Math 151 Week in Review
Monday Oct 18, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
8. A point traces a curve whose parametric
equations are given by the following:
x(t) =
−t3
t3
− 6t2 + 20t, y(t) =
+ 5t2 + 5.
3
3
Section 3.8
Given s(t) is a position function
(a) When is the object moving to the
right? To the left?
• v(t) = s′ (t)
(b) When is the object moving up? Down?
• a(t) = v ′ (t) = s′′ (t)
(c) Does the particle ever stop? If so, when
and where?
1. The figure below shows the graphs
of f , f ′ and f ′′ .
Identify each curve
and
explain
your
answer
choices.
(d) Find the parametric and cartesian
equations for the line tangent to the
curve at t = 3.
Section 3.10
9. A plane flying horizontally at an altitude of
1 km passes directly over a radar station at
a speed of 600 km/hr.
2. Given y = t4 − 4t3 + 2 is the position of an
object at time t (y is measured in feet and
t is measured in seconds) find:
(a) the acceleration of the object when t =
1.
(b) the position and velocity of the object
when the acceleration is zero.
√
3. Given f (x) = 3x + 1, compute f 3 (0).
4. Given p(x) = ax2 + bx + c, find a, b, and c
such that p(2) = 4, p′ (2) = 3, and p′′ (2) = 4.
5. Given f (x) = xex , what is f (50) (x)?
6. Suppose the position of a particle at time
t is given by r(t) = hsin(2t), cos(2t)i. Find
the velocity, speed, and acceleration of the
π
particle when t = .
6
Section 3.9
7. A curve is given parametrically by
x = t3 − 3t2 , y = t3 − 3t.
(a) Find the equation of the line tangent to
the curve at the point where t = −2.
(b) Find all the points on the curve where
the tangent line is horizontal or vertical.
(a) Find the rate at which distance from
the plane to the station is increasing
when it is 2 km away from the station.
(b) Suppose the plane continutes to fly at
600 km/hr but begins to climb at an
angle of 30◦ as soon as it reaches the
station. At what rate is the distance
from the plane to the radar station increasing one minute later?
10. A filter in the shape of a cone is 10cm high
and has a radius of 20 cm at the top. A solution is poured in at the rate of 2 cm3 /min.
Find the rate at which the height of the
residue is increasing when the height is 2
cm.
11. The length of a rectangle is increasing at a
rate of 2 feet per second, while the width
is increasing at a rate of 1 foot per second.
When the length is 5 feet and the width is
3 feet, how fast is the area increasing? How
fast is the perimeter increasing?
12. A water trough is 8 m long and has a crosssection in the shape of an isosceles trapezoid
that is 20 cm wide at the bottom, 60 cm
wide at the top, and has height 40 cm. If
the trough is being filled with water at the
rate of 0.1 m3 /min how fast is the water
level rising when the water is 10 cm deep?
13. A ladder 10 feet long rests against a vertical
wall. If the bottom of the ladder slides away
from the wall at a speed of 2 ft/s, how fast
is the angle between the top of the ladder
and the wall changing when the angle is π/4
radians?