Exercise 5.1
Question 1. Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5
Note: Any polynomial function, e.g, y = f(x) is continuous for all x
Question 2. Examine the continuity of the function f(x)= 2×2^{-1} at x=3.
Question 3. Examine the following functions for continuity
Question 4. Prove that the function f (x) = xn is continuous at x = n, where n is a positive integer.
Question 5.
Question : Find all points of discontinuity of f in Q 6 to Q 12, where f is defined by
Question 13.
Question : Discuss the continuity of the function f, where f is defined by
Question 17. Find the relationship between a and b so that the function f is defined by
Question 18.
Question 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
Question 20. Is the function defined by f (x) = x^{2} – sin x + 5 continuous at x = pi?
Question 21. Discuss the continuity of the following functions:(a) f (x) = sin x + cos x(b) f (x) = sin x – cos x (c) f (x) = sin x . cos x
Question 22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Question 23. Find all points of discontinuity of f, where
Question. 24. Determine if f is defined by
Note: If f(x) and g(x) are two continuous functions in their domain, Then their product’s also a continuous function in their common domain.
Question 25.
Note: Difference of two continuous functions is continuous in their common domain.
Question: Find the values of k so that the function f is continuous at the indicated point in Questions 26 to 29.
Question 30. Find the values of a and b such that the function defined by
Question 31. Show that the function defined by f (x) = cos (x2) is a continuous function.
Sol. Now, f (x) = cosx2, let g (x) = cosx and h (x) = x2.\ goh(x) = g (h (x)) = cos x2. Now g and h both are continuous ” x Î R.f (x) = goh (x) = cos x2 is also continuous at all x E R
Note: A composite function of two continuous function in D is also a continuous function in D.
Question 32. Show that the function defined by f (x) = |cos x| is a continuous function.
Sol. Let g (x) = |x| and h (x) = cos x , f (x) = goh (x) = g (h (x)) = g (cosx) = |cos x |Now g (x) = |x| and h (x) = cos x both are continuous for all values of x E R.
:. (goh) (x) is also a continuous function for all value of R.
Hence, f (x) = goh (x) = |cos x| is continuous for all values of x ER.
Question 33. Examine that sin |x| is a continuous function.
Sol. Let g (x) = sin x, h (x) = |x| , goh (x) = g (h(x)) = g (|x|) = sin |x| = f (x)
Now g (x) = sin x and h (x) = |x| both are continuous for all x E R.
:. f (x) = goh (x) = sin |x| is continuous at all x ÎR.
Question 34. Find all the points of discontinuity of f defined by f (x) = |x| – |x + 1|.
Exercise 5.2
Question : Differentiate the functions with respect to x in Questions 1 to 8
Question 9. Prove that the function f given by f (x) = |x – 1|, x E R is not differentiable at x = 1.
Question 10. Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Exercise 5.3
Question. Find dy/dx in the following :
Exercise 5.4
Question. Differentiate the following w.r.t.x :
Exercise 5.5
Question. Differentiate the functions given in Questions 1 to 11 w.r.to x
Question. Find dy/dx of the functions given in Questions 12 to 15
Question 16. Find the derivative of the function given by
Question 17. Differentiate (x^{2}– 5x + 8) (x^{3} + 7x + 9) in three ways mentioned below :
- by using product rule
- by expanding the product to obtain a single polynomial.
- by logarithmic differentiation.
Do they all give the same answer?
Question 18. If u, v and w are functions of x then show that
Exercise 5.6
If x and y are connected parametrically by the equations given in Questions 1 to 10, without eliminating the parameter. Find dy/dx .
Question 11.
Exercise 5.7
Question. Find the second-order derivatives of the functions given in questions 1 to 10
Question 11.
Question 12.
Question 13. If y = 3 cos (log x) + 4 sin (log x), show that x^{2} y_{2} + xy_{1} + y = 0
Question 14.
Question 15.
Question 16.
Question 17.
Exercise 5.8
Question 1. Verify Rolle’s theorem for the function f(x) = x^{2} + 2x – 8, x E[– 4, 2]
Sol. Now f (x) = x^{2} + 2x – 8 is a polynomial.
:. it is continuous and derivable in its domain x E R.
Hence it is continuous in the interval [– 4, 2] and derivable in the interval (– 4, 2) f (–4) = (– 4)^{2} + 2 (– 4) – 8 = 16 – 8 – 8 = 0 and f (2) = 22 + 4 – 8 = 8 – 8 = 0
Conditions of Rolle’s theorem are satisfied.
Can you say something about the converse of Rolle’s theorem from these examples?
Question 2. Examine if Rolle’s theorem is applicable to any of the following functions. Can you say something about the converse of Rolle’s theorem from these examples?
Hint: Greatest integral function f(x) = [x] is neither continuous nor differentiable at any integral value of x
Sol. (i) In the interval [5, 9], f (x) = [x] is neither continuous nor derivable at x = 6, 7, 8Hence Rolle’s theorem is not applicable
(ii) f (x) = [x] is not continuous and derivable at –1, 0, 1. Hence Rolle’stheorem is not applicable.
(iii) f (x) = (x^{2} – 1) Þ, f (1) = 1 – 1= 0 and f (2) = 22 – 1 = 3 not equal to f (1)Though it is continuous and derivable in the interval [1, 2]. Rolle’s theorem is not applicable, since f (1) not equal to f (2).
In case of converse if f’ (c) = 0, c E [a, b] then conditions of rolle’s theorem are not true.
(i) f (x) = [x] is the greatest integer less than or equal to x.
:. f (x) = 0, But f is neither continuous nor differentiable in the interval [5, 9].
(ii) Here also, though f¢ (x) = 0 , but f is neither continuous nor differentiable in the interval [–2, 2].(iii) f (x) = x^{2} – 1, f’ (x) = 2x. Here f’ (x) is not zero in the [1, 2].
Question 3.
Question 4. Verify Mean Value Theorem, if f (x) = x^{2} – 4x – 3 in the interval [a, b], where a = 1 and b = 4
Question 5. Verify Mean Value Theorem, if f (x) = x^{3} – 5×2 – 3x in the interval [a, b], where a = 1 and b = 3. Find all c E (1, 3) for which f ‘(c) = 0
Question 6. Examine the applicability of Mean Value theorem for all three functions given in the above exercise 2.
Sol. (i) f(x) = [x] for x Î[5, 9] , f (x) = [x] in the interval [5, 9] is neither continuous, nor differentiable. Hence value theorem is not applicable.
(ii) f (x) = [x], for x Î [–2, 2] , Again f (x) = [x] in the interval [–2, 2] is neither continuous, nor differentiable. Hence value theorem is not applicable.
(iii) f (x) = x^{2} – 1 for x E [1, 2], It is a polynomial. Therefore it is continuous in the interval [1, 2] and differentiable in the interval (1, 2)
Miscellaneous Exercise
Differentiate w.r.t.x the function in Questions 1 to 11
Question 12.
Question 13.
Question 14.
Question 15.
constant independent of a and b.
Question 16.
Question 17.
Question 18. If f (x) = |x|^{3}, show that f¢¢ (x) exists for all real x and find it.
Question 19.
Question 20. Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines
Question 21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.
Sol. Consider the function f (x) = |x| + |x – 1|f (x) is continuous everywhere, but it is not differentiable at x = 0 and x = 1
Question 22.
Question 23.
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