Tutorial-2
Mathematics-1 (MATH F111)
Q.1 Solve the initial value problem for the
Differential equation:
d2 r
= et i − e−t j + 4e2t k
dt2
Initial conditions:
r(0) = 3i + j + 2k and
dr
dt
= −i + 4j
t=0
Q.2 A particle traveling in a straight line is located at the point (1, −1, 2) and has speed 2 at time t = 0.
The particle moves toward the point (3, 0, 3) with constant acceleration 2i + j + k. Find its position vector
r(t) at time t.
Q.3 Find the arc length parameter along the curve from the point where t = 0 by evaluating the integral
Z t
s=
|v(τ )| dτ.
0
Then, find the lenght of the indicated portion of the curve
r(t) = (et cos t)i + (et sin t)j + et k, −ln4 ≤ t ≤ 0
Q.4 To illustrate that the lenght of a smooth space curve does not depend on the parametrization you use
to compute it, calculate the length of one turn of the helix r(t) = (cos t)i + (sin t)j + tk, 0 ≤ t ≤ 2π, with
the following parametrizations.
a. r(t) = (cos 4t)i + (sin 4t)j + 4tk, 0 ≤ t ≤ π2
b. r(t) = cos 2t i + sin 2t j + 2t k, 0 ≤ t ≤ 4π
c. r(t) = (cos t)i − (sin t)j − tk, −2π ≤ t ≤ 0.
Q.5 Show that the curvature of a smooth curve r(t) = f (t)i + g(t)j defined by twice differentiable functions
x = f (t) and y = g(t) is given by the formula
κ=
|ẋÿ − ẏẍ|
3
.
(ẋ2 + ẏ 2 ) 2
The dots in the formula denote differentiation with respect to t, one derivative for each dot.
Q.6 Show that the parabola y = ax2 , a ̸= 0, has its largest curvature at its vertex and has no minimum
curvature. (Note: Since the curvature of a curve remains the same if the curve is translated or rotated, this
result is true for any parabola.)
Q.7 Find an equation for the circle of curvature of the curve r(t) = ti + (sin t)j at the point
curve parametrizes the graph of y = sin x in the xy-plane.)
1
π
2,1
. (The