Math 1210-001 Lab 6
Spring 2016; due March 3 at the beginning of lab
Name _____________________________________________UID______________________________
This lab covers sections 2.9, 3.1-3.2. There are also two related rates problems, since that topic takes a
while to digest and you asked for us to revisit it.
1) (16 points) Given the following functions, find dy and then compute its numerical value at x = 0, for
dx = 0.1.
1a) y = x2 C x K 3.
1b) y = sin x C cos x
1c) y =
10
5
1 K x2
1d) y = x2 K 6 x C 2
1Cx .
2) (14 points) Find an approximation for the following expressions using differentials. Compare with the
"exact" calculator values after you are done.
2a)
2b)
15.99
3
125.5
3) (10 points) A right circular cylinder is exactly 12 inches long, and its diameter is measured as 6 inches,
G 0.005 inches.
3a) Calculate its estimated volume, together with an error estimate based on differentials. Make sure to use
correct units.
3b) The relative error in a measurement is defined as the absolute value of the possible error (called
absolute error), divided by the actual value. (In this case, the actual value would be the estimated volume.)
Find the relative error in this case, expressed as a percent.
4) (25 points) Let f x =Kx3 C 3 x.
4a) Find the critical points and deduce the extreme values for f on the interval K2, 3 .
4b) Now consider the natural domain of all real numbers, for the same cubic f x . Use first derivative
information to find the intervals on which f x is increasing, and on which it is decreasing. (We did an
example like this on Tuesday February 23. The topic is covered in depth in section 3.2, and in our notes
from Wednesday which we finish on Friday.)
4c) Use second derivative information to find the intervals on which f x is concave up and concave
down. (We did an example like this on Tuesday February 23. The topic is covered in depth in section 3.2,
and in our notes from Wednesday which we will finish on Friday.)
4d) Use the information from 4abc to sketch the graph of f x =Kx3 C 3 x on the interval K2, 3 . Your
graph should reflect the information about where f is increasing and decreasing, as well as where it is
concave up and concave down.
10
K2
0
K1
1
2
x
y K10
K20
3
5) (15 points) Find, if possible, the maximum and minimum values for the specified functions and
domains. Use critical point analysis.
5a) f x = 6 x K 4 x, 0 % x % 4.
5b) f q = cos2 2 q , 0 % q % p.
5c) f x =
2x
2
x C4
, x R 0.
(Related rates revisit)
6a) (easier) (10 points) Suppose x t and y t are two functions related by the identity
x2 y = 100.
(For example, this could happen if there was a rectangular prism with height y t , and with square cross
section having side lengths x t , if the dimensions of the prism were changing in a way that kept the prism
volume equal to 100.)
Find x# t at an instant when y = 4, y# t =K1.
6b) (harder)(10 points) A light in a lighthouse 1 kilometer offshore from a straight shoreline is rotating at
1
2 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point
2
kilometer from the point oposite the lighthouse?