Ncert Solutions for Class 12
Continuity and Differentiability
Miscellaneous Exercise
Exercise 5.1
Exercise 5.2
Exercise 5.3
Exercise 5.4
Exercise 5.5
Exercise 5.6
Exercise 5.7
Exercise 5.8
Continuity and Differentiability
Miscellaneous Exercise
Differentiate w.r.t. x the function in Exercises 1 to 11.
3. (left(5xright)^{3cos{2}x}).
4. ({sin}^{-1}{left(xsqrt xright)}),(0le xle1).
5. (frac{{cos}^{-1}{frac{x}{2}}}{sqrt{2x+7}}),(-2<x<2).
7. (left(log{x}right)^{log{x}}), (x>1).
8. (cos{left(acos{x}+bsin{x}right)}), for some constant a and b.
9. (left(sin{x}-cos{x}right)^{left(sin{x}-cos{x}right)}), (frac{pi}{4}<x<frac{3pi}{4}).
10. (x^x+x^a+a^x+a^a), for some fixed a>0 and x>0.
11. (x^{x^2-3}+left(x-3right)^{x^2}), for x>3.
13. Find (frac{dy}{dx}), If (y={sin}^{-1}{x}+{sin}^{-1}{sqrt{1-x^2}}),(-1le xle1).
17. If (x=aleft(cos{t}+tsin{t}right) ) and (y=aleft(sin{t}-tcos{t}right)), find (frac{d^2y}{dx^2}).
18. If (fleft(xright)=left|xright|^3), show that f”(x) exists for all real x and find it.
Continuity and Differentiability
Exercise 5.1
1. Prove that the function (f(x)=5x–3) is continuous at (x=0), at (x=–3) and at (x=5).
2. Examine the continuity of the function (f(x)=2x^2–1) at (x=3).
3. Examine the following functions for continuity.
(i) (f(x)=x-5)
(ii) (f(x)=frac{1}{x-5}),(x≠5)
(iii) (f(x)=frac{x^2-25}{x+5}),(x≠5)
(iv) (f(x)=|x-5|)
4. Prove that the function (f(x)=x^n) is continuous at (x=n), where n is a positive integer.
5. Is the function f defined by (f(x)=begin{cases}x, & text{if} & x≤1\ 5, & text{if} & x>1 end{cases}) continuous at (x=0)? At (x=1)? At (x=2)?
Find all points of discontinuity of f, where f is defined by:
6. (f(x)=begin{cases}2x+3, & text{if} & x≤2\2x-3, & text{if} & x>2 end{cases})
7. (f(x)=begin{cases}|x|+3, & text{if} & x≤-3\-2x, & text{if} & -3<x<3\ 6x+2, & text{if} & xge 3 end{cases})
8. (f(x)=begin{cases}frac{|x|}{x}, & text{if} & xne 0\0, & text{if} & x=0 end{cases})
9. (f(x)=begin{cases}frac{x}{|x|}, & text{if} & x<0\-1, & text{if} & xge 0 end{cases})
10. (f(x)=begin{cases}x+1, & text{if} & xge 1\x^2+1, & text{if} & x<1 end{cases})
11. (f(x)=begin{cases}x^3-3, & text{if} & xle 2\x^2+1, & text{if} & x>2 end{cases})
12. (f(x)=begin{cases}x^{10}-1, & text{if} & xle 1\x^2, & text{if} & x>1 end{cases})
13. Is the function defined by (f(x)=begin{cases}x+5, & text{if} & xle 1\x-5, & text{if} & x>1 end{cases}) a continuous function ?
Discuss the continuity of the function f, where f is defined by:
14. (f(x)=begin{cases}3, & text{if} & 0≤x≤1\4, & text{if} & 1<x<3\ 5, & text{if} & 3≤x≤10end{cases})
15. (f(x)=begin{cases}2x, & text{if} & x<0\0, & text{if} & 0≤x≤1\ 4x, & text{if} & x>1end{cases})
16. (f(x)=begin{cases}-2, & text{if} & x≤-1\2x, & text{if} & -1<x≤1\ 2, & text{if} & x>1end{cases})
17. Find the relationship between a and b so that the function f defined by:
(f(x)=begin{cases}ax+1, & text{if} & xle 3\bx+3, & text{if} & x>3 end{cases}) is continuous at x=3.
18. For what value of λ is the function defined by (f(x)=begin{cases}λ(x^2-2x), & text{if} & xle 0\4x+1, & text{if} & x>0 end{cases}) continuous at x=0? What about continuity at x=1?
19. Show that the function defined by (g(x)=x–[x]) is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.
20. Is the function defined by (f(x)=x^2 –sinx+5) continuous at (x=π)?
21. Discuss the continuity of the following functions:
(a) (f(x)=sinx+cosx)
(b) (f(x)=sinx–cosx)
(c) (f(x)=sinx.cosx)
22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
23. Find all points of discontinuity of f, where
(f(x)=begin{cases}frac{sinx}{x}, & text{if} & x< 0\x+1, & text{if} & xge 0 end{cases})
24. Determine if f defined by (f(x)=begin{cases}x^2 sin(frac{1}{x}), & text{if} & xne 0\0, & text{if} & x=0 end{cases}) is a continuous function?
25. Examine the continuity of f, where f is defined by (f(x)=begin{cases}sinx-cosx, & text{if} & xne 0\-1, & text{if} & x=0 end{cases}).
Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.
26. (f(x)=begin{cases}frac{k cosx}{pi – 2x}, & text{if} & xne frac{pi}{2}\3, & text{if} & x=frac{pi}{2} end{cases}) at (x=frac{pi}{2}).
27. (f(x)=begin{cases}kx^2, & text{if} & xle 2 \3, & text{if} & x>2 end{cases}) at (x=2).
28. (f(x)=begin{cases}kx+1, & text{if} & xle pi \cosx, & text{if} & x>pi end{cases}) at (x=pi).
29. (f(x)=begin{cases}kx+1, & text{if} & xle 5 \3x-5, & text{if} & x>5 end{cases}) at (x=5).
30. Find the values of a and b such that the function defined by: (f(x)=begin{cases}5, & text{if} & x≤2\ax+b, & text{if} & 2<x<10\21, & text{if} & xge 10end{cases}) is a continuous function.
31. Show that the function defined by (f(x)=cos(x^2)) is a continuous function.
32. Show that the function defined by (f(x)=|cosx|) is a continuous function.
33. Examine that (sin|x|) is a continuous function.
34. Find all the points of discontinuity of f defined by (f(x)=|x|–|x+1|).
Continuity and Differentiability
Exercise 5.2
Differentiate the functions with respect to x in Exercises 1 to 8.
1. (sin(x^2+5))
2. (cos(sinx))
3. (sin(ax+b))
4. (sec(tan√x ))
5. (frac{sin(ax+b)}{cos(cx+d)})
6. (cos x^3. sin^2(x^5))
7. (2sqrt{cot(x^2)})
8. (cos(√x))
9. Prove that the function f given by (f(x)=|x–1|, x∈R) is not differentiable at (x=1).
10. Prove that the greatest integer function defined by (f(x)=[x], 0<x<3) is not differentiable at x=1 and x=2.
