Math 165 - Assignment #3
Name:
Due 7/8/2013
SHOW YOUR WORK!
This assignment covers sections 6 - 8 from chapter 2. Make sure you justify your answer clearly! Simply writing the
final result does not mean you receive full credit. You must also show you understand the procedure.
1.- Find
d3 y
dx3
where y =
x
x−1 .
2.- Find f 00 (2) for the following functions.
(a) f (x) = x cos( πx ).
(b) f (x) =
(x−1)2
x+1 .
3.- If f (x) = −x3 +
15 2
2 x
+ 42x − 100, find the value of f 00 at each zero of f 0 , that is, at each point c where f 0 (c) = 0.
3
2
4.- The positions of two objects, P1 and P2 , on a coordinate line at the end of t seconds are given by s1 = t3 − 35
2 t +
5 2
t3
211t − 9 and s2 = − 3 − 2 t + 11t − 7, respectively. When do the two objects have the same velocity?
5.- For the following function, find
dy
dx
through implicit differentiation assuming it exists.
√
π x y + 5 x = x y2 + y3
6.- Find the equation of the tangent line at the indicated point.
p
2y − 1 + x y 3 = 100,
1
1
, 1)
at the point ( −
3 3π
2
7.- Find the equation of the normal line (line perpendicular to the tangent line) to the curve 3 (x − 1)2 − y 2 =
(x − 1)2 + y 2 at the point (2, 1).
8.- Use implicit differentiation twice to find
d2 y
dx2
at (1,5) if (x + 1)2 + (y − 1)2 = 20.
9.- A metal disk expands during heating. If its radius increases at the rate of
of one of its faces increasing when its radius is 10 inches?
1
2π
inch per second, how fast is the area
10.- Water is pumped at a uniform rate of 2 liters per minute into a tank shaped like a conical or pyramidal frustum.
The tank has altitude 100 centimeters and the area of the lower and upper base is of 202 and 402 centimeters,
respectively. How fast is the water level rising when the depth of the water is 30 centimeters? Hint: √
The volume of
a conical or pyramidal frustum of altitude h and lower and upper base B1 and B2 is V = 13 h · (B1 + B1 B2 + B2 ).