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“Plotting graphs using derivatives” - Generalization. Sketch the graph of the function. Find the asymptotes of the graph of the function. Graph of the derivative of a function. Additional task. Explore the function. Name the intervals of decreasing function. Independent work students. Expand knowledge. Lesson on consolidating the material learned. Assess your skills. Maximum points of the function.
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Lesson and presentation on the topic: "Function y=cos(x). Definition and graph of the function"
Additional materials
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Teaching aids and simulators in the Integral online store for grade 10
Algebraic problems with parameters, grades 9–11
Software environment "1C: Mathematical Constructor 6.1"
What we will study:
1. Definition.
2. Graph of a function.
3. Properties of the function Y=cos(X).
4. Examples.
Definition of the cosine function y=cos(x)
Guys, we have already met the function Y=sin(X).
Let's remember one of the ghost formulas: sin(X + π/2) = cos(X).
Thanks to this formula, we can claim that the functions sin(X + π/2) and cos(X) are identical, and their function graphs coincide.
The graph of the function sin(X + π/2) is obtained from the graph of the function sin(X) by parallel translation π/2 units to the left. This will be the graph of the function Y=cos(X).
The graph of the function Y=cos(X) is also called a sine wave.
Properties of the function cos(x)
- Let's write down the properties of our function:
- The domain of definition is the set of real numbers.
- The function is even. Let's remember the definition even function. A function is called even if the equality y(-x)=y(x) holds. As we remember from the ghost formulas: cos(-x)=-cos(x), the definition is fulfilled, then cosine is an even function.
- The function Y=cos(X) decreases on the segment and increases on the segment [π; 2π]. We can verify this in the graph of our function.
- The function Y=cos(X) is limited from below and from above. This property follows from the fact that
-1 ≤ cos(X) ≤ 1 - The smallest value of the function is -1 (at x = π + 2πk). Highest value function is equal to 1 (at x = 2πk).
- The function Y=cos(X) is a continuous function. Let's look at the graph and make sure that our function has no breaks, this means continuity.
- Range of values: segment [- 1; 1]. This is also clearly visible from the graph.
- Function Y=cos(X) is a periodic function. Let's look at the graph again and see that the function takes the same values at certain intervals.
Examples with the cos(x) function
1. Solve the equation cos(X)=(x - 2π) 2 + 1
Solution: Let's build 2 graphs of the function: y=cos(x) and y=(x - 2π) 2 + 1 (see figure). _2.jpg)
y=(x - 2π) 2 + 1 is a parabola shifted to the right by 2π and upward by 1. Our graphs intersect at one point A(2π;1), this is the answer: x = 2π.
2. Plot the function Y=cos(X) for x ≤ 0 and Y=sin(X) for x ≥ 0
Solution: To build the required graph, let's build two graphs of the function in “pieces”. First piece: y=cos(x) for x ≤ 0. Second piece: y=sin(x)
for x ≥ 0. Let us depict both “pieces” on one graph.
_3.jpg)
3. Find the largest and smallest value of the function Y=cos(X) on the segment [π; 7π/4]
Solution: Let's build a graph of the function and consider our segment [π; 7π/4]. The graph shows that the highest and lowest values are achieved at the ends of the segment: at points π and 7π/4, respectively.
Answer: cos(π) = -1 – the smallest value, cos(7π/4) = the largest value.
_4.jpg)
4. Graph the function y=cos(π/3 - x) + 1
Solution: cos(-x)= cos(x), then the desired graph will be obtained by moving the graph of the function y=cos(x) π/3 units to the right and 1 unit up.
_5.jpg)
Problems to solve independently
1)Solve the equation: cos(x)= x – π/2.2) Solve the equation: cos(x)= - (x – π) 2 - 1.
3) Graph the function y=cos(π/4 + x) - 2.
4) Graph the function y=cos(-2π/3 + x) + 1.
5) Find the largest and smallest value of the function y=cos(x) on the segment.
6) Find the largest and smallest value of the function y=cos(x) on the segment [- π/6; 5π/4].
Now we will look at the question of how to build graphs trigonometric functions multiple angles ωx, Where ω - some positive number.
To graph a function y = sin ωx Let's compare this function with the function we have already studied y = sin x. Let's assume that when x = x 0 function y = sin x takes the value equal to 0. Then
y 0 = sin x 0 .
Let us transform this relationship as follows:
Therefore, the function y = sin ωx at X = x 0 / ω takes the same value at 0 , which is the same as the function y = sin x at x = x 0 . This means that the function y = sin ωx repeats its meanings in ω times more often than the function y = sin x. Therefore, the graph of the function y = sin ωx obtained by "compressing" the graph of the function y = sin x V ω times along the x axis.
For example, the graph of a function y = sin 2x obtained by “compressing” a sinusoid y = sin x twice along the x-axis.
Graph of a function y = sin x / 2 is obtained by “stretching” the sinusoid y = sin x twice (or “compressing” it by 1 / 2 times) along the x axis.
Since the function y = sin ωx repeats its meanings in ω
times more often than the function
y = sin x, then its period is ω
times less than the period of the function y = sin x. For example, the period of the function y = sin 2x equals 2π/2 = π
, and the period of the function y = sin x / 2
equals π
/
x/ 2
= 4π .
It is interesting to study the behavior of the function y = sin ax using the example of animation, which can be very easily created in the program Maple:
Graphs of other trigonometric functions of multiple angles are constructed in a similar way. The figure shows the graph of the function y = cos 2x, which is obtained by “compressing” the cosine wave y = cos x twice along the x-axis.
Graph of a function y = cos x / 2 obtained by “stretching” the cosine wave y = cos x doubled along the x axis.
In the figure you see the graph of the function y = tan 2x, obtained by “compressing” the tangentsoids y = tan x twice along the x-axis.
Graph of a function y = tg x/ 2 , obtained by “stretching” the tangentsoids y = tan x doubled along the x axis.
And finally, the animation performed by the program Maple:
Exercises
1. Construct graphs of these functions and indicate the coordinates of the points of intersection of these graphs with the coordinate axes. Determine the periods of these functions.
A). y = sin 4x/ 3 G). y = tan 5x/ 6 and). y = cos 2x/ 3
b). y= cos 5x/ 3 d). y = ctg 5x/ 3 h). y=ctg x/ 3
V). y = tan 4x/ 3 e). y = sin 2x/ 3
2. Determine the periods of functions y = sin (πх) And y = tg (πх/2).
3. Give two examples of functions that take all values from -1 to +1 (including these two numbers) and change periodically with period 10.
4 *. Give two examples of functions that take all values from 0 to 1 (including these two numbers) and change periodically with a period π/2.
5. Give two examples of functions that take all real values and vary periodically with period 1.
6 *. Give two examples of functions that accept all negative values and zero, but not accepted positive values and change periodically with a period of 5.