MATH 151 Engineering Mathematics I
Week In Review
Fall, 2015, Problem Set 6 (Section 3.6, 3.7, 3.8, 3.9, and 3.10)
JoungDong Kim
dy
by implicit differentiation.
dx
√
√
(a) xy − 2x = y
1. Find
(b) y 5 + 3x2 y 2 + 5x4 = 12
1
(c) x sin y + cos 2y = cos y
(d) xy = cot(xy)
2
2. Find
dx
dy
(a) y 4 + x2 y 2 + yx4 = y + 1
(b) (x2 + y 2 )2 = ax2 y
3
3. Find an equation of the tangent line to the curve at the given point.
(a)
x2 y 2
+
= 1,
9
36
(b) y 2 = x3 (2 − x),
√
(−1, 4 2)
(1, 1)
4
4. Show that the given curves are orthogonal.
x2 − y 2 = 5,
4x2 + 9y 2 = 72
5
5. Find the tangent vector of unit length at t = 1 to the curve given by r(t) = (1 + t3 )i + t2 j.
6. The vector function r(t) represents the position of a particle at time t. Find the velocity and
speed at the given value of t.
(a) r(t) = h4 cos t, 3 sin ti ,
t = π/3
(b) r(t) = (t4 + 7t)i + (t2 + 2t)j,
t=1
6
7. Find a formula for f (n) (x).
(a) f (x) =
1
(1 − x)2
(b) f (x) =
1
3x3
7
8. (1) Sketch the curve traced by the vector equation. (2) Find r′ (t) and r′′ (t). (3) Sketch the
position vector r(t), the tangent vector r′ (t), and r′′ (t) for the given value of t.
(a) r(t) = (1 + t)i + t2 j,
(b) r(t) = h2 cos t, 3 sin ti ,
t=1
t = π/3
8
9. If x = 1 − t3 , y = t2 − 3t + 1, find an equation of the tangent line corresponding to t = 1.
10. If x = t(t2 − 3), y = 3(t2 − 3), find the point(s) on the curve where the tangnet is horizontal or
vertical.
9
11. If x2 + 3xy + y 2 = 1 and
dx
dy
= 2, find
when y = 1.
dt
dt
10
12. If a snowball melts so that its surface area decreases at a rate of 1cm2 /min, find the rate at which
the diameter decreases when the diameter is 10cm.
11
13. A spotlight on the ground shines on a wall 12m away. If a man 2m tall walks from the spotlight
toward the building at a speed of 1.6m/s, how fast is his shadow on the building decreasing when
he is 4m from the building?
12