limits n quad_X-1_1.doc
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Yukti Educational Services Pvt Ltd. Assignment: Limits and Quadratic Date: 11 th September, 2009 1. (R-3.19-11)If Δ= tan x tan( x + h ) tan( x +2 h) tan( x +2 h) tan x tan( x + h) tan( x + h) tan( x +2 h ) tan x , find lim h →0 Δ h 2 2. (R-3.19-13)If α, β are the roots of ax 2 + bx + c , then evaluate (i) lim x → β 1− cos( ax 2 + bx + c ) ( x − β) 2 (ii) lim x → 1 α 1− cos( cx 2 + bx + a ) (1− αx) 2 3. (R-3.21-16)Evaluate: lim n →∞ n 2 1− cos 1 n ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1− cos 1 n ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1− cos 1 n ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ......... ∞ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ 4. (R-4.9-14) If f ( x)= (1+ |sin x |) a |sin x| ,− π 6 < x <0 b , x =0 e tan2x tan3x ,0 < x < π 6 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ is continuous at x=0, find the values of a and b. 5.(R-4.30-16)If f ( x)= min{| x |,| x −2|,2− | x −1|} draw the graph of f(x) and discuss the continuity and differentiability. 6. (R-4.31-18)If f ( x)= | x |−1 , then draw the graph of f(x) and fof(x) and also discuss their continuity and differentiability. Also, find the derivative of [ fof ( x)] 2 at x=3/2 7. (R-5.7-9)If y = f 2x −1 x 2 +1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ andf '( x)= sin 2 x , find dy dx . 8. (R-5.13-30) If y 3 −3 ax 2 + x 3 = 0, thenprovethaty 2 +2 a 2 x 2 y −5 =0 9. (R-5.12-25) Given F ( x)= f ( x) φ ( x) and f '( x) φ '( x)= c , prove that F '' F = f '' f + φ '' φ + 2 c fφ 10. (R-20.9-19) If P ( x)= ax 2 + bx + c andQ( x)= −ax 2 + bx + c, whereac ≠0 then show that the equation P(x).Q(x) = 0 has at least two real roots. 11. (R-20.10-20) Show that if p,q,r, and s are real numbers and pr = 2(q+s), then at least one of the equations x 2 + px + q =0andx 2 + rx + s =0 , has real roots.
Transcript of limits n quad_X-1_1.doc
Yukti Educational Services Pvt Ltd.
Assignment: Limits and Quadratic Date: 11th September, 2009
1. (R-3.19-11)If
€
Δ =tan x tan(x + h) tan(x + 2h)
tan(x + 2h) tan x tan(x + h)tan(x + h) tan(x + 2h) tan x
, find
€
limh→0
Δh2
2. (R-3.19-13)If
€
α,β are the roots of
€
ax 2 + bx + c , then evaluate
(i)
€
limx→β
1− cos(ax 2 + bx + c)(x −β )2
(ii)
€
limx→1
α
1− cos(cx 2 + bx + a)(1−αx)2
3. (R-3.21-16)Evaluate:
€
limn→∞n2 1− cos 1
n ⎛ ⎝ ⎜
⎞ ⎠ ⎟ 1− cos 1
n ⎛ ⎝ ⎜
⎞ ⎠ ⎟ 1− cos 1
n ⎛ ⎝ ⎜
⎞ ⎠ ⎟...............∞
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
4. (R-4.9-14) If
€
f (x) =
(1+ | sin x |)a
|sin x| ,− π6
< x < 0
b ,x = 0
etan 2xtan 3x ,0 < x < π
6
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
is continuous at x=0, find the
values of a and b. 5.(R-4.30-16)If
€
f (x) = min{| x |, | x − 2 |, 2− | x −1 |}, draw the graph of f(x) and discuss the continuity and differentiability.
6. (R-4.31-18)If
€
f (x) = | x | −1 , then draw the graph of f(x) and fof(x) and also discuss their continuity and differentiability. Also, find the derivative of
€
[ fof (x)]2 at x=3/2
7. (R-5.7-9)If
€
y = f 2x −1x 2 +1 ⎛ ⎝ ⎜
⎞ ⎠ ⎟and f '(x) = sin2 x, find dy
dx.
8. (R-5.13-30)
€
If y 3 − 3ax 2 + x 3 = 0, then prove that y2 + 2a2x 2y−5 = 0
9. (R-5.12-25) Given
€
F(x) = f (x)φ(x)and f '(x)φ'(x) = c , prove that
€
F ' 'F
= f ' 'f
+ φ' 'φ
+ 2cfφ
10. (R-20.9-19) If
€
P(x) = ax 2 + bx + c andQ(x) = −ax 2 + bx + c, where ac ≠ 0 then show that the equation P(x).Q(x) = 0 has at least two real roots.
11. (R-20.10-20) Show that if p,q,r, and s are real numbers and pr = 2(q+s), then at least one of the equations
€
x 2 + px + q = 0and x 2 + rx + s = 0, has real roots.
12. (R-20.10-24) If a<b<c<d, then show that the roots of the equation (x-a)(x-c)+2(x-b)(x-d)=0 are real and distinct.
