MindMap Gallery 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 21:50:54Rumi: 10 dimensions of spiritual awakening. When you stop looking for yourself, you will find the entire universe because what you are looking for is also looking for you. Anything you do persevere every day can open a door to the depths of your spirit. In silence, I slipped into the secret realm, and I enjoyed everything to observe the magic around me, and didn't make any noise. Why do you like to crawl when you are born with wings? The soul has its own ears and can hear things that the mind cannot understand. Seek inward for the answer to everything, everything in the universe is in you. Lovers do not end up meeting somewhere, and there is no parting in this world. A wound is where light enters your heart.
Chronic heart failure is not just a problem of the speed of heart rate! It is caused by the decrease in myocardial contraction and diastolic function, which leads to insufficient cardiac output, which in turn causes congestion in the pulmonary circulation and congestion in the systemic circulation. From causes, inducement to compensation mechanisms, the pathophysiological processes of heart failure are complex and diverse. By controlling edema, reducing the heart's front and afterload, improving cardiac comfort function, and preventing and treating basic causes, we can effectively respond to this challenge. Only by understanding the mechanisms and clinical manifestations of heart failure and mastering prevention and treatment strategies can we better protect heart health.
Ischemia-reperfusion injury is a phenomenon that cellular function and metabolic disorders and structural damage will worsen after organs or tissues restore blood supply. Its main mechanisms include increased free radical generation, calcium overload, and the role of microvascular and leukocytes. The heart and brain are common damaged organs, manifested as changes in myocardial metabolism and ultrastructural changes, decreased cardiac function, etc. Prevention and control measures include removing free radicals, reducing calcium overload, improving metabolism and controlling reperfusion conditions, such as low sodium, low temperature, low pressure, etc. Understanding these mechanisms can help develop effective treatment options and alleviate ischemic injury.
Rumi: 10 dimensions of spiritual awakening. When you stop looking for yourself, you will find the entire universe because what you are looking for is also looking for you. Anything you do persevere every day can open a door to the depths of your spirit. In silence, I slipped into the secret realm, and I enjoyed everything to observe the magic around me, and didn't make any noise. Why do you like to crawl when you are born with wings? The soul has its own ears and can hear things that the mind cannot understand. Seek inward for the answer to everything, everything in the universe is in you. Lovers do not end up meeting somewhere, and there is no parting in this world. A wound is where light enters your heart.
Chronic heart failure is not just a problem of the speed of heart rate! It is caused by the decrease in myocardial contraction and diastolic function, which leads to insufficient cardiac output, which in turn causes congestion in the pulmonary circulation and congestion in the systemic circulation. From causes, inducement to compensation mechanisms, the pathophysiological processes of heart failure are complex and diverse. By controlling edema, reducing the heart's front and afterload, improving cardiac comfort function, and preventing and treating basic causes, we can effectively respond to this challenge. Only by understanding the mechanisms and clinical manifestations of heart failure and mastering prevention and treatment strategies can we better protect heart health.
Ischemia-reperfusion injury is a phenomenon that cellular function and metabolic disorders and structural damage will worsen after organs or tissues restore blood supply. Its main mechanisms include increased free radical generation, calcium overload, and the role of microvascular and leukocytes. The heart and brain are common damaged organs, manifested as changes in myocardial metabolism and ultrastructural changes, decreased cardiac function, etc. Prevention and control measures include removing free radicals, reducing calcium overload, improving metabolism and controlling reperfusion conditions, such as low sodium, low temperature, low pressure, etc. Understanding these mechanisms can help develop effective treatment options and alleviate ischemic injury.
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