Notes from Calculus I. I created these notes to help get me through Calculus I. My notes end somewhere around the end of Volume 1 of the OpenStax Textbook, because that's all we used in class.
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Calculus
Derivative as a limit
Difference quotient
Difference quotient with increment h
Differentiation rules
Constant rule
Power rule
Sum rule
Constant multiple rule
Product Rule
Quotient rule
Chain Rule
Trig derivatives
Inverse Function Theorem
Let be a function that is both invertible and differentiable. Let
be the inverse of . For all satisfying
Alternatively, if is the inverse of , then
Inverse trig derivatives
Inverse function theorem
Power rule with rational exponents
Derivative of inverse sine function
Derivative of inverse cosine function
Derivative of inverse tangent function
Derivative of inverse cotangent function
Derivative of inverse secant function
Derivative of invers cosecant function
Implicit differentiation
To perform implicit differentiation on an equation that defines a function
implicitly in terms of variable use the following steps:
Take the derivative of both sides of the equation. Keep in mind that is a
function of . Consequently, whereas ,
because we must use the chain
rule to differentiate with respect to .
Rewrite the equation so that all terms containing are on the
left hand side and that all terms that don't contain are on
the right.
Factor out on the left.
Solve for by dividing both sides of the eauation by an
appropriate algebraic equation.
Derivatives of exponential and log functions
Solving related rates
Assign symbols to all variables involved in the problem. Draw a figure if
applicable.
State, in terms of the variables, the information that is given, and the rate
to be determined.
Fine an equation relating the variables introduced in step 1.
Using the chain rule, differentiate bothe sides of the equation found in step
3 with respect to the independent variable. This new equation will relate the
derivatives.
Substitute all known values into the equation from Step 4, then solve for the
unknown rate of change.
Linear Approximation
can be approximated around point with the following:
the differential for can be written as:
Fermat's Theorem(Not his last one though)
If has a local extremum at and is differentiable at c, then .
Rolle's Theorem
If a real valued function is continuous on a proper closed interval
differentiable on open interval and , then there exists at
least one point in the open interval such that
Mean Value Theorem
Let be continuous over closed interval and differentiable over the
open interval . There exists a point such that
Mean Value Corrollaries
Corollary 1
Let be differentible over interval . If for all ,
then is constant for all
Corollary 2
If and are differentiable over an interval and for
all then for some constant .
Corollary 3
Let be a differentiable function over interval .
if for all then is increasing on that interval.
if for all then is decreasing on that interval.
Limits at infinity
Limit at infinity
We say a function has a Limit at infinity if there exists a real number
such that for all , there exists such that
for all . In that case, we write
Infinite Limit at infinity
We say a function has an infinite limit at infinity and write
if for all , there exists an such
that
for all
(See text chapter 4.6 for negitive infinities)
L'Hôpital's rule
Suppose and are differentiable functions over an open interval
containing , except possibly at . if and , then
assuming the limit on the right exists or is or . This result
also holds if we are considering one sided limits, or if and
Newton's Method
To approximate a root of function , start with an initial estimate of the
root, , then iteratively apply the following formula
Antiderivatives
Definition
A function is an antiderivative of the function if
for all in the domain of .
General Form
Let be an antiderivative of over an interval . Then,
for each constant , the function is also an antiderivative of
over ;
if is an antiderivative of over , there is a constant for
which over
In other words, the most general form of the antiderivative of over is
Definition
Given a function , the indefinite integral of , denoted
is the most general antiderivative of . If is an antiderivative of ,
then
The expression is called the integrand and the variable x is the
variable of integration
