Key Concepts from Calculus I, First Half

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.

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:

1. 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 .
2. 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.
3. Factor out on the left.
4. Solve for by dividing both sides of the eauation by an appropriate algebraic equation.

Derivatives of exponential and log functions

Solving related rates

1. Assign symbols to all variables involved in the problem. Draw a figure if applicable.
2. State, in terms of the variables, the information that is given, and the rate to be determined.
3. Fine an equation relating the variables introduced in step 1.
4. 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.
5. 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 .

1. if for all then is increasing on that interval.
2. 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,

1. for each constant , the function is also an antiderivative of over ;
2. 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

These notes are based on the OpenStax Calculus Textbook