1.2.2 Derivatives of Exponentials and Logarithms.
In this section we establish the following two formulas and use them to find derivatives of various
functions involving exponentials and logarithms.
Proposition 1.
(1)
[ ln x ] =
(2)
[ ex ] = e x
Proof. Le
1.3. Inverse Trigonometric Functions
In this section we look at the derivatives of the inverse trigonometric functions and the corresponding
integrals.
1
x
y
Inverse Sine. y = sin-1x = arcsin x is the inverse function to x = sin y
with the restriction - y
1.5. L'Hospital's Rule
L'Hospital's rule expresses the limit of a quotient as the limit of the quotient of the derivatives in the
case where either the numerator and denominator both approach zero or both approach infinity. In
symbols
(1)
=
provided eithe
2
Integration Methods
Integration is different from differentiation in the following sense. If a function can be expressed in
terms of elementary function then so can the derivative. However, this is not true about integration.
For example, the integral e
1.2.3 Integrals Involving Exponentials and logarithms.
In this section we look at integrals involving exponentials and logarithms. From the derivative
formulas of exponentials and logarithms we obtain the following integral formulas.
Proposition 1.
(1)
dx
1.4 Hyperbolic Functions
The hyperbolic functions are combinations of exponential functions that have certain similarities to the
trigonometric functions. They are useful for doing certain integrals and they also arise is the solution of
problems involvin
1.2. Exponentials and Logarithms.
1.2.1 Exponentials and logarithms to the base e.
When we take derivatives and do integrals of exponentials and logarithms it is convenient if they are
expressed in base e. This is similar to when we take derivatives and d
2.3 Trigonometric Substitutions
This method works on integrals involving a square root of a quadratic. The simplest cases are , and
where a is some number. In the first case we can substitute x = a sin or x = a tanh . In the second
case substitute x = a t
1
Some Important Functions
This chapter is devoted to derivatives and integrals of exponential, logarithmic, inverse trigonometric
and hyperbolic functions. These functions are useful in a variety of applications.
1.1 Inverse Functions.
Many important fun
Math 116 - Assignment 1
Due:
Thursday, September 25. Nothing accepted after Monday, September 29. This is
worth 15 points. 2 points off for being late (after the 2nd).
Work by yourself. I don't want to see 30 identical copies of the same work. I
don't eve
4.3 Conic Sections
These are the three types of curves that one obtains by slicing a cone by a plane. They are
parabolas
ellipses
hyperbolas
We are already familiar with parabolas and hyperbolas. The graphs of quadratic functions y = ax2 + bx +
c are para
2.5 Numerical Integration
Many integrals can not be expressed in terms of elementary functions. This is the case, for example,
with dx. However, if we have a definite integral like dx, then one can obtain a very good numerical
approximation using one of m