Calculus Answers

Use the mean value theorem to show that x is smaller than sin inverse x for x is greater than 0

1. A certain lot is in the shape of a parabola bounded by a slant line. The parabola is defined by the equation y = x^2 – 2 and the line at y = x. If the lot is bounded by the coordinates at
x = 2 and x = -1 then what is the area of the lot?

Apply Simpson’s rule, as well as the trapezoidal rule, for evaluating integration limit 4 to 8 dx/x to obtain an approximate value of l n 2 . [Take 4 subdivisions of the interval.]

If f(x,y) = (x^1/4+y^1/4)/(x^1/5+y^1/5), then show that x×df/dx +y ×df/dy = 1/20 f(x,y) stating the result used.

True or false, why? (1) the level curves of z=y/x are hyperbolas. (2) lim (x,y) tends to (0,0) x^2-y^2/x^2+y^2 does not exist. (3) the set {(x,y,z)/x>0,y>0,z<0} is domain in R^3. (3) the work done by the force F(x,y)=(-y,x) in moving a particle along the boundary of the ellipse 9x^2+4-y^2=36 is 6 (5) the function f(x,y,z) = e^xyz is integrable over [0,1]×[0,1]×[0,1].

Show that 2z =(ax+y)^2 + B , where a, b are arbitrary constants is a complete integral of px+qy-q^2 = 0

Solve the following equation by Jacobi’s method x^2 du/dx -(du/dx)^2 - (du/dz)^2 =0

If f(x,y)= (x^1/4+y^1/4)/(x^1/5+y^1/5),then show that x×∂f/∂x+y×∂f/∂y=1/20×f(x,y)stating the results used.

.Find the length of the arc of the hyperbolic spiral rθ = a lying between r= a and r= 2a

If y= tan inverse x , obtain an equation showing the relationship between y n+2, y n+1 and yn