Math 1210-001 Lab 4
Spring 2016; due February 18 at the beginning of lab
Name _____________________________________________UID______________________________
1) (25 points) Chain rule. Some of these problems will help with your WebWork assignment due this
Friday.
1a) Compute Dx 4 x2 K 3
K5
.
1b) We'll see in section 2.7 that the power rule
Dx xn = n xn K 1
holds for every rational number n, not just for the integers. For example
Dx
x = Dx x
1
2
1
1 K
1
= x 2 =
2
2 x
1
. This is the sort of derivative that we were able to compute using the limit
2
definition of derivative, but as you recall it was quite messy. The chain rule gives a much better way to get
the shortcut formula: Let g x = x , for x O 0. Then we have the identity
g x 2 = x for all x O 0 .
Thus
Dx g x 2 = Dxx.
1
Use the chain rule to expand the equality above, solve for g# x and deduce the power rule with n = .
2
is the power rule with n =
1
from 1b, together with the chain rule. (This
2
problem is similar to #13, 15 on your WebWork assignment.)
1c) Find Dx
4 x2 K 3
, using the power rule with n =
1d) Give another explanation of the quotient rule shortcut, by writing
f x
= f x $g x K1
g x
and using the product rule and the chain rule to compute the derivative that way.
2) (20 points) Trig function derivatives and identities.
2a) Use the quotient rule and the derivative shortcuts for sin x and cos x , to find the derivative
shortcut for Dx cot x .
2b) The double angle identity for cos arises from the addition angle formula expansion for cos x C x :
Since
cos a C b = cos a cos b K sin a sin b
we can set a = b = x to get the double angle identity
cos 2 x = cos2 x K sin2 x .
Differentiate both sides of this identity using the chain rule on each side, and obtain the double angle
formula for sin 2 x .
3) (15 points) Applied trig and chain rule. A wheel centered at the origin and of radius 15 cm is rotating
counterclockwise at a constant rate of 3 revolutions per second. A point P is at 15, 0 at time t = 0.
3a) What are the coordinates of P at time t seconds?
3b) At what rate is P rising (or falling) at time t = 0.5 seconds?
4) (25 points) Derivative rules review. Below are graphs of functions f x and g x . I copied them out
of section 2.5 of our text, from page 123. Use the graphical information and derivative rules to
approximate the indicated expressions. Assume that each function value and derivative value you require
is either an integer or half an integer
4a)
fK2 g # 5
4b)
fg # 5
4c)
g+f # 0
4d)
f+g # 1
4e)
f
g
4f)
f 3 # 5 , i.e. Dx f x
4g)
gK3 # 1 , i.e. Dx g x
# 2
3
, evaluated at x = 5.
K3
, evaluated at x = 1.
5) (15 points) Higher order derivatives. The first derivative function f# x tells us the slope function for
the graph of the function f x . So, for example, on intervals where f# x O 0 the slopes on the original
graph are positive, so the values of f x are increasing as x increases. The second derivative function
f## x = Dxf# x tells us the rate of change of the slope function, which also gives us information about
the shape of the graph of f. In the following pictures determine which curves are the graphs of the original
function, its first derivative, and its second derivative.
5a)
5b)
5c)
(Optional) Extra Credit:
1) (20 points) Pythagorean Theorem. When a region of the plane is uniformly scaled by a factor of R, then
the area of the rescaled region is R2 times the area of the original region. One way to understand this fact
is to think of the original region filled with (possibly infinitely many) rectangles of varying sizes, with the
area of the region equal to the sum of the areas of the rectangles. When the region is expanded by the
factor of R, the side lengths of each of those rectangles is multiplied by R , so that the product of the side
lengths is multiplied by R2 . Since this is true for all the rectangles, the area of the rescaled region will be
R2 times the area of the original one.
Exercise P) Use similar triangles in the picture below (There are 4 of them!) and the way area changes
under rescaling, to verify the Pythagorean Theorem
a2 C b2 = c2
for right triangles.
(Optional) Extra Credit:
2) (40 points) Angle addition formulas for sine and cosine. These arise as a special case of the formulas
one uses to describe how to rotate points in the plane, about the origin. For convenience, consider a point
x, y in the first quadrant, and an acute rotation angle q. In the diagram below, the star point coordinates
are represented vertically as
x
y
and represented with the star symbol.
Exercise R1) The goal of this exercise is to find the coordinates of the second star symbol, which shows
where the original star point ends up after rotation by angle θ about the origin. Find the final coordinates in
stages, as indicated below. Note that the original solid rectangle is rotated to the rectangle with dashed line
edges.
a) The solid circle is originally located at
x
. Find the coordinates of the solid circle it is rotated to.
0
Note that the part of the xKaxis between the origin and the original solid circle is rotated to a dashed line
segment at angle q from the x K axis. This segment is the hypotenuse of a right triangle that has its base
along the x K axis and and vertical side. (Triangle not shown.) Find the side lengths of that triangle, then
use geometry to deduce the coordinates of the rotated solid circle.
b) Similarly, find the coordinates of the rotated solid square.
c) The horizontal and vertical displacements from the rotated solid circle to the rotated star are the same as
those from the origin to the rotated solid square, because the shorter dashed line segments are parallel and
of equal length. Use this information to find the coordinates of the rotated star from those of the rotated
solid circle and square.
Exercise R2) If, in the original diagram, the star point
x
y
was on the unit circle at an angle a from
positive horizontal axis, i.e. if
x
y
=
cos a
sin a
,
then after rotation by the angle q the new star point will be at an angle a C q from the positive horizontal
axis, so it will have coordinates
cos a C q
.
sin a C q
Compare these coordinates with the ones you found in R2 to deduce the angle addition formulas
cos a C q = cos a cos q K sin a sin q
sin a C q = cos a sin q C sin a cos q .