MATHEMATICS 54
Exercise Set 4
D E √ 1. Consider the space curve given by ~R(t) = cos 2t, 5t, sin 2t . (a) Find the moving trihedral at the point (1, 0, 0). (b) Give an equation of the osculating, rectifying and normal planes at the point (1, 0, 0). 2. A particle is moving with position function ~R(t) = hln t, tan(t − 1), 2ti. (a) Determine the tangential and normal components of the acceleration vector at (0, 0, 2). (b) Find the radius of curvature of the curve defined by ~R (t) at t = 1. 3. A projectile is fired from the ground at an angle of 45◦ with the ground and reached its maximum height in 1 second. What is the range (in feet) of the projectile? 4. Reparametrize the curve represented by ~R(t) = h5t, 4 sin 3t, 4 cos 3ti with respect to arclength s measured from the point (0, 0, 4) in the direction of increasing t. 5. Determine if the following limits exist. If they do, give the value of the limit. (a)
lim
( x,y)→(0,0)
x 2 − y2 x 2 + y2
(b)
lim
( x,y)→(+∞,+∞)
x+y x 2 + y2
6. Find the indicated partial derivatives. (a) Let u = rs2 ln t, r = x2 , s = 4y + 1, t = xy3 . Find
∂u . ∂y
∂z if e xy cos yz + 2 = eyz sin xz. ∂x ˆ x 2 (c) Determine f xy (1, ln 3) if f ( x, y) = eyt dt.
(b) Find
y
7. Find a Cartesian equation of the form ax + by + cz + d = 0 of the plane tangent to z =
y2 x2 + + 3 at (−4, 3, 5). 16 9
8. Approximate (1.02)3 · (0.97)2 using the linearization of z = x3 y2 at (1, 1). 9. The length, width, and height of a rectangular box are increasing at the rates of 1 in/s, 2 in/s, and 3 in/s, respectively. At what rate is the volume increasing when the length is 2 in, the width is 3 in, and the height is 6 in? (Use differentials) 10. Show that f ( x, y) = x2 + y2 is differentiable at (0, 0).