"Monotonicity of Functions" Concept and Properties of Functions PPT

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"Monotonicity of Functions" Concept and Properties of Functions PPT

Part One: Explanation of Curriculum Standards

1. Understand the definitions of increasing and decreasing functions.

2. Understand the meaning of monotonicity of functions and master the method of proving the monotonicity of functions by using definitions.

3. Be able to use definitions or images to find the monotonic interval of a function, and be able to use the monotonicity of a function to solve related problems.

Monotonicity of functions PPT, part 2: independent preview

1. Definition of increasing and decreasing functions

1. (1) Draw the graphs of the functions f(x)=x, f(x)=x2 and observe their graphs. How do the graphs rise and fall?

Tip: According to the three steps of the list method: list → draw points → connect the lines, the image of the two functions is as follows.

The graph of function f(x)=x rises from left to right; the graph of function f(x)=x2 falls on the left side of the y-axis and rises on the right side of the y-axis; the function y=- The graph of x2 is rising on the left side of the y-axis and falling on the right side of the y-axis.

(2) How to use the function analytical formula f(x)=x2 to describe the change of the function value f(x) as the value of the independent variable x changes?

Tip: On (-∞,0], as the value of the independent variable x increases, the function value f(x) gradually decreases; on (0,+∞), as the value of the independent variable x increases is large, the function value f(x) gradually increases.

(3) Use the changes in x and f(x) to describe that when x takes values ​​from small to large in a given interval, the function value increases sequentially? What if the function value decreases sequentially?

Tip: On the given interval, ∀x1, x2, and x1 f(x2).

(4) In the definition of increasing function, change "When x1x2, all f(x1)>f(x2)" ,Is this okay?

Tip: Yes. The definition of increasing function: When x1 < f(x1)>f(x2)” also has the same inequality sign ">", and the steps are also consistent. Therefore, we can call it simply: the step-by-step increasing function.

2. Fill in the form

3. Do it

(1)f(x)=-2x-1 on (-∞,+∞) is ___________. (Fill in “increasing function” or “decreasing function”)

(2) f(x)=x2-1 is___________ on the interval [0,+∞). (Fill in "increasing function" or "decreasing function")

Answer: (1) Decreasing function (2) Increasing function

2. Monotonicity and Monotone Interval of Functions

1.(1) Is "the function y=f(x) is an increasing function on the interval D" the same as "the monotonically increasing interval of the function y=f(x) is D"?

Tip: It’s different. “The monotonically increasing interval of function y=f(x) is D”, indicating that interval D is all the monotonically increasing intervals of function y=f(x); and “function y=f(x) is in the interval D "is an increasing function", as long as the function increases on the interval D, which is a subset of the entire monotonically increasing interval.

(2) The domain of function y= is (-∞,0)∪(0,+∞). The graph is monotonically decreasing in the first and third quadrants respectively. Can we say that the monotonically decreasing interval of function y= is (-∞,0)∪(0,+∞)?

Tip: No. Discontinuous monotonic intervals must be written separately and connected with "," or "and" in the middle. They cannot be connected with the symbol "∪".

(3) When writing a monotonic interval of a function, can it only be written as an open interval?

Tip: No. For a certain point, since its function value is a definite constant and there is no monotonicity at all, when writing a monotonic interval, you can include the endpoints or not include the endpoints, but for some points that are not The endpoints of intervals within the definition domain must be removed when writing. Therefore, when writing a monotonic interval, you might as well agree to "close if it can be closed, and open if it needs to be opened."

2. Fill in the blanks

If the function y=f(x) monotonically increases (or monotonically decreases) on the interval D, then the function y=f(x) is said to have (strict) monotonicity in this interval, and the interval D is called y=f(x ) is a monotonic increasing (or decreasing) interval.

Monotonicity of functions PPT, part 3: inquiry learning

Determine the monotonic interval of a function

Example 1 Find the monotonic interval of the following function, and indicate whether it is an increasing or decreasing function on the monotonic interval:

(1)y=3x-2;(2)y=-1/x.

Analysis: If the function is a function we are familiar with, the monotonic interval should be given directly. Otherwise, a sketch of the function should be drawn first, and then the monotonic interval should be given by combining the "rise and fall" of the image.

Solution: (1) The monotonic interval of the function y=3x-2 is R, which is an increasing function on R.

(2) The monotonic interval of the function y=- is (-∞,0), (0,+∞), and it is an increasing function on (-∞,0) and (0,+∞).

Reflection and Insight 1. The geometric meaning of function monotonicity: On a monotonic interval, if the image of the function "rises", the function is an increasing interval; if the image of the function "falls", the function is a decreasing interval. Therefore, with the help of the function It is an intuitive and effective method to find the monotonic interval of a function through the image. In addition to this method, the definition method can also be used to find the monotonic interval, that is, the monotonic interval can be found by the definition of increasing and decreasing functions. Find the monotonic interval After the interval, if the monotonic interval is not unique, it can be separated by ",".

2. Monotonicity of linear, quadratic functions and inverse proportional functions:

(1) The monotonicity of the linear function y=kx+b (k≠0) is determined by the coefficient k: when k>0, the function is an increasing function on R; when k<0, the function on R is subtraction function.

Monotonicity of functions PPT, part 4: thinking analysis

Mistake caused by confusing the two concepts of "monotone interval" and "monotone on the interval"

Typical example: If the monotonically decreasing interval of function f(x)=x2+2(a-1)x+4 is (-∞,4], then the value set of real number a is _________.

The axis of symmetry of the graph of the wrong solution function f(x) is the straight line x=1-a. Since the function decreases monotonically in the interval (-∞,4], 1-a≥4, that is, a≤-3. Therefore, the value set of the real number a is {a|a≤-3}.

What are the errors in the above problem-solving process? What are the reasons for the errors? How do you correct them? How to prevent them?

Tip: In the misunderstanding, the monotonic interval is mistaken for being monotonic on the interval.

Positive answer: Because the monotonic decreasing interval of the function is (-∞,4], and the symmetry axis of the function graph is the straight line x=1-a, so 1-a=4, that is, a=-3. Therefore, the real number a The value set of is {-3}.

Answer:{-3}

Monotonicity of functions PPT, Part 5: Practice in class

1. If the domain of function f(x) is (0,+∞) and satisfies f(1)

A. It is an increasing function B. It is a decreasing function

C. Increase first and then decrease D. Monotonicity cannot be determined

Analysis: 1, 2, and 3 are not arbitrary values ​​and cannot be used as a basis for judging the monotonicity of a function.

Answer:D

2. The graph of function y=f(x),x∈[-4,4] is as shown in the figure, then all monotonically decreasing intervals of function y=f(x) are ()

A.[-4,-2] B.[1,4]

C.[-4,-2]and[1,4] D.[-4,-2]∪[1,4]

Answer:C

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Update Time: 2024-09-27

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