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Geometric Applications of Differential Calculus of Functions of One Variable
This is a mind map about the geometric application of differential calculus of functions of one variable. The main contents include the concepts of extreme value and maximum value, the discrimination of monotonicity and extreme value, the concepts of concavity and inflection point, asymptote, maximum value or taking Value range.
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Geometric Applications of Differential Calculus of Functions of One Variable
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- Outline
Geometric applications of integral calculus of one variable
The concept of extreme value and maximum value
★The premise for the existence of extreme value must be that both sides are defined.
The extreme point is not necessarily the maximum point, and the maximum point is not necessarily the extreme point.
y=e to the power of x has a maximum point in [0, ∞], but no extreme point
y=3x-x to the third power
There are extreme points but no maximum points
The maximum point inside the interval must be the extreme point
The points inside the interval are not extreme points, so they must not be the maximum points either.
Discontinuity points can also be extreme points
All four types of discontinuity points can be used to reach extreme points, as long as the left and right sides are defined.
Discrimination between monotonicity and extreme values
Judgment of monotonicity
Extreme point
It does not have to be differentiable at the extreme point, as long as it is differentiable in the neighborhood of this point
Necessary conditions (Fermat)
f(x) is differentiable at x=x0 and takes an extreme value at point x0
Then there must be f′(x0)=0
first sufficient condition
First verify the continuity before we can find it. The premise is that it is continuous at this point and f′(x0) changes sign in the decentered neighborhood of x0.
second sufficient condition
f(x) is second-order differentiable at x=x0 and f′(x0)=0, f′′(x0)≠0
f′′(x0)>0, f(x0) is a minimum value which can be proved by the limit definition and local sign preservation property.
On the contrary, the maximum value
third sufficient condition
The m-order derivative of f(x0)=0, the n-order derivative of f(x0)≠0
When n is an even number, the n-order derivative is >0, and at x=x0, it takes the minimum value.
When n is an even number, the n-order derivative <0, and takes the maximum value at x=x0
You can also use the definition to identify
The concept of concavity and inflection point
The function value at the line connecting two points < the function value at the midpoint of the two points on the curve
convex
Function value at the line connecting two points > Function value at the midpoint of the two points on the curve
Concave
Inflection point definition
The inflection point only needs to be continuous
It has nothing to do with whether it is derivable or not.
Concave and convex in no particular order
The inflection point is on the curve
Write (x0,f(x0))
Extreme points refer to points on the definition domain
Extreme values are function values
Distinguish concave and convex type
Second derivative > 0, concave
Second derivative <0, convex
Inflection point judgment
necessary conditions
f′′(x0) exists, and point (x0, f(x0)) is the inflection point on the curve
Then f′′(x0)=0
first sufficient condition
The premise is that the second derivative changes sign if it is continuous in the decentered neighborhood.
second sufficient condition
f''(x0)＝0, f'''(x0)≠0, then (x0,f(x0)) is the inflection point
third sufficient condition
The mth derivative of f(x0)=0
When n is an odd number, the nth derivative ≠0
(x0,f(x0)) is the inflection point
You can also use the definition to identify
Exam questions
5.5 Proof of monotonicity
The relationship between f' and f
Think of Lagrange's mean value theorem
The second derivative > 0 near a point
A curve near a point is a concave curve
Identify extreme values according to the definition of extreme values
Asymptote
plumb asymptote
no definition point
Define the endpoints of the interval
piecewise function piecewise point
limx tends to x0 =∞ (or limx tends to x0-=∞), then x=x0 is a vertical asymptote
horizontal asymptote
limx tends to ∞=y1, then y=y1 is a horizontal asymptote
limx tends to -∞=y2, then y=y2 is a horizontal asymptote
If = the same asymptote, y = y0 is a horizontal asymptote
oblique asymptote
limx tends to ∞, limf(x)/x=a1 (a cannot=0.) lim[f(x)-a1x]=b1
For y=a1x b1 is an oblique asymptote
similar
Exam questions
step
1. Find undefined points and endpoints
2. Is there a plumb asymptote approaching this point?
3. Tends to infinity, whether there is a level
4. Compared with x, is there any oblique gradient?
5.8 and Exercise 2.6
The same method of finding limits★★
x tends to infinity, 1/x=0
lne to the power of x × (e to the power of -x 1) = x ln (e to the power of -x 1)
1-(1-1/n) to the kth power~k/n
Maximum value or value range
Find the maximum and minimum values (range) of the continuous function f(x) on the closed interval [a,b]
The point where the first derivative is zero
Points where the first derivative does not exist, underivable points
endpoint
Big loophole
When finding the derivative of the piecewise point of a piecewise function, you must use the derivative definition to find the derivative.
Find the maximum value or value range of the continuous function f(x) in the open interval (a,b)
stationary point
non-derivable point
The right limit of the left endpoint, the left limit of the right endpoint
Same goes for ±infinity
If you encounter practical problems of finding the maximum and minimum values, first establish the objective function, and then convert it into an optimal value problem after determining the definition interval.
In particular, if the actual problem under consideration has a maximum or minimum value. The objective function has a unique extreme point, so it must be the maximum point. Find the n largest term under nth root.
The extreme point inside the interval must be the maximum point
Make function graphs
① Determine the domain of the function and check whether it has parity or evenness
② Find the first-order derivative and the second-order derivative
The undefined point of f(x)
Point f'(x)=0
Points where f'(x) does not exist
Point f′′(x)=0
③Make a form
④Determine the asymptote
⑤ Make function graphs
Definition of derivatives and local sign preservation using limits
5.1
Worth doing several times
5.5
4.11
Exam questions can be used to practice calculations
5.3
Implicit function extreme value
Let the first derivative equal 0
Find the relationship between y and x
Find x
Find the second derivative at this point
Points where the first derivative does not exist can also be extreme values
5.2
5.8
The arithmetic mean is greater than the geometric mean
Promote to five times
5.9
common factor method
For the first derivative which is difficult to calculate = 0
5.10
Examination of computing ability
Take logarithmic changes, add and subtract
Believe in your own computing skills
Extreme point
First derivative = 0
The first derivative does not exist
inflection point
Second derivative = 0
The second derivative does not exist
Don’t enter the wrong expression at the end of the turning point
Concave and convex intervals are separated by commas
The only minimum value is the minimum value
Pay attention to the approximation