DIFFERENTIATION
Raw- 2
1) If y= √sin√x then dy/dx is
a) 1/(2√sin√x)
b) (√cos√x)/2x
c) 1/(2√cos√x)
d) (√cos√x)/(4√x √sin√x)
2) If f(x)= cos⁻¹[(1- (logx)²)/(1+ (logx)²)], then the value of f'(e) is
a) 2/e b) 1/e c) 1 d) 1/e²
3) If y= a cos mx - b sin mx, then d²y/dx² is
a) - m²y b) m²y c) -my d) my
4) If y= ₓeˣ, then dy/dx is
a) y(logx + eˣ)
b) ylogx( 1/2 + eˣ)
c) yeˣ(logx + 1/x)
d) yeˣ(logx + x)
5) If 2ˣ + 2ʸ= 2ˣ⁺ʸ, then dy/dx at x= y= 1 is
a) 0 b) -1 c) 1 d) 2
6) If x= sin⁻¹t, y= log(1- t²), 0≤ t < 1, then the value of d²y/dx² at t= 1/3 is
a) -9/4 b) -9/8 c) 9/4 d) 9/8
7) If y= {x + √(1+ x²)}ⁿ, then (1+ x²) d²y/dx² + x dy/dx is
a) -y b) n²y c) - n²y d) 2n²y
8) If siny + ₑ- x cosy = e Then the value of dy/dx at (1,π) is
a) 0 b) 1 c) e d) -1
9) If x= 2 cos t + cos2t and y= 2 sin t - sin2t, then dy/dx at t=π/4 is
a) -(√2+1) b) √2 c) (√2- 1) d) 1- √2
10) If logx = z, then x² d²y/dx² is
a) d²y/dz² b) d²y/dz² + dy/dz d) d²y/dz² - 2dy/dz
Raw- 1
1) ₑtan⁻¹{(y- x²)/x²} then dy/dx is
a) 2x[1+ tan(logx)]+ x sec²(logx)
b) x[1+ tan(logx)]+ sec²(logx)
c) 2x[1+ tan(logx)]+ x² sec²(logx)
d) 2x[1+ tan(logx)]+ sec²(logx)
2) If x= a cos⁴θ, y= a sin⁴θ then dy/dx at θ= 3π/4 is
a) 0 b) 1 c) -1 d) -2
3) d/dx(xˣ) is
a) xˣ(1- logx)
b) xˣ logx
c) xˣ⁺¹(1+ logx)
d) xˣ(1+ logx)
4) The differential coefficient of ₑx³ w.r.t. logx is
a) ₑx³ b) 3x³ ₑx³ c) 3x² ₑx³ d) 3x² ₑx³ + 3x²
5) The second derivative of a sin³ t w.r.t. a cos³t at t=π/4 is
a) 2 b) 1/12a c) 4√2/3a d) 0
6) The derivative of the function f(x)= 3|x +2| at the point, x= -3 is
a) -3 b) 3 c) 0 d) does not exist
7) If y= √[x + √{x + √(x+....... ∞ Then the value of dy/dx is
a) x/(2y -1)
b) 2/(2y -1)
c) 1/(2y -1)
d) x/(y -1)
8) If y= sinx + eˣ, then the value of d²x/dy² is
a) (sinx - eˣ)/(cosx + eˣ)
b) 1/(eˣ - sinx)
c) (sinx - eˣ)/(cosx + eˣ)²
d) (sinx + eˣ)/(cosy + eˣ)³
9) If sin⁻¹x + sin⁻¹y =π/2, then dy/dx is
a) x/y b) -x/y c) y/x d) - y/x
10) If dx/dy = u and d²x/dy² = c, then d²y/dx² is
a) - v/u² b) v/u² c) -v/u³ d) v/u³
CONTINUITY & DIFFERENTIABLE
1) Show that f(x)= x³ is continuous at x= 2.
2) Show that f(x)=[ x] is not continuous at x= n, where n is any integer.
3) Show that f(x)={ x, if x is an integer
0, if x is not integer
is discontinuous at each integral value of x.
4) Show that f(x)={x/|x|, when x ≠ 0,
1, when x= 0
is discontinuous at x= 0.
5) If f(x)={ (x²-1)/(x -1) for x≠ 1
2 for x= 1
Show that f(x) is continuous at x= 1.
6) Determine the value of k for which the function
f(x)={ (sin5x)/3x, if x≠ 0
k, if x= 0
is continuous at x=0.
7) Show that the function is continuous at x= 0
f(x)={ x sin(1/x), when x ≠ 0
0, when x = 0
8) Let f(x)={ (sinx)/x + cosx, when x ≠ 0
2, when x= 0
Show that f(x) is continuous at x= 0.
9) Show that the function is discontinuous at x= 0
f(x)={ (sin²ax)/x, when x≠ 0
1, when x= 0
Redefine the function in such a way that it becomes continuous at x= a.
10) Is the function, f(x)= (3x + 4 tanx)/x continuous at x= 0 ? If not, how many the function be defined to make it continuous at this point ?
11) Discuss the continuity of the function
f(x)={ 3x -2, when x≤ 0
x +1, when x> 0 at x= 0.
12) Discuss the continuity of the function
f(x)={ - x, when x≤ 0
x, when 0< x ≤1
2-x, when 1< x<2
1, when x> 2
at each of the point x= 0,1,2.
13) Show that the function
f(x)={ 2x, if x< 2
2, if x= 2
x², if x> 2
has a removable discontinuity at x= 2.
CONTINUOUS FUNCTIONS
1) Let f(x)={ x if x≥ 1
x² if x< 1
Is a continuous function? Why ?
2) Prove that f(x)= |x | is a continuous function.
3) Discuss the continuity of the function.
f(x)= {2x -1, if x< 0;
2x +1, if x≥ 0.
4) Discuss the continuity of the function
f(x)= { (sinx)/x, if x < 0;
(x+1), if x≥ 0.
5) Discuss the continuity of the function
f(x)={ x/|x|, if x ≠ 0;
0, if x= 0.
6) Locate the point of discontinuity of the function
f(x)= { (x⁴-16)/(x -2), if x≠ 2
16, if x= 2.
7) Determine the value of k so that the function
f(x)={ kx², if x< 2;
3, if x> 2 is continuous.
8) Let f(x)={ 1, if x≤ 3;
ax+ b, if 3<x<5;
7, if 5≤ x.
Find the values of a and b so that f(x) is continuous.
9) Show that the function f(x)= √(x⁴+3) is continuous at each point.
10) Show that the function f(x)= |sinx + cosx| is continuous at x=π.
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