Calculus 3 – Help!

This is a take home test for calculus 3! I’m having trouble on it, and I have no idea where to start! I’m willling to pay! First come first serve! So please let me know when you have gotten all or some done! Thanks!

Harold Washington College

Math

2

09

Test 2

Name: Date: 2/2

1

/2012

(1) Show that the curvature of the helix r(t) =< a cos t,a sin t,bt > with a,b 6= 0 is

a

a2 + b2
, and also that the unit tangent vector of this curve makes a constant angle

with the z-axis whose value is θ = cos−

1

b

√
a2 + b2

.

(2) Show that the representation of r(s) =<

s +
√

s2 + 1

2

,

1

2(s +

√
s2 + 1)

,

√
2(ln(s +

√
s2 + 1)

2
>

is the arc length parametrization of the curve. In other words, show that

∥∥∥∥dr(s)ds
∥∥∥∥ = 1.

Hint. Define u = s +
√
s2 + 1 and apply the chain rule.

(

3

) Show that the parametrization of the curve r(t) =< et cos t,et sin t,et > with

respect to the arc length measured from the point t = 0 in the direction of increasing

t is r(s) =

(
s +
√

3

√

3

)
< sin

(
ln(

s +
√

3
√

3
)

)
, cos

(
ln(
s +
√
3
√
3
)

)
, 1 >

(

4

) Show that the curve r(t) =< t, 1 + t

t
,
1 − t2

t
> lies in a plane.

(5) Show that the curvature along the curve r(t) =< t− sin t, 1 − cos t, t > is

|κ| =

√
1 + 4 sin4 t

2√
(1 + 4 sin2 t

2
)3

1

(6) Show that the torsion along the curve r(t) =< t− sin t, 1 − cos t, t > is

τ = −
1

1 + 4 sin4 t
2

(7) For the curve r(t) =< 3t−t3, 3t2, 3t+t3 > use the formulas |κ| =
‖r′ (t) ×r′′ (t)‖
‖r′ (t)‖3

and τ =
|r′ (t) ×r′′ (t) ·r′′′ (t) |
‖r′ (t) ×r′′ (t)‖2

to show that |κ| = τ =
1

3(1 + t2)2
.

(8) Show that the torsion of the helix r(t) =< a cos t,a sin t,bt > with a > 0 and

b 6= 0 is
b

a2 + b2
.

(9) (a) Show that the the equation of the osculating plane for the curve r(t) =<

t,t2, t3 > is 3x− 3y + z = 1.

(10) For the helix r(t) =< a cos t,a sin t,bt > with a,b 6= 0 and t = π
3
, find the

following.

(a) The tangent vector, the normal vector, and the binormal vector.

(b)The tangent line, the normal line, and the binormal line.

(c) The rectifying plane, the normal plane, and the osculating plane.

(11) Two particles travel along the space curves r1(t) =< t,t 2, t3 > and

r2(t) =< 1 + 2t, 1 + 6t, 1 + 14t >. Do the particles collide? where? Do their paths

intersect? where?

2

(12) Find the intersection point of the helix r1(t) =< cos t, sin t, t > and the curve

r2(t) =< (1 + t), t 2, t3 >, and find the angle of intersection of these curves at that

point.

(13) Evaluate the following limit

lim
t→0

⟨
sin 10t

sin 3t
, csc t− cot t,

et −e−t − 2t
t− sin t

⟩
=

(14) At what point do the curves

r1(t) =
⟨
t, 1 − t, 3 + t2

⟩
and r2(s) =

⟨
3 −s,s− 2,s2

⟩
intersect?

(15) For the space curve r(t) =< t +

t3

3
, t−

t3

3
, t2 > determine (a) the unit tangent

vector, (b) the curvature, (c) the principal normal, (d) the binormal vector, (e) the

torsion.

(16) Find the length of the curve r(t) =< 3 cos t, 4 cos t, 5 cos t > from t = 0

to t = 2π.

(17) Evaluate the following integrals

(a)

∫
< sin(ln x),

√
t2 − 9
t3

,
t2 − t + 6
t3 + 3t

> dt =

(b)

∫
<

1√
(1 + t2) ln(x +

√
1 + t2)

,
t−
√

tan−1 t

1 + 4t2
,

1
√
t(1 + t)

> dt =
3

(18) If r(t) =< 3x2(1 + x)

4
3

4
−

9x(1 + x)
7
3

1

4
+

27(1 + x)
10
3

140
,

(2×2 − 1)
√
x2 + 1

3×3
,

2(3ax− 2b)
√

(ax + b)3

15a2
>, then

r
′
(t) =< x2(1 + x)

1
3 ,

1

x4
√
x2 + 1

, x
√
ax + b >.

(19) If r(t) =< x2

4
+
x sin 2x

4
+

cos 2x

8
,

sin3 x

3
−

sin5 x

5
, x3 cos x−3×2 sin x−6x cos x+

6 sin x >, then

r
′
(t) =< x cos2 x, sin2 x cos3 x, −x3 sin x >.

4