Math 234 BL1 - Exam 2 Review
Know how to: 2.1: Inspect a graph to determine the increasing/decreasing and concavity behavior of a function. 2.2: Relate the first and second derivative to increasing/decreasing and concavity behavior of a function. Fin
MTH 234 Groupwork 1 (1.19.16) - Algebra Review
Solutions
Each layer of mathematics uses the layer below it to solve its problems.
First you learn numbers, the fundamental building blocks of mathematics.
Then you learn arithmetic, in which you use numbers
MTH 234 Groupwork 6 - 3.1
1. Consider the function f (x) =
x2 + x 6
x2
(a) Use a table to numerically estimate the right-hand limit lim
x2+
x
f (x)
2.1
5.1
2.01
5.01
2.001
5.001
lim f (x) = 5
x2+
(b) Use a table to numerically estimate the left-hand limit
MTH 234 Groupwork 9
SOLUTIONS
In class we explored taking derivatives of various types of functions, and came up with
the following set of rules:
Constant Rule
Constant Multiple Rule
Power Rule
Sum/Difference Rule
Product Rule
Quotient Rule
f (x) = k, The
MTH 234 Groupwork 2 - 1.2, 2.1 (Solutions)
Definitions
A function from set A to set B is a rule that assigns each element of
set A to exactly 1 element of set B.
An even function is one where f (x) = f (x) and an odd function
is one where f (x) = f (x)
MTH 234 Groupwork 4 - 2.5, 2.6
Exponent and Lograithm Review
For a > 0, a 6= 1, and x > 0 we define the logarithm function to be:
y = loga (x) means ay = x
We choose this definition to create a function which is the inverse to the
exponential function.
1.
MTH 234 Groupwork 5
Solutions
1. Salmonella bacteria, found on almost all chicken and eggs, grow rapidly
in a nice, warm place. If just a few hundred bacteria are left on the cutting board when a chicken is cut up, and they get into the potato salad,
the
MTH 234 Groupwork 10 - 4.3
SOLUTIONS
The chain rule is a derivative law used to take derivatives of composite
functions (functions which are plugged into other functions).
Chain Rule Version I: Suppose you have a function that you write as
a composite: h(
Math 234 GW 8 - 3.4, 3.5
1. Using the limit definition of derivative, find the derivative function,
f 0 (x), of the following functions. Show all your beautiful algebra.
(a) f (x) = 2x
f (x + h) f (x)
2(x + h) 2x
= lim
h0
h0
h
h
2x + 2h 2x
= lim
h0
h
2h
=
MTH 234 Groupwork 7 - 3.2, 3.3
1. The cost to transport a mobile home depends on the distance, x in
miles that the home is moved. Let C(x) represent the cost to move a
mobile home x miles. One firm charges as follows.
Cost per mile
$4.00
$3.00
$2.50
Dista
p
(ii) g(x) =
x (Hint: Multiply by the conjugate
p
x+ h+
p
x.)
f (x + h) f (x)
h! 0
h
p
p
x+ h
x
lim
h! 0
h
p
p
p
p
x+ h
x
x+ h+ x
p
lim
p
h! 0
h
x+ h+ x
x+ h x
lim p
(Work out the details)
p
h ! 0 h( x + h +
x)
x+ h x
lim p
p
h ! 0 h( x + h +
x)
h
lim p
Mock Midterm 1A
Note: The problems on this mock midterm were not necessarily selected to allow them to
be easy to work without a calculator. The problems on the real midterm will not require a
calculator.
(1) (a) Give the definition of the derivative.
(b)
MTH 234 Groupwork 3 - 2.2, 2.3
Solutions
Understanding Parabolas as Translations
Standard Form: f (x) = ax2 + bx + c
Vertex Form: f (x) = a(x h)2 + k
1. For each of the functions below, write the equation of the new function
after performing the following