cases} x+1, & \text{ if } x\ge 1 \\ {{x}^{2}}+1, & \text{ if } x < 1 \end{cases} }

Question 4. { \displaystyle f(x)= \begin{cases} {{x}^{10}}-1, & \text{ if } x\le 1 \\ {{x}^{2}}, & \text{ if } x>1 \end{cases} }

Question 5. { \displaystyle f(x)= \begin{cases} \frac{x}{|x|}, & \text{ if } x<0 \\ -1, & \text{ if } x\ge 0 \end{cases} }

Question 6. { \displaystyle f(x)= \begin{cases} {{x}^{3}}-3, & \text{ if } x\le 2 \\ {{x}^{2}}+1, & \text{ if } x>2 \end{cases} }

Question 7. { \displaystyle f(x)= \begin{cases} x+5, & \text{ if } x\le 1 \\ x-5, & \text{ if } x>1 \end{cases} }

Question 8. { \displaystyle f(x)= \begin{cases} |x|+3, & \text{ if } x \le -3 \\ -2x, & \text{ if } -3 < x < 3 \\ 6x+2, & \text{ if } x \ge 3 \end{cases} }

Question 9. { \displaystyle f(x)= \begin{cases} 3, & \text{ if } 0\le x\le 1 \\ 4, & \text{ if } 1 < x < 3 \\ 5, & \text{ if } 3\le x\le 10 \end{cases} }

Question 10. { \displaystyle f(x)= \begin{cases} 2x, & \text{ if } x<0 \\ 0, & \text{ if } 0\le x\le 1 \\ 4x, & \text{ if } x>1 \end{cases} }

Question 11. { \displaystyle f(x)= \begin{cases} -2, & \text{ if } x\le -1 \\ 2x, & \text{ if } -1 < x \le 1 \\ 2, & \text{ if } x > 1 \end{cases} }

Question 12. { \displaystyle f(x)= \begin{cases} \frac{{{e}^{x}}-1}{\log (1+2x)}, & x\ne 0 \\ 7, & x=0 \end{cases} }

Question 13. { \displaystyle f(x)= \begin{cases} \frac{\sin x}{x}, & \text{ if } x<0 \\ x+1, & \text{ if } x\ge 0 \end{cases} }

Question 14. { \displaystyle f(x)= \begin{cases} {{x}^{2}}\sin \frac{1}{x}, & \text{ if } x\ne 0 \\ 0, & \text{ if } x=0 \end{cases} }

Question 15. { \displaystyle f(x)= \begin{cases} \sin x-\cos x, & \text{ if } x\ne 0 \\ -1, & \text{ if } x=0 \end{cases} }

Question 16. { \displaystyle f(x)= \begin{cases} \frac{\sin 3x}{\tan 2x}, & x < 0 \\ \frac{3}{2}, & x = 0 \\ \frac{\log (1+3x)}{{{e}^{2x}}-1}, & x > 0 \end{cases} }

Questions Discussed:

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

Question 1. { \displaystyle f(x) = \begin{cases} k{{x}^{2}},& \text{ if } x \le 2 \\ 3,& \text{ if } x > 2 \end{cases} }

Question 2. { \displaystyle f(x) = \begin{cases} kx+1,& \text{ if } x \le \pi \\ \cos x, & \text{ if } x > \pi \end{cases} }

Question 3. { \displaystyle f(x) = \begin{cases} kx+1,& \text{ if } x \le 5 \\ 3x-5, & \text{ if } x > 5 \end{cases} }

Question 4. { \displaystyle f(x) = \begin{cases} \frac{{{x}^{2}}-2x-3}{x+1}, & x\ne -1 \\ k, & x=-1 \end{cases} }

Question 5. { \displaystyle f(x) = \begin{cases} \frac{k\cos x}{\pi -2x}, & \text{ if } x\ne \frac{\pi }{2} \\ 3, & \text{ if } x = \frac{\pi }{2} \end{cases} }

Question 6. { \displaystyle f(x) = \begin{cases} 5,& \text{ if } x\le 2 \\ ax+b,& \text{ if } 2 < x < 10 \\ 21,& \text{ if } x \ge 10 \end{cases} }

Question 7. { \displaystyle f(x) = \begin{cases} \frac{1-\cos 2x}{2{{x}^{2}}}, & x\ne 0 \\ k, & x=0 \end{cases} }

Question 8. { \displaystyle f(x) = \begin{cases} \frac{\log (1+ax)-\log (1-bx)}{x}, & x\ne 0 \\ k, & x=0 \end{cases} }

Question 9. { \displaystyle f(x) = \begin{cases} \frac{{{\sin }^{2}}kx}{{{x}^{2}}}, & x\ne 0 \\ 1, & x=0 \end{cases} }

Question 10. { \displaystyle f(x) = \begin{cases} \frac{1-\cos 4x}{{{x}^{2}}}, & x<0 \\ k, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x > 0 \end{cases} }

Question 11. { \displaystyle f(x) = \begin{cases} \frac{x}{|x|+2{{x}^{2}}}, & x\ne 0 \\ k, & x=0 \end{cases} }

Questions Discussed:

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

Question 12. { \displaystyle f(x) = \begin{cases} \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{cases} }

Question 13. { \displaystyle f(x) = \begin{cases} \frac{1-\cos kx}{x\sin x}, & x \ne 0 \\ \frac{1}{2}, & x = 0 \end{cases} }

Question 14. { \displaystyle f(x) = \begin{cases} \frac{x-4}{|x-4|}+a, & x < 4 \\ a+b, & x = 4 \\ \frac{x-4}{|x-4|}+b, & x > 4 \end{cases} }

Question 15. { \displaystyle f(x) = \begin{cases} \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{cases} }

Question 16. { \displaystyle f(x) = \begin{cases} \frac{\sqrt{1+px}-\sqrt{1-px}}{x}, & -1 \le x < 0 \\ \frac{2x+1}{x-2}, & 0 \le x \le 1 \end{cases} }

Question 17. { \displaystyle f(x) = \begin{cases} a \sin \frac{\pi }{2}(x+1), & x \le 0 \\ \frac{\tan x-\sin x}{{{x}^{3}}}, & x > 0 \end{cases} }

Question 18. { \displaystyle f(x) = \begin{cases} \frac{\log (1+3x)-\log (1-2x)}{x}, & x \ne 0 \\ k, & x = 0 \end{cases} }

Question 19. { \displaystyle f(x) = \begin{cases} \frac{1-\cos 10x}{{{x}^{2}}}, & x < 0 \\ a, & x = 0 \\ \frac{\sqrt{x}}{\sqrt{625+\sqrt{x}}-25}, & x > 0 \end{cases} }

Questions Discussed:

Check whether the following functions are differentiable at indicated points:

Question 1. { \displaystyle f(x) = |x| \text{at } x = 0 }

Question 2. { \displaystyle f(x) = {{x}^{2}} \text{ at } x = 1 }

Question 3. { \displaystyle f(x) = x|x| \text{ at } x = 0 }

Question 4. { \displaystyle f(x) = \begin{cases} x-1, & x < 2 \\ 2x-3, & x \ge 2 \end{cases} \text{at } x = 2 }

Question 5. { \displaystyle f(x) = \begin{cases} {{x}^{2}}\sin \left( \frac{1}{x} \right), & x \ne 0 \\ 0, & x = 0 \end{cases} \text{ at } x = 0 }

Question 6. { \displaystyle f(x) = |x-2| \text{ at } x = 2 }

Questions Discussed:

Check whether the following functions are differentiable at indicated points:

Question 7. { \displaystyle f(x) = {{x}^{2}}+2x+7 \text{ at } x = 3 }

Question 8. { \displaystyle f(x) = [x] \text{ at } x = 1 } and { \displaystyle x=4 }

Question 9. { \displaystyle f(x) = \begin{cases} x[x], & 0 \le x < 2 \\ (x-1)x, & 2 \le x < 3 \end{cases} } at { \displaystyle x = 2 }

Question 10. { \displaystyle f(x) = \begin{cases} {{x}^{2}}\sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases} } at { \displaystyle x = 0 }

Question 11. Discuss the differentiability of { \displaystyle f(x) = |x-1|+|x-2| }

Question 12. If { \displaystyle f(x) = \begin{cases} {{x}^{2}}+3x+a, & x \le 1 \\ bx+2, & x > 1 \end{cases} } is differentiable everywhere, find the values of a and b.

Questions Discussed:

NCERT Exercise 5.1:

Question 1. Prove that the function { \displaystyle f(x) = 5x – 3 } is continuous at { \displaystyle x = 0 } , at { \displaystyle x = – 3 } and at { \displaystyle x = 5 } .

Question 2.Examine the continuity of the function { \displaystyle f(x) = 2x^2 - 1 } at { \displaystyle x = 3 } .

Question 3.Examine the following functions for continuity.

(a) { \displaystyle f(x) = x - 5 }

(b) { \displaystyle f(x) = \frac { 1 }{ x - 5 }, x \ne 5 }

(c) { \displaystyle f(x) = \frac { x^2 - 25 }{ x + 5 }, x \ne -5 }

(d) { \displaystyle f(x) = | x - 5 | }

Question 4. Prove that the function { \displaystyle f(x) = x^n } is continuous at { \displaystyle x = n } , where n is a positive integer.

Question 5. Is the function f defined by { \displaystyle f(x) = \begin{cases} x, & \text{ if } x \le 1 \\ 5, & \text{ if } x > 1 \end{cases} } continuous at { \displaystyle x = 0 } ? At { \displaystyle x = 1 } ? At { \displaystyle x = 2 } ? .

Question 17. Find the relationship between a and b so that the function f defined by { \displaystyle f(x) = \begin{cases} ax+1, & \text{ if } x \le 3 \\ bx+3, & \text{ if } x > 3 \end{cases} } is continuous at { \displaystyle x = 3 } .

Question 18. for what value of { \displaystyle \lambda } is the function defined by { \displaystyle f(x) = \begin{cases} \lambda {x^2 - 2x}, & \text{ if } x \le 0 \\ 4x+1, & \text{ if } x > 0 \end{cases} } continuous at { \displaystyle x = 0 } ? What about continuity at { \displaystyle x = 1 } ?

Question 20. Is the function defined by { \displaystyle f(x) = x^2 - \sin x + 5 } continuous at { \displaystyle x = \pi } ?

Questions Discussed:

NCERT Exercise 5.1:

Question 19. Show that the function defined by { \displaystyle g(x) = x – [x] } is discontinuous at all integral points. Here { \displaystyle [x] } denotes the greatest integer less than or equal to x.

Question 21. Discuss the continuity of the following functions:

(a) { \displaystyle f(x) = \sin x + \cos x }

(b) { \displaystyle f(x) = \sin x – \cos x }

(c) { \displaystyle f(x) = \sin x . \cos x }

Question 22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Question 31. Show that the function defined by { \displaystyle f(x) = \cos (x^2) } is a continuous function.

Question 32. Show that the function defined by { \displaystyle f(x) = | \cos x | } is a continuous function.

Question 33. Examine that { \displaystyle \sin | x | } is a continuous function.

Question 34. Find all the points of discontinuity of f defined by { \displaystyle f(x) = | x | – | x + 1 | } .