[" For the one-dimensional motlon,described by "x=t-sin t],[x(t)>0" for all "t>0" ,"],[v(t)>0" for all "t>0" ."],[[mu(t)>0" for as "l=0" ,"],[" vasetween "0" sand "2" ."]]

A function y=f(x) is given by x=(1)/(1+t^(2)) and y=(1)/(t(1+t^(2))) for all t>0 then f is

For the function f(x)=int_(0)^(x)(sin t)/(t)dt where x>0

A particle executes the motion described by x(t) = x_0 (1-e^(-gammat) , t le 0, x_0 > 0 . Find maximum and minimum vlaues of x(t), v(t), a (t). Show that x (t) and a (t) increase with time and v (t) decreases with time.

if f(x)oversetxunderseto(int)e^(t^2)(t-2)(t-3)dt for all x in(0,oo) , then

A particle executes the motion described by x (t) =x_(0) (w-e^(gamma t) , t gt- 0, x_0 gt 0 . (a) Where does the particle start and with what velocity ? (b) Find maximum and minimum values of x (t0 , a (t). Show that x (t) and a (t0 increase with time and v (t) decreases with time.

A long taut string is plucked at its centre. The pulse travelling on it can be described as y (x, t) = e^(-(x+2t)^(2))+e^(-(x-2t)^(2)) . Draw the shape of the string at time t=0 , a short time after t=0 and a long time after t=0 .

For x , t in R let p_(t) (x)= ( sin t) x^(2) - (2 cost ) x + sin t be a family of quadratic polynomials in x with variable coefficients. Let A(t) = int_(0)^(1) p_(t) ( x)dx , Which of the following statements are true? (I) A(t) lt 0 for all t. (II) A(t) has infinitely many critical points. (III) A(t) = 0 for infinitely many t. (IV) A'(t) lt 0 for all t.

If f(x) = int_(0)^(x) e^(t^(2)) (t-2) (t-3) dt for all x in (0, oo) , then