MindMap Gallery Core knowledge module

Core knowledge module

This is a mind map about the core knowledge module, about the application of derivatives and calculus, and the main contents include: application topics, in-depth expansion modules, and tool kit for checking and filling in missings.

Edited at 2025-03-08 21:50:54

Mason·Carter

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Core knowledge module

Mason·Carter

Recent works View more works>>

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Outline

POWER SERIES
- 12
Uyie Fakhri
CFA Level 2 Review - Derivatives
- 32
PlotWizard
Derivative
- 123
PlotWizard
Differential Calculus of Functions of One Variable
- 60
PlotWizard
Differential calculus of multivariate functions
- 30
PlotWizard
ordinary differential equations
- 282
- 1
- 1
PlotWizard
limit and continuity
- 100
- 1
- 1
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Carboxylic acids and their derivatives
- 20
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Mathematics
- 31
3zz

Core knowledge module

Application topics

Functional properties analysis

Monotonic determination flow chart

The relationship between positive and negative derivative and monotonic

When the derivative is greater than 0, the function increases monotonically

When the derivative is less than 0, the function is monotonically decreasing

Decision steps

Calculate derivatives

Analyze derivative symbols

Draw monotonic conclusions

Double conditions for judging extreme points

First-order derivative test

The derivative of 0 may be an extreme point

Derivative symbol changes can determine the extreme value point

Second-order derivative test

The second derivative is greater than 0 as the local minimum value

The second derivative is less than 0. The local maximum value is

Most valued application question template

Practical problem modeling

Transforming practical problems into functional models

Determine the domain of the function

Solve the most value

Find the critical point of the function

Calculate the function value of the boundary value and critical point

Compare the maximum and minimum values

Geometric Applications

The tangent equation speed calculation formula

Point tangent equation

The tangent equation is y y1 = m(x x1)

where m is the tangent slope and (x1, y1) is the tangent coordinate

Special treatment of normal equations

Normals are vertical lines of tangents

The normal slope is the negative reciprocal of the tangent slope

Preliminary curvature

Curvature definition

Curvature is a measure of the degree of curve

Curvature K = y'' / (1 (y')^2)^(3/2)

Calculation method

Calculate the curvature of the curve by derivative

Used to analyze the concaveness of curves

In-depth expansion module

High-order derivatives

Physical significance of second-order derivative

Describe the concaveness of function graphs

When the second derivative is greater than 0, the function graph is concave upward

When the second derivative is less than 0, the function graph concaves downward

Links to kinematics

Describe acceleration in physics

The second derivative represents the rate of change of velocity

Leibniz formula memory formula

Promotion of the Law of Product

Used to find the higher order derivative of the product of two functions

Memory formulas help quick memory formulas

Application example

Calculate the higher order derivative of the product of a specific function

For example, find the third derivative of (x^2*sinx)

Differential concept

Geometric comparison between Δy and dy

Δy represents the actual change in the function value

Δy = f(x Δx) f(x)

dy represents the linear approximate change of function value

dy = f'(x)Δx

Geometric meaning

dy is the vertical coordinate change of the tangent of the function image at point x

Δy is the actual vertical coordinate change of the function image in the interval x, x Δx

Differential approximation calculation error

Error estimation

Error is related to higher order derivatives

The error size is proportional to the high power of Δx

Error analysis

When Δx is small enough, the error can be ignored

Used to approximate the variation of function values

Toolkit for checking missing tools

High-frequency wrong questions

Derivatability judgment of segmented functions

Derivative continuity of segmented points

Derivatability judgment of segment functions at segment points

Need to check whether the left and right derivatives are equal

Typical error analysis

Ignore derivative continuity checks for segmented points

Apply derivative formula incorrectly

Implicit function to obtain the leak-derivation issue

Implicit function derivation rules

Find the derivative of x at the same time by finding both sides of the derivative equation

Pay attention to applying chain rules when deriving y

Common Errors

Some items are missing when searching for derivation

The chain rule is not applied correctly

Confusion of derivative denominator for parametric equations

Calculation of derivatives of parametric equations

Derivative parameters for parameter equations

Then use the chain rule to find the derivative

Error analysis

Confused denominator during derivation

Causes error in the derivation result