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Calculus Mind Map

Types of Functions

Functions

  • Continuous
  • No holes
  • No jumps
  • Limit exists
  • Is always defined
  • Discontinuous
  • Has a hole/jump
  • Can be undefined
  • Limit does not exist at a certain point

Extreme Value Points

Extreme value points (EVPs) are the turning points on a derivative function.

EVPs

EVP Graph

Concavity

  • Concavity refers to the shape of the function, up or down, particularly in reference to the second derivative.

Concavity

Concavity Graph

Concavity

Types of Tangents

  • Tangent Exists
  • Continuous function
  • Can or cannot be differentiable
  • Tangent Does Not Exist
  • Discontinuous function
  • Not differentiable

Tangent Types

Tangents

  • Tangents are straight lines that pass through a single point on a curve. They depend on the slope of the curve at that point.

Tangents

Tangent Line

Tangent line

Differentiability

Differentiability

  • Differentiable
  • Derivative exists
  • Non-vertical tangent
  • Not Differentiable
  • Derivative does not exist
  • Undefined/vertical tangent

Differentiable or Not?

Example

Derivatives

  • A derivative is how the slope of a curve is calculated at any point. It can also be used to find the instantaneous rates.

Derivatives

Implicit Differentiation

  • Implicit differentiation is a form of differentiation that relates the derivative of a value to x or t.

Implicit Differentiation

Related Rates

Related rates

  • Related rates manipulate implicit differentiation to find the rate that a value is changing at any point in time.

Applications to the Real World

  • Physics
  • Related Rates
  • Instantaneous Rates
  • Optimization

Applications

Derivative Rules

Derivative Rules

  • Use derivative rules to find the derivative of a function with at least two variables.

  • Product Rule
  • Quotient Rule
  • Chain Rule

Quotient Rule

Product Rule

Product Rule

(f g)'= f' g + f g'

Chain Rule

Chain Rule

Short cuts & Power Rules

Power Rules

  • The power rule can be a quick and easy way to change a term with just one variable into a derivative. This acn be very helpful when dealing with derivtives.

Instantaneous Rates

Related rates

  • The rate of change at a particular moment. Same as the value of the derivative at a particular point. For a function, the instantaneous rate of change at a point is the same as the slope of the tangent line. That is, it's the slope of a curve.
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