Continuity and Differentiability — Class 12 solved problems

44 problems from this chapter, each solved step by step.

Find dy/dx if y = (ln(x))^(cos(x)).
(dy)/(dx) = (ln x)^(cos x) ( -sin x · ln(ln x) + (cos x)/(x ln x))
Differentiate the following w.r.t. x. (i) cos^-1(sin x) (ii) tan^-1(sin x / (1 + cos x)) (iii) sin^-1(2^(x+1) / (1 + 4^x))
(i) (-cos x)/(|cos x|) (or -1 if cos x > 0, 1 if cos x < 0) (ii) (1)/(2) (iii) (2^(x+1) ln 2)/(1 + 4^x)
If x = t² and y = t³, then (d² y)/(dx²) is equal to:
(3)/(4t)
Find (d y)/(d x), if x=a(θ+sin θ), y=a(1-cos θ).
(d y)/(d x) = tan((θ)/(2))
Differentiate the following w.r.t. x. (i) cos ^(-1)(sin x) (ii) tan ^(-1)((sin x)/(1+cos x)) (iii) sin ^(-1)((2^(x+1))/(1+4^x))
(i) -1 (ii) (1)/(2) (iii) (2^(x+1) log 2)/(1+4^x)
Let the function f(x)=(x²+1)|x²-ax+2|+cos|x| be not differentiable at the two points x=α=2 and x=β. Then the distance of the point (α, β) from the line 12x+5y+1
3
If the function f(x) = (2)/(x) sin (k_1 + 1) x + sin (k_2 - 1) x, & x < 0 4, & x = 0 (2)/(x) log_e ( (2 + k_1 x)/(2 + k_2 x)), & x > 0 is continuous at x = 0, t
10
Find (d y)/(d x), if x^((2)/(3))+y^((2)/(3))=a^((2)/(3))
(dy)/(dx) = -((y)/(x))^((1)/(3))
Find the derivative of f(x)=(x+1)/(x)
f'(x) = -(1)/(x²)
30. If (sin x)^y = y^(cos x), then find (dy)/(dx). (differentiation, implicit differentiation, logarithmic differentiation, chain rule, trigonometric functions)
The derivative (dy)/(dx) is (-y(sin x ln y + y cot x))/(y ln(sin x) - cos x).
Discuss the continuity of the function f given by f(x)=x³+x²-1.
The function f(x)=x³+x²-1 is continuous for all real numbers.
Check the continuity of the function f given by f(x)=2 x+3 at x=1.
The function f(x) = 2x+3 is continuous at x=1.
For α,β, R, if _x 0 x²sin(α x)+( -1)e^(x²)sin(2x)-β x=3, then β+ -α is equal to:
7
Let m and n be the number of points at which the function f(x)=maxx, x³, x^5,, x^(21), x R is not differentiable and not continuous, respectively. Then m + n is
3
Discuss the continuity of the function defined by f(x)= r x+2, if x<0 -x+2, if x>0
The function f(x) is continuous for all x R except at x=0.
Discuss the continuity of the function f given by f(x)= x, & if x ≥ 0 x², & if x<0
The function f(x) is continuous for all real numbers x.
Find (d y)/(d x), if x=a t², y=2 a t.
(d y)/(d x) = (1)/(t)
For a positive constant a find (d y)/(d x), where y=a^(i+(1)/(t)), and x=(t+(1)/(t))^a
(d y)/(d x) = (a^(t+(1)/(t)) ln a)/(a(t+(1)/(t))^(a-1))
Differentiate w.r.t. x, the following function: (i) √(3 x+2)+(1)/(√(2 x²+4)) (ii) log _7(log x)
(i) (3)/(2√(3x+2)) - (2x)/((2x²+4)³/2) (ii) (1)/(x log x log 7)
Find (d y)/(d x), if y^(x)+x^(y)+x^(x)=a^(b).
(dy)/(dx) = - (y^x ln y + x^y (y)/(x) + x^x (ln x + 1))/(y^x (x)/(y) + x^y ln x)
If the function f(x) = (tan(tan x) - sin(sin x))/(tan x - sin x) is continuous at x = 0, then f(0) is equal to:
2
Find (d y)/(d x), if x=a cos θ, y=a sin θ.
(d y)/(d x) = -cot θ
Find all continuous functions f: ℝ → ℝ such that f(x+y) = f(x) + f(y) for all x,y ∈ ℝ.
The continuous functions f: R → R satisfying f(x+y) = f(x) + f(y) are of the form f(x) = cx for some constant c R.
Differentiate x^(sin x), x>0 w.r.t. x.
(dy)/(dx) = x^(sin x) ((sin x)/(x) + cos x log x)
Show that the function f given by f(x)= x³+3, & if x ≠ 0 1, & if x=0 is not continuous at x=0.
The function f(x) is not continuous at x=0 because lim_x → 0 f(x) = 3 while f(0) = 1, and 3 ≠ 1.
Differentiate sin ² x w.r.t. e^(cos x).
-(2 cos x)/(e^(cos x))
Let f: R - 0 → R be a function such that f(x) - 6 f((1)/(x)) = (35)/(3x) - (5)/(2). If the lim_x → 0 ((1)/(α x) + f(x)) = β; α, β R, then α + 2β is equal to
-4
Differentiate the following w.r.t. x: (i) e^(-x) (ii) sin (log x), x>0 (iii) cos ^(-1)(e^(x)) (iv) e^(cos x)
(i) -e^(-x) (ii) (cos (log x))/(x) (iii) (-e^x)/(√(1 - e²x)) (iv) -sin x · e^(cos x)
Let f: R→ R be a twice-differentiable function such that (sin xcos y) [f(2x+2y)-f(2x-2y) ]=(cos xsin y) [f(2x+2y)+f(2x-2y) ] for all x,y R. If f'(0)= 12, then t
-3
Find the derivative at x=2 of the function f(x)=3 x.
The derivative of the function f(x)=3x at x=2 is 3.
Discuss the continuity of the function f given by f(x)=|x| at x=0.
The function f(x)=|x| is continuous at x=0.
Differentiate a^(x) w.r.t. x, where a is a positive constant.
(d)/(dx)(a^x) = a^x ln a
Prove that the function f(x) = x sin(1/x) for x ≠ 0 and f(0) = 0 is continuous everywhere but not differentiable at x = 0.
The function f(x) = x sin(1/x) for x ≠ 0 and f(0) = 0 is continuous everywhere but not differentiable at x = 0.
Let f(x) = 3x, & x < 0 min 1 + x + [x], x + 2[x], & 0 ≤ x ≤ 2 5, & x > 2 where [.] denotes greatest integer function. If α and β are the number of points, where
5
The number of points of discontinuity of the function f(x) = (x²)/(2) - √x, x [0, 4], where · denotes the greatest integer function, is _____
8
Examine whether the function f given by f(x)=x² is continuous at x=0.
The function f(x)=x² is continuous at x=0.
Let f: R → R be a continuous function satisfying f(0) = 1 and f(2x) - f(x) = x for all x R. If lim_n→∞ [f(x) - f(x/2^n)] = G(x) then Σ_r=1^(10) G(r²) is equal t
385
Find f^()(x) if f^()(x)=(sin x)^(sin x) for all 0<x<π.
f^()(x) = (sin x)^(sin x) cos x [ln (sin x) + 1]
Show that the sequence f_n(x) = x^n converges pointwise but not uniformly on [0,1].
The sequence f_n(x) = x^n converges pointwise to f(x) = 0 & if 0 ≤ x < 1 1 & if x = 1 on [0,1]. It does not converge uniformly on [0,1] because lim_n → ∞ _x [0,
If y=sin ^(-1) x, show that (1-x²) (d² y)/(d x²)-x (d y)/(d x)=0.
The given equation is proven: (1-x²) (d² y)/(d x²)-x (d y)/(d x)=0.
Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points where the function f(x) = [x] + |x - 2|, -2 < x < 3, is
8
Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y R. Then Σ_n=1^(100) log_e f(n) is equal to:
2525
Discuss the continuity of the function f defined by f(x)=(1)/(x), x ≠ 0.
The function f(x) = (1)/(x) is continuous for all x R, x ≠ 0. It is discontinuous at x=0.
Find all the points of discontinuity of the function f defined by f(x)= rr x+2, & if x<1 0, & if x=1 x-2, & if x>1
The function f(x) is discontinuous at x=1.

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