Differentiation — Page 1 — Tinkutara Math Forum

1

ans

0

cmts

Question 228758

1

ans

0

cmts

Question 227073

1

ans

0

cmts

A Segment of a sphere has radius r and maximum height h.Prove that its volume ((𝛑h)/6)(h^2 +3r^2 )

1

ans

0

cmts

Differentiate 20sin (x+3)cos (x^2 /2)

1

ans

0

cmts

Differentiate x^x^x

0

ans

0

cmts

Question 226340

0

ans

0

cmts

Question 225651

2

ans

0

cmts

∫ x^x dx

0

ans

0

cmts

for all n ∈ Z , Show that τ ( ϕ ( n )) ≥ ϕ (τ (n ))

1

ans

0

cmts

L { tsin((√t) )}=?

1

ans

0

cmts

lim_(x→0) ((2log(1+x)−((x(3x+2))/((x+1)^2 )))/x^3 )

0

ans

1

cmts

L 60 1. y=x^2 +5x Find the equation of a line with the slope of 7 that touches y=x^2 +5x. [Sol.] Let f(x)=x^2 +5x Then f′(x)=2x+5 Since 2x+5=7⇒x=1 then the point is (1, 1^2 +5∙1)=(1, 6) So the equation of a line is y−6=7(x−1)⇒y=7x−1 2. y=ax^2 +bx (2, 2) a, b Find the values of constants a, b that the slope of the line that touches (2, 2) and y=ax^2 +bx is 5. [Sol.] Let f(x)=ax^2 +bx Then f′(x)=2ax+b and build two equations to solve for a and b { ((f(2)=a∙2^2 +b∙2=4a+2b=2)),((f′(2)=2a∙2+b=4a+b=5)) :} Solving for a, b gives a=2, b=−3 3. y=x^3 −3x^2 −1 Find the equation of a line that is drawn, touches y=x^3 −3x^2 −1. [Sol.] The line of the equation is y−(a^3 −3a^2 −1)=(3a^2 −6a)(x−a) Calculating gives y=(3a^2 −6a)x−(3a^2 −6a)a+(a^3 −3a^2 −1) y=(3a^2 −6a)x+(−3a^3 +6a^2 )+(a^3 −3a^2 −1) y=(3a^2 −6a)x+(−2a^3 +3a^2 −1) −2a^3 +3a^2 −1=0 a=−(1/2) or a=2 ...a=−3x, a=((15)/4)x

1

ans

0

cmts

if lim_(x→0) (((sin2x)/x^3 )+(a/x^2 )+b)=1 find a and b without using LHopial rule

0

ans

0

cmts

Prove: ∫_0 ^∞ ((cos(nx)cos(p arctan x))/((1+x^2 )^(p/2) ))=(π/2) ((n^(p−1) e^(−n) )/(Γ(p))) (p>0)

0

ans

1

cmts

(1) [ax^3 +bx^2 +cx+d]_x ′ (2) [x(x−a)^2 ]_x ′ (3) [(x^2 −x)(x^2 −4)]_x ′ (4) [(x+2)(x−5)(x−1)]_x ′

0

ans

1

cmts

lim_(x→∞) 4x+(√(16x^2 −3x))

1

ans

0

cmts

y=(8^x /((in8)^3 )) find (d^6 y/dx^6 )

0

ans

2

cmts

name the following compound

0

ans

1

cmts

If V be a function of x and y, prove that (∂^2 V/∂x^2 )+(∂^2 V/∂y^2 )=(∂^2 V/∂r^2 )+(1/r) (∂V/∂r)+(1/r^2 ) (∂^2 V/∂θ^2 ), where x=r cos θ , y=rsin θ

0

ans

0

cmts

Find the general solution of the differential equation x^2 (d^3 y/dx^3 ) + x(d^2 y/dx^2 )−6(dy/dx)+6(y/x)=((x ln x+1)/x^2 ),[x>0]

1

ans

0

cmts

(211) Find the derivative of Δx, where Δx= determinant (((f_1 (x)),(φ_1 (x)),(Ψ_1 (x))),((f_2 (x)),(φ_2 (x)),(Ψ_2 (x))),((f_3 (x)),(φ_3 (x)),(Ψ_3 (x)))) and f_1 (x) ,f_2 (x), f_3 (x),φ_1 (x), etc. are different functions of x.

1

ans

0

cmts

prove that (π/(16)) < ∫_0 ^( 1 ) (√((x(1−x))/(sin(πx)+cos(πx)+2))) dx<(π/8)

2

ans

0

cmts

If f(x,y)=(((x^2 +y^2 )^n )/(2n(2n−1)))+xφ((y/x))+Ψ((y/x)), then using Euler′s theorem on homogenous functions,show that x^2 ((δ^2 f)/(δx^2 ))+2xy((δ^2 f)/(δxδy))+y^2 ((δ^2 f)/(δy^2 ))=(x^2 +y^2 )^n

1

ans

0

cmts

Prove that; (d/dx) (((sin^( 2) x)/(1+cot x)) + ((cos^( 2) x)/(1+tan x))) = −cos 2x

0

ans

1

cmts

Question 219451

3

ans

2

cmts

Question 219243

2

ans

0

cmts

ζ(α)=Σ_(n=1) ^(+∞) (1/n^α )

1

ans

0

cmts

Question 216800

4

ans

0

cmts

Prove that ^3 (√((√5)+2)) −^3 (√((√5)−2)) =1

1

ans

1

cmts

without using LHopital rule evalute lim_(x→0) ((ln(1−x)−sin(x) )/(1−cox^2 (x)))

1

ans

0

cmts

(dx/dx)

0

ans

0

cmts

u_n = Σ_(k=n+1) ^(2n) (1/k) and v_n = Σ_(k=n) ^(2n−1) (1/k) • show that u_n and v_n are adjacent use ln(x+1) ≤ x and x≤−ln(1−x) and • show that u_n ≤ Σ_(k=n+1) ^(2n) (ln(k)−ln(k−1)) hence deduce that u_n ≤ ln2 • show that v_n ≥ Σ_(k=n) ^(2n−1) (ln(k+1)−ln(k)) hence deduce that v_n ≥ln2