1. ## 3 Calculus Questions

1. A point moves along the curve y = x^2 + 1 so that the x-coordinate is increasing at the constant rate of 3/2 units per second. The rate, in units per second, at which the distance from the origin is changing when the point has coordinates (1,2) is equal to...
2.(multiple choice) The area bounded by the curve y = 1/x+1, the axes, and the line x = e - 1 equal to: a) 1 - (1/e^2) b) ln(e - 1) C) 1 D) 2 or E) 2 (square root e + 1) - 2
3. A solid is cut out of a sphere of radius 2 by two parallel planes each 1 unit from the center. The volume of this solid is.....

Thanks for any help!

2. Originally Posted by turtle
1. A point moves along the curve y = x^2 + 1 so that the x-coordinate is increasing at the constant rate of 3/2 units per second. The rate, in units per second, at which the distance from the origin is changing when the point has coordinates (1,2) is equal to...
dx/dt = 3/2 units/s

dy/dt = 2x*dx/dt = 3x units/s

The overall rate of change will be:
ds/dt = sqrt{(dx/dt)^2 + (dy/dt)^2} units/s
(Think of this as a speed with x and y components.)

Thus
ds/dt = sqrt{9/4 + 9x^2} units/s

Thus
ds/dt(1, 2) = sqrt{9/4 + 9*1^2} units/s = sqrt{45/4} units/s = (3/2)*sqrt{5} units/s

-Dan

3. Hello, turtle!

Here's #1 . . .

1. A point moves along the curve y = x² + 1 so that the x-coordinate
is increasing at the constant rate of 3/2 units per second.

The rate, in units per second, at which the distance from the origin is changing
when the point has coordinates (1,2) is equal to __

The distance from the origin is given by: . .= .x² + y² . [1]

Since y = x² + 1, we have: . .= .x² + (x² + 1)² .= .x^4 + 3x² + 1

Differentiate with respect to time: .2z(dz/dt) .= .(4x³ + 6x)(dx/dt)

. . . . . . . . . . . . . dz . . . x(2x² + 3) . .dx
. . and we have: . --- .= . ------------- · --- . [2]
. . . . . . . . . . . . . dt . . . . . . z . . . . . dt

With the point (1,2), substitute into [1]: . .= .1² + 2² . . z = √5
And we are told that: .dx/dt = 3/2

. . . . . . . . . . . . . . .dz . . . . 1(2·1² + 3) . 3 . . . . 3√5
Substitute into [2]: .--- . = . -------------- · -- . = . -----
. . . . . . . . . . . . . . .dt . . . . . . . .√5 . . . .2 . . . . . 2