Rowan Alyssa Arave
Assignment HW6 Derivatives III due 02/16/2012 at 11:00pm MST
1. (1 pt) From Rogawski ET 2e section 3.5, exercise 9.
Calculate the second and the third derivative of y = 9x − 4x
y00 =
000
y =
math1210spring2012-3
8. (1 pt)
Consider the curve x5 + 5xy + y5 = 7
The equation of the tangent line to the curve at the point (1, 1)
has the form y = mx + b where
and b =
m=
2. (1 pt) Suppose that the equation of motion for a particle
(where s is in meters and t in seconds) is
9. (1 pt) For the equation given below, evaluate y0 at the point
(−1, −1).
s = 2t 3 − 3t
4y2 − 2x2 − 2 = 0.
.
(a) Find the velocity and acceleration as functions of t.
Velocity at time t =
Acceleration at time t =
(b) Find the acceleration after 1 second.
Acceleration after 1 second:
y0 at (−1, −1) =
10. (1 pt) For the equation given below, evaluate y0 at the
point (−2, −2).
(c) Find the acceleration at the instant when the velocity is 0.
Acceleration:
(2 + y)3 + 2y = x − 2.
3. (1 pt) Let f (x) = 9x2 /(2 − 5x). Then
y0 at (−2, −2) =
f 0 (x) =
11. (1 pt) From Rogawski ET 2e section 3.10, exercise 11.
Calculate the derivative of y with respect to x.
and
f 00 (x) =
x2 y + 2xy2 = x + y
dy
dx =
4. (1 pt) Let f (x) = (3 − 5x2 )6 .
Then f (0) is
12. (1 pt) Find the slope of the tangent line to the curve (a
lemniscate)
f 0 (0) is
is
f 000 (0) is
2(x2 + y2 )2 = 25(x2 − y2 )
f 00 (0)
at the point (−3, 1).
√
5. (1 pt) Let f (x) = x2 + 3.
Then f 0 (x) is
f 0 (2) is
,
f 00 (x) is
and f 00 (2) is
m=
13. (1 pt) Find the slope of the tangent line to the curve
p
p
1x + 2y + 2xy = 10.7
at the point (6, 4).
Answer:
7. (1 pt)
Find the given derivative by finding the first few derivatives
and observing the pattern that occurs.
14. (1 pt) If f (x) = arcsin3 (6x + 7), then
f 0 (x) =
D103 cos 7x
Note: The inverse of sin(x) can be entered as arcsin(x) or
asin(x)
Note: Read ”Dn f (x)” as ”the nth derivative of f (x)”
1
(Show the student hint after 0 attempts: )
par f Hint: The law of cosines for a triangle is [ c2 = a2 + b2 −
2abcos(heta)]where(heta)istheanglebetweenthesideso f lengthaandb.
15. (1 pt) Let
f (x) = x3 tan−1 (5x)
f 0 (x) =
NOTE: The WeBWorK system will accept arctan(x) but not
tan−1 (x) as the inverse of tan(x).
19. (1 pt) Let A be the area of a circle with radius r. If
find dA
dt when r = 1.
16. (1 pt) Gravel is being dumped from a conveyor belt at a
rate of 40 cubic feet per minute. It forms a pile in the shape of
a right circular cone whose base diameter and height are always
equal. How fast is the height of the pile increasing when the
pile is 15 feet high?
dr
dt
= 4,
Answer:
20. (1 pt) At noon, ship A is 30 nautical miles due west of
ship B. Ship A is sailing west at 16 knots and ship B is sailing
north at 24 knots. How fast (in knots) is the distance between
the ships changing at 3 PM? (1 knot is a speed of 1 nautical mile
per hour.)
Recall that the volume of a right circular cone with height h
and radius of the base r is given by
1
V = πr2 h
3
Answer:
(Show the student hint after 1 attempts: )
Note: See number 23 on pg 261 for a picture of this.
17. (1 pt) A price p (in dollars) and demand x for a product
are related by
Note: Draw yourself a diagram which shows where the ships
are at noon and where they are ”some time” later on. You will
need to use geometry to work out a formula which tells you
how far apart the ships are at time t, and you will need to use
distance = velocity ∗ time to work out how far the ships have
travelled after time t.
2x2 − 1xp + 50p2 = 26000.
If the price is increasing at a rate of 2 dollars per month when
the price is 20 dollars, find the rate of change of the demand.
21. (1 pt) The length of a rectangle is increasing at a rate of
7cm/s and its width is increasing at a rate of 4cm/s. When the
length is 40cm and the width is 10cm, how fast is the area of the
rectangle increasing?
Answer (in cm2 /s):
Rate of change of demand =
18. (1 pt) A plane flying with a constant speed of 24 km/min
passes over a ground radar station at an altitude of 14 km and
climbs at an angle of 20 degrees. At what rate, in km/min is the
distance from the plane to the radar station increasing 2 minutes
later?
22. (1 pt) A street light is at the top of a 17.0 ft. tall pole.
A man 6.5 ft tall walks away from the pole with a speed of 4.5
feet/sec along a straight path. How fast is the tip of his shadow
moving when he is 47 feet from the pole?
Your answer:
Hint: Draw a picture and use similar triangles.
Answer:
c
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