Prove that r'' = r' dr'/dr ? where ' represents derivative wrt time.
Prove that r'' = r' dr'/dr ? where ' represents derivative wrt time.
Prove ▽f(r) = (f,(r)/r)r, where r = Vx2 + y2 + ~2 and f'(r)-df/dr is assum ed to exist.
Find the G.S. of the DE: xy' + y = 3x2 Prime denotes derivative WRT X. (Hint: guess a P.S. yı = Axa)
Find the particular antiderivative of the following derivative that satisfies the given condition. dR/dt=111/ t4, R(1)=25
d 4. Use the definition of the derivative to show that dr (3x2 - 4) = 62. You must use the definition of the derivative to receive credit. The height A (in feet) of an object shot into the air from a tall building is given by the function 10) = 510 + 801 – 1662, where t is the time elapsed in seconds. (a) Write a formula for the velocity of the object as a function of time t....
1. The position of a particle is described as r=xi+yj, where i and j are the coordinate unit vectors in 2D Cartesian coordinates a?d x and y are the coordinates of a particle. The velocity can be calculated as v=dr/dt. Find an expression for v, by taking the derivative of r with respect to time. 2. a=2.00 t, where t is the time, in seconds, and a is the acceleration in meters per second squared. s=0 and v=0 at t=0....
Prove that the total gravitational potential energy can be written as ρ(z)φ(z)dr, where ρ(z) and φ(z) are the density and gravitational poi.(.mi.ial ai. locai;ion x in space, Π iptXtively Use the above formula for W to obtain the total gravitational energy of the Plummer sphere with the graviational potential: φ(r) = (r2+p)
1. Let ơ E Aut(R), where R denotes the field of real numbers. a) Prove that if a > b then σ(a) > σ(b) ( . (b) Prove that o is a continuous function. (c) Prove that ơ must be the identity function. Therefore Aut(R)-(1). (see problem 7 on pg. 567 for more details for each step). 1. Let ơ E Aut(R), where R denotes the field of real numbers. a) Prove that if a > b then σ(a) >...
*3. Suppose that f:R2 >R is such that .) dy dr-1 and R LR l i f (x,y) dx dy--1. What range of values can I Lndrn2 possibly have? (Here R LR m2 refers to the Lebesgue measure in R2) Prove your assertion.
Show that −1 ≤ r ≤ 1, where r represents Pearson's correlation coefficient. Note : Show the formulas and calculations used.
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' : [0,00) + R is also differentiable, with f(0) = f'(0) = 0. Suppose also that [f"(x) < 1 for all € [0, 0). a) Show how the Mean Value Theorem can be used to prove that f(x) <r? for all x € (0,00). b) Show how the Cauchy Generalized MVT can be used to prove a stronger statement: |f(7) < 2 for all 2...