Math 165 - Assignment #2
Name:
Due 7/1/2013
SHOW YOUR WORK!
This assignment covers sections 1 - 5 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.- Following Example 2 on 2.1 find the slopes of the tangent lines to the curve y = 21 x2 − x+1 at the points where
x = −2, −1, 0, 1, 2.
2.- Suppose that the revenue R(n) in dollars from producing n iphones is given by R(n) = 0.04n − 0.0001n2 . Find the
instantaneous rates of change of revenue when n = 100 and n = 1000.
3.- The radius of a spherical balloon is increasing at the rate of
find the rate of change in the volume at time t = 5.
1
2
inch per second. If the radius is 0 at the time t = 0,
4.- Using the derivative definition on page 100, find the following derivatives. Note that here f 0 refers to the derivative
of f with respect to the variable x.
(a) f 0 (4) if f (x) =
5
2x+1 .
(b) f 0 (γ) if f (x) = αx + β.
(c) f 0 (t) if f (x) =
7x+8
x+4 .
5.- The given limit is a derivative, but of what function and at what point? Justify your answer using the limit
definition of the derivative.
(3 + h)3 + 3(3 + h) − 36
h→0
h
lim
6.- The graph of a function y = f (x) is given. Use this graph to sketch the graph of y = f 0 (x).
f (x)
y = f (x)
x
7.- Find
dy
dx
(a) y =
using the rules of section 2.3. Be specific as to which rules you’re applying
5β
3x6 .
(b) y = 2x13 −
1 −7
14 x
− πx−9 .
(c) y = (2x3 − 3x)(x2 + 5x − 1).
(d) y =
6
3x2 −4x2 .
8.- Using the rules of section 2.3, find the equation of the tangent line to y =
1
x3 −5
at the point (1, −1/4).
9.- Using what we have learned up to section 2.4, find the derivative of the following functions involving trigonometric
functions.Justify each step clearly.
(a) y =
sin(x)+cos(x)
.
sin(x)
(b) y =
1−cos(x)
.
x
(c) y = csc3 (x).
10.- Use the trigonometric identity cos(2x) = 1 − 2 sin2 (x) along with the Product Rule to find
11.- Find all points on the graph of y = cot2 (x) where the tangent line is horizontal.
d
dx 2 cos(2x).
12.- Find
dy
dx .
(a) y =
1
(2x3 −x2 +7x)8 .
(b) y = sin3
(c) y =
x2
1−2x
cos(x)
sin(2x)
4
.
.
(d) y = cos
sin cos(α x)
!
.
13.- Find the equation of the tangent line to the graph of y = 3 − x2 cos(2x) at ( π4 , 3). Where does this line cross the
x-axis?