Partial Fractions
Partial Factions
In algebra we learn how to find the common denominator of two expressions and then combine
like terms. Partial fractions are all about going backwards.
Example 1
Consider the rational function
3x + 2
3x + 2
P(x) =
=
2
x
The Derivative of the Natural Logarithm
Derivation of the Derivative
Our next task is to determine what is the derivative of the natural logarithm. We begin with the
inverse definition. If
y = ln x
then
ey = x
Now implicitly take the derivative of both si
Advanced Substitution
Substituting x
Example
Find
Solution
Notice first that the square root is what makes this problem difficult. Hence we let
u = x-1
du = dx
This seems not to get rid of the x term, but notice that if we add 1 to both sides, we get
x =
MATH 116 PRACTICE MIDTERM 2
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Integrate the following functions
A)
Solution
We use integration by parts here
u =
Numerical Integration
Trapezoidal Rule
As we saw before, often we can not evaluate a definite integral because it is too difficult or
because we do not have an algebraic expression for the integrand. We saw that the Midpoint
estimate was one way to approx
Name
MATH 116 PRACTICE MIDTERM 1
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Find the derivative of the following functions
A. f(x) = ln(ln x)
Solution
We
The Derivative of the Exponential
Derivation of the Derivative
Before we derive the derivative, we need a few preliminary remarks. Recall that the number e is
defined by the limit as x approaches 0 of
f(x) = (1 + x)1/x
If we substitute x for x and let x b
Math 116 Practice Midterm 3
Problem 1
Integrate the following. If the integral is improper and diverges, state this. If the integral is
improper and converges evaluate it.
A.
Solution
We use partial fractions. We write
x-1
A
=
x2 - 4
B
+
x-2
x+2
Clearing
Exponential Growth
The Exponential Differential Equation
An equation is called a differential equation if it is an equation that contains derivatives. In this
section, we will consider differential equations of the type
dy/dt = ky
This differential equati
Integration by Parts
Derivation of Integration by Parts
Recall the product rule:
(uv)' = u' v + uv'
or
uv' = (uv)' - u' v
Integrating both sides, we have that
uv' dx
=
=
uv -
(uv)' dx -
u' v dx
u' v dx.
Theorem: Integration by Parts
Let u and v be differe
LOGARITHMS
The inverse of the exponential function- The Natural Logarithm
The graph of
y = ex
clearly shows that it is a one to one function, hence an inverse exists. We call this inverse
the natural logarithm. and write it as
y = ln x
Below is the graph
Math 116 Practice Final Key
Please work out each of the given problems. Credit will be based on the steps that you show
towards the final answer. Show your work
Problem 1 Find the derivative of the following functions
A. f(x) = ln(ln(1 - x)
Solution
We us
Integrals of Trigonometric Functions
Six Important Integrals Involving Trig Functions
In the previous discussion we gave the derivatives of the six basic trig functions. Since integrals
are antiderivatives, we can read the table backwards and arrive at si
Exponentials
Example of an Exponential Function
A biologist grows bacteria in a culture. If initially there were three grams of bacteria, after one
day there were six grams of bacteria, and after two days, there were twelve grams, how many
grams will ther
e
Definition of e
There is a special base of an exponential that plays a particularly important role in mathematics.
One way of defining e is with the compound interest formula
A = P(1 + r/n)nt
where A corresponds to the amount in the account after t year
Derivatives of the Trigonometric Functions
Derivative of f(x) = sin(x)
First note that angles will always be given in radians. Degrees and calculus never go
together. If you ever hear the word "Degree" used in this class the appropriate question to
ask is
Improper Integrals
Integrals with Infinite Limits
Up until now, all the integrals that we have performed have had finite limits and integrands with
no asymptotes. Now we will investigate integrals with infinite limits and integrands with
vertical asymptot