MATHEMATICS 54 Exercise Set 4

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) = 〈ln t, tan(t− 1), 2t〉.

(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 1second. What is the range (in feet) of the projectile?

4. Reparametrize the curve represented by ~R(t) = 〈5t, 4 sin 3t, 4 cos 3t〉 with respect to arclength s measured fromthe 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)

x2 − y2

x2 + y2(b) lim

(x,y)→(+∞,+∞)

x + yx2 + y2

6. Find the indicated partial derivatives.

(a) Let u = rs2 ln t, r = x2, s = 4y + 1, t = xy3. Find∂u∂y

.

(b) Find∂z∂x

if exy cos yz + 2 = eyz sin xz.

(c) Determine fxy(1, ln 3) if f (x, y) =ˆ x

yeyt2

dt.

7. Find a Cartesian equation of the form ax + by + cz + d = 0 of the plane tangent to z =x2

16+

y2

9+ 3 at (−4, 3, 5).

8. Approximate (1.02)3 · (0.97)2 using the linearization of z = x3y2 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 6in? (Use differentials)

10. Show that f (x, y) = x2 + y2 is differentiable at (0, 0).