Test III
Math 261, Section 1
Instructor: Sam Watson
March 27, 2009
Name:
Please solve the following problems in the space provided. You may use a scientific calculator, but you will not need one. You must show work to support your answer.
1. (15 points) A particle moves on a coordinate line so that it has position s(t) = t3 − t2
on the interval 0 ≤ t ≤ 2, with s measured in meters and t in seconds.
(a)
Find the particle’s displacement and average velocity over the interval [0, 2].
∆x =
(b)
vave =
Find the particle’s speed and acceleration at the time t = 0.5.
speed =
(c)
a=
When does the particle change direction?
2. (12 points) Calculate the following derivatives.
(a)
d
dx
(b)
d
dx
(c)
d
dx
sin( x )
sec( x )
cot( x )
3. (15 points) Calculate the following derivatives.
(a)
d
3
20
(9x − 1)
dx
(b)
d √
4
x−1
dx
(c)
d
dx
sin(5x/2)
4. (8 points) For each value of x, identify whether the function shown below is differentiable at that value of x. (Circle “Y” if it is differentiable, and “N” if it is not). In each case,
either state why the function is not differentiable, or (if it is differentiable) estimate the
derivative.
5
4
3
2
1
−2
−1
0
−1
−2
−3
1
2
3
4
5
6
7
x
diff? if not, why? if so, estimate.
−1 YN
0
YN
1
YN
2
YN
3
YN
4
YN
5
YN
6
YN
5. (10 points) Find the slope of a line tangent to the graph of x3 − y2 + 7 = xy at the point
(2, 3).
d2 y
6. (10 points) Find 2 where y is related to x implicitly by the equation
dx
x2 + y2 = 4.
7. (8 points) A spherical balloon is being inflated at a rate of 20 cubic centimeters per
second. What is the rate of change of the radius when the radius is 12 centimeters? (Recall
that the volume of a sphere is given by V = 43 πr3 .) You may express your answer as a
fraction in terms of π or as a decimal.
8. (8 points) A 10 foot ladder is slipping down a wall in such a way that the rate of change
of x (the distance from the wall to the ladder’s point of contact on the ground, see figure)
is 2.4 meters per second. What is the rate of change of θ when y = 6?
y
x
θ
√
3
9. (10 points) Find the linearization
L
(
x
)
of
the
function
f
(
x
)
=
x about x = 8. Use
√
3
your answer to estimate 7.5. Express your answer as a mixed number.
L( x ) =
√
3
7.5 ≈
10. (4 points) In this problem, we will differentiate the inverse of the sine function. Recall
that the function arcsin( x ) is the inverse function of sine, i.e. it takes an input x from
[−1, 1] and outputs the angle between −π/2 and π/2 whose sine is x. In other words, it
“undoes” sine. Notice that if y = arcsin( x ), then sin(y) = x. Use this equation to find y0 ,
and then express your answer in terms of x.