Let consider γ :I→R^2 a parametric curve
1)Prove that if a<b and γ(a)≠γ(b) then there exist t_0 ∈]a,b[
such as γ′(t_0 ) is colinear to γ(b)−γ(a)
2)Show that if γ is regular and the function f :I→R t→f(t)=∣∣γ(t)−O(0,0) ∣∣ is maximal in t_0 ∈I
Then ∣K_γ (t_0 )∣≥(1/(f(t_0 )))
Let consider α : I→R^2 a parametric curve defined as
∀ t∈I α(t)=(((t^2 −1)/(t^3 −1)) ,((2t)/(t^3 −1)))
Prove that for a,b,c∈I
α(a),α(b),α(c) are on the same lign iff abc=a+b+c+1
1 what is the order and the degree of the differential equation
(d^3 Z/dt^3 ) +((dZ/dt))^4 − y^5 =e^(−2x) .
2) 2x(d^5 y/dx^5 ) + 5x^2 ((dy/dx) )^4 − xy^2 = 0.
please kindly help me with the solutions to these question?very urgent please (1) if dy=x^3 dx. find the equation of y in terms of x
if the curve passes through (1,1)
2) Given that the volume v(t) of cell at a time t changes according to
((dV(t))/dt)= sin t, with v(t)=4. find v(t)
3) Given (dP/dt) + 3P = 0. determine P (t)
if p(0)= 4
4) Radium decomposes at a rate proportion to the amount
present. if the half−life of the radium is
1000 years. what is the percentage lost in 100 years?
5) Calculate the pressure of a gas after phase
transition at 171°K from 101.3kPa
pressure at 472°K taking R = 0.1886kJ\kgK and L = 35.73kg
calculate f(a)=∫_0 ^1 (√(x^2 +ax+1))dx and g(a)=∫_0 ^1 ((xdx)/(√(x^2 +ax+1)))
with ∣a∣<2
2)find the value of ∫_0 ^1 (√(x^2 +(√2)x+1))dx and ∫_0 ^1 ((xdx)/(√(x^2 +(√2)x+1)))