Independent work on exponential functions. Exponential function - properties, graphs, formulas
10th grade" width="271" height="129 src="/>
Independent work on the topic “Solving equations and systems of equations (repetition).”
Option 1.
1)https://pandia.ru/text/78/476/images/image006_16.gif" width="99" height="24 src=">.gif" width="179" height="44 src=" >.gif" width="99" height="51 src=">
Independent work on the topic “Solving inequalities.” Repetition.
Option 1.
1)https://pandia.ru/text/78/476/images/image012_10.gif" width="64" height="27 src=">.gif" width="100" height="41 src=" >.gif" width="72" height="27 src=">.gif" width="52" height="41 src=">.gif" width="189" height="24 src=">.
Option 1.
The function is given https://pandia.ru/text/78/476/images/image022_7.gif" width="33" height="20 src=">.gif" width="33" height="20 src=" >.gif" width="171" height="51 src=">
a) Find https://pandia.ru/text/78/476/images/image023_5.gif" width="43" height="20 src=">.gif" width="68" height="32 src= ">.
Independent work on the topic “Function”. Repetition.
Option 3.
The function is given https://pandia.ru/text/78/476/images/image022_7.gif" width="33" height="20 src=">.gif" width="43" height="20 src=" >.gif" width="156" height="51 src=">
a) Find https://pandia.ru/text/78/476/images/image023_5.gif" width="43" height="20 src=">.gif" width="68" height="32 src= ">.
b) Plot a graph of this function.
c) Indicate for this function D(y), E(y), intervals of increase and decrease.
Independent work on the topic “Function”. Repetition.
Option 5.
The function is given https://pandia.ru/text/78/476/images/image022_7.gif" width="33" height="20 src=">.gif" width="43" height="20 src=" >, DIV_ADBLOCK535">
Independent work on the topic “Function”. Repetition.
Option 6.
The function is given https://pandia.ru/text/78/476/images/image022_7.gif" width="33" height="20 src=">.gif" width="33" height="20 src=" >.gif" width="131" height="24">.
2. Find the domain of definition of the function https://pandia.ru/text/78/476/images/image035_4.gif" width="89 height=53" height="53">
4.
Solve a set of inequalities: 
Additional task.
Solve the system of equations: 
VII - IX classes"
Option 2.
1.
Solve the equation 
.
2. Find the domain of definition of the function https://pandia.ru/text/78/476/images/image040_3.gif" width="91 height=53" height="53">
4.
Solve the system of inequalities: 
Additional task.
Solve the system of equations:
Test on the topic “Repetition of algebra course material VII - IX classes"
Option 3.
1.
Solve the equation 
.
2. Find the domain of definition of the function https://pandia.ru/text/78/476/images/image044_3.gif" width="89" height="75">
4. Solve a set of inequalities: https://pandia.ru/text/78/476/images/image037_4.gif" width="137 height=48" height="48">
Test on the topic “Repetition of algebra course material VII - IX classes"
Option 4.
1.
Solve the equation 
.
2. Find the domain of definition of the function https://pandia.ru/text/78/476/images/image048_3.gif" width="108" height="56">
4.
Solve the system of inequalities: 
Additional task.
Solve the system of equations:
Option 1.
1. Compare the numbers: a) and ; b) and ; c) and https://pandia.ru/text/78/476/images/image056_2.gif" width="48" height="24 src=">.gif" width="107" height="43 src= ">.
Independent work on the topic “ Exponential function»
Option 2.
1. Compare the numbers: a) and ; b) and ; c) and https://pandia.ru/text/78/476/images/image068_2.gif" width="65" height="49 src=">.gif" width="107" height="43 src= ">.
3. Construct graphs of functions: a); b); V).
Independent work on the topic “Exponential Equations”
Option 1.
Solve equations:
1)https://pandia.ru/text/78/476/images/image075_2.gif" width="136" height="24 src=">.gif" width="147" height="33 src=" >.gif" width="161" height="24 src="> .
Independent work on the topic “Exponential inequalities”
Option 1.
Solve inequalities:
1) https://pandia.ru/text/78/476/images/image081_2.gif" width="144" height="21 src=">.gif" width="61" height="48 src=" >.gif" width="88" height="28 src=">.
Option 1.
1. Construct a graph of the function.
2. Solve the equations: a), b).
3. Solve inequalities: a); b) .
4. Solve the system of equations:
Test on the topic “Exponential function”
Option 2.
1. Construct a graph of the function.
2. Solve the equations: a), b).
3. Solve inequalities: a) 
; b) .
4. Solve the system of equations:
Option 1.
1. Calculate: a); b); V); G).
2..gif" width="147" height="24 src=">.
Independent work on the topic “The concept of logarithm”
Option 2.
1. Calculate: a); b); V); G).
2..gif" width="161" height="27 src=">.
Option 1.
2..gif" width="87" height="44 src=">.
Independent work on the topic “Basic properties of the logarithm”
Option 2.
1. Find if it is known that .
2..gif" width="113" height="45 src=">.
Independent work on the topic “Logarithmic function”
Option 1.
Find the domain of definition of each function:
1)https://pandia.ru/text/78/476/images/image118_0.gif" width="97" height="27 src=">.gif" width="147" height="28 src=" >.gif" width="192" height="31 src=">.
Option 1.
Graph the function:
1)https://pandia.ru/text/78/476/images/image124_0.gif" width="81" height="27 src=">.gif" width="75" height="27 src=" >.
Independent work on the topic “Graph of a logarithmic function”
Option 2.
Graph the function:
1)https://pandia.ru/text/78/476/images/image128_0.gif" width="99" height="28 src=">.gif" width="81" height="29 src=" >.
Option 1.
Independent work on the topic “Inverse function”
Option 2.
a) Find the inverse function of the given one,
b) Indicate the domain of definition and range of values of the inverse function,
c) Construct graphs of this function and its inverse in the same coordinate system.
Independent work on the topic “Inverse function”
Option 3.
a) Find the inverse function of the given one,
b) Indicate the domain of definition and range of values of the inverse function,
c) Construct graphs of this function and its inverse in the same coordinate system.
Independent work on the topic “Inverse function”
Option 4.
a) Find the inverse function of the given one,
b) Indicate the domain of definition and range of values of the inverse function,
c) Construct graphs of this function and its inverse in the same coordinate system.
Option 1.
1. Calculate: a); b) ; V) 
; G); d) 
; e).
2. Find X, If 
.
3..gif" width="93" height="27">.
Test on the topic: “Logarithm”.
Option 2.
1. Calculate: a); b) ; V) 
; G); d) 
; e).
2. Find X, If 
.
3..gif" width="91" height="27">.
5. Find the function inverse to the function , . Specify the domain of definition and range of values of the inverse function.
Option 1.
1)https://pandia.ru/text/78/476/images/image168_0.gif" width="117" height="24 src=">.gif" width="131" height="48 src=" >.
Independent work on the topic “Logarithmic equations”
Option 2.
1)https://pandia.ru/text/78/476/images/image172_0.gif" width="125" height="41 src=">.gif" width="133" height="40 src=" >.
Option 1.
1)
, 2), 3),
4) https://pandia.ru/text/78/476/images/image179_0.gif" width="93 height=20" height="20">.
Independent work on the topic “Logarithmic inequalities”
Option 2.
4) https://pandia.ru/text/78/476/images/image184.gif" width="92 height=20" height="20">.
Test work on the topic “ logarithmic equations and inequalities"
Option 1.
1. Solve the equations: a); b); V) 
.
2. Solve the system of equations: 
3. Solve inequalities: a) 
; b) 
.
4..gif" width="159" height="29">; b); c) 
.
2. Solve the system of equations: 
3. Solve inequalities: a) 
; b) 
.
4..gif" width="25" height="41 src=">.gif" width="77" height="41">; b).
4..gif" width="109" height="21 src=">..gif" width="36" height="19 src=">.
Option 2.
1. Express the magnitude of the angles 560 in radian measure; 1700.
2..gif" width="37" height="41 src=">.
3. Indicate the sign of the number: a); b).
4..gif" width="100" height="21 src=">..gif" width="29" height="19 src=">.
Independent work on the topic “Basics of trigonometry”
Option 3.
1. Express the magnitude of the angles 720 in radian measure; 1400.
2..gif" width="36" height="41 src=">.
3. Indicate the sign of the number: a) 
; b).
4..gif" width="29" height="19 src=">, if it is known that https://pandia.ru/text/78/476/images/image221.gif" width="27" height ="41 src=">.gif" width="123" height="48">; b).
4..gif" width="36" height="19 src=">, if it is known that https://pandia.ru/text/78/476/images/image226.gif" width="497" height ="24">.
2. Simplify the expression: .
3..gif" width="527" height="24">.
2. Simplify the expression: .
3..gif" width="497" height="24">.
2. Simplify the expression: .
3..gif" width="527" height="24">.
2. Simplify the expression: 
.
3..gif" width="192" height="24">.
2. Prove the identity: 
.
Option 2.
1. Calculate: 
.
2. Prove the identity:.
3. Convert into a product:.
Independent work on the topic “Sum and difference of trigonometric functions”
Option 3.
1. Calculate: .
2. Prove the identity: 
.
3. Convert into a product: .
Independent work on the topic “Sum and difference of trigonometric functions”
Option 4.
1. Calculate: 
.
2. Prove the identity:.
3. Convert into a product: .
Option 1.
1.
Simplify the expression: 
.
2.
Calculate 
.
3.
Calculate 
.
4. Calculate.
5. Convert into a work https://pandia.ru/text/78/476/images/image255.gif" width="109" height="17 src=">..gif" width="16 height=13" height= "13">.
2. Draw a graph of the function 
.
Test on the topic “Trigonometric transformations”
Option 2.
1.
Simplify the expression: 
.
2. Simplify the expression: .
3.
Calculate 
.
4. Calculate.
5.
Convert to work 
.
Optional task.
1..gif" width="43" height="17 src=">and the smallest value.
2. Draw a graph of the function 
.
Test on the topic “Trigonometric transformations”
Option 3.
1. Calculate.
2. Calculate.
3.
Calculate 
.
4. Calculate.
5.
Convert to work 
.
Optional task.
1..gif" width="43" height="17 src=">and nai higher value.
2. Draw a graph of the function .
Test on the topic “Trigonometric transformations”
Option 4.
1. Calculate.
2. Simplify the expression: https://pandia.ru/text/78/476/images/image274.gif" width="280" height="47">.
4. Calculate.
5. Convert to work: .
Optional task.
1..gif" width="43" height="17 src=">and the smallest value.
2. Draw a graph of the function .
Option 1.
Solve equations:
1)https://pandia.ru/text/78/476/images/image278.gif" width="153" height="21 src=">.gif" width="109" height="45 src=" >.gif" width="284" height="48 src=">
Independent work on the topic “Equation cosx=a”
Option 3.
Solve the equations: , periodic with main period 6. Moreover, belonging to the interval
5. Write down all solutions to the equation 
, belonging to the interval.
6. Write down all solutions to the inequality 
, belonging to the interval.
Provides reference data on the exponential function - basic properties, graphs and formulas. The following issues are considered: domain of definition, set of values, monotonicity, inverse function, derivative, integral, power series expansion and representation using complex numbers.
ContentProperties of the Exponential Function
The exponential function y = a x has the following properties on the set of real numbers ():
(1.1)
defined and continuous, for , for all ;
(1.2)
for a ≠ 1
has many meanings;
(1.3)
strictly increases at , strictly decreases at ,
is constant at ;
(1.4)
at ;
at ;
(1.5)
;
(1.6)
;
(1.7)
;
(1.8)
;
(1.9)
;
(1.10)
;
(1.11)
,
.
Other useful formulas.
.
Formula for converting to an exponential function with a different exponent base:
When b = e, we obtain the expression of the exponential function through the exponential:
Private values
, , , , .
y = a x for different values of the base a. The figure shows graphs of the exponential function
y (x) = a x
for four values degree bases: a = 2
, a = 8
, a = 1/2
and a = 1/8
. 1
It can be seen that for a > 0
< a < 1
the exponential function increases monotonically. The larger the base of the degree a, the stronger the growth. At
the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.
Ascending, descending
| The exponential function for is strictly monotonic and therefore has no extrema. Its main properties are presented in the table. 1 | y = a x , a > 0 < a < 1 | |
| y = ax, | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Domain | 0 < y < + ∞ | 0 < y < + ∞ |
| Range of values | Monotone | monotonically increases |
| monotonically decreases 0 | Zeros, y = | Zeros, y = |
| No 0 | Intercept points with the ordinate axis, x = 1 | Intercept points with the ordinate axis, x = 1 |
| + ∞ | 0 | |
| 0 | + ∞ |
y=
Inverse function
The inverse of an exponential function with base a is the logarithm to base a.
.
If , then
.
Differentiation of an exponential function
To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the differentiation rule complex function.
To do this you need to use the property of logarithms
and the formula from the derivatives table:
.
Let an exponential function be given:
.
We bring it to the base e:
Let's apply the rule of differentiation of complex functions. To do this, introduce the variable
Then
From the table of derivatives we have (replace the variable x with z):
.
Since is a constant, the derivative of z with respect to x is equal to
.
According to the rule of differentiation of a complex function:
.
Derivative of an exponential function
.
Derivative of nth order:
.
Deriving formulas > > >
An example of differentiating an exponential function
Find the derivative of a function
Intercept points with the ordinate axis, x = 3 5 x
Solution
Let's express the base of the exponential function through the number e.
3 = e ln 3
Then
.
Enter a variable
.
Then
From the table of derivatives we find:
.
Because the 5ln 3 is a constant, then the derivative of z with respect to x is equal to:
.
According to the rule of differentiation of a complex function, we have:
.
Answer
Integral
Expressions using complex numbers
Consider the function complex number z:
f (z) = a z
where z = x + iy; 2 = - 1
.
i
Let us express the complex constant a in terms of modulus r and argument φ:
Then
.
a = r e i φ The argument φ is not uniquely defined. IN
φ = φ general view,
0 + 2 πn where n is an integer. Therefore the function f(z)
.
is also not clear. Its main significance is often considered Independent work on the topic
"Exponential function". Independent work contains 2 options with three tasks each. Self-study texts are divided into three levels of difficulty. Each option task corresponds to its own level of difficulty. Created independent work in the text editor Microsoft Word. For convenience, the correct answers are given.
View document contents
“Independent work “Exponential function””
Republic of Belarus Government agency
education "Lyceum of Novopolotsk"
Independent work in mathematics, algebra section
Topic: Exponential function
Prepared by: Konovalyonok
Olga Vladimirovna,
higher mathematics teacher
Option 1
1. Compare: 
2) 
1) and 
And
a) the value of a;
b) domain of definition;2
Option 1
1) 
1) and 
2) 
1) and 
Option 2. The figure shows a graph of a function given by the formula
And
a) the value of a;
on set D. Specify for it:
c) set (area) of values;
d) intervals of increase (decrease);
e) coordinates of the points of intersection of the graph with the Oy axis;
e) value at points x1= -1 and x2= 1;
g) the largest and smallest values.
3. Indicate the natural domain of definition of expression (a1):
1. 1) ; 2)
Option 12
Topic: Exponential function, its properties and graph.
Target: Check the quality of mastering the concept of “exponential function”; to develop skills in recognizing the exponential function, using its properties and graphs, teaching students to use analytical and graphical forms of recording the exponential function; provide a working environment in the classroom.
Equipment: board, posters
Lesson form: class lesson
Lesson type: practical lesson
Lesson type: lesson in teaching skills and abilities
Lesson Plan
1. Organizational moment
2. Independent work and verification homework
3. Problem solving
4. Summing up
5. Homework
During the classes.
1. Organizational moment :
Hello. Open your notebooks, write down today’s date and the topic of the lesson “Exponential Function”. Today we will continue to study the exponential function, its properties and graph.
2. Independent work and checking homework .
Target: check the quality of mastering the concept of “exponential function” and check the completion of the theoretical part of the homework
Method: test task, frontal survey
As homework, you were given numbers from the problem book and a paragraph from the textbook. We won’t check your execution of numbers from the textbook now, but you will hand in your notebooks at the end of the lesson. Now the theory will be tested in the form of a small test. The task is the same for everyone: you are given a list of functions, you must find out which of them are indicative (underline them). And next to the exponential function you need to write whether it is increasing or decreasing.
higher mathematics teacher Answer B) D) - exponential, decreasing | Option 2 Answer D) - exponential, decreasing D) - exponential, increasing |
Option 3 Answer A) - exponential, increasing B) - exponential, decreasing | Option 4 Answer A) - exponential, decreasing IN) - exponential, increasing |
Now let’s remember together which function is called exponential?
A function of the form , where and , is called an exponential function.
What is the scope of this function?
All real numbers.
What is the range of the exponential function?
All positive real numbers.
Decreases if the base of the power is greater than zero but less than one.
In what case does an exponential function decrease in its domain of definition?
Increasing if the base of the power is greater than one.
3. Problem solving
Target: to develop skills in recognizing an exponential function, using its properties and graphs, teach students to use analytical and graphical forms of writing an exponential function
Method: demonstration by the teacher of solving typical problems, oral work, work at the blackboard, work in a notebook, conversation between the teacher and students.
The properties of the exponential function can be used when comparing 2 or more numbers. For example: No. 000. Compare the values and if a) 
..gif" width="37" height="20 src=">, then this is a rather complicated job: we would have to take the cube root of 3 and 9, and compare them. But we know that it increases, this in its own way turn means that as the argument increases, the value of the function increases, that is, we just need to compare the values of the argument and , it is obvious that 
(can be demonstrated on a poster showing an increasing exponential function). And always, when solving such examples, you first determine the base of the exponential function, compare it with 1, determine monotonicity and proceed to compare the arguments. In the case of a decreasing function: when the argument increases, the value of the function decreases, therefore, we change the sign of inequality when moving from inequality of arguments to inequality of functions. Next, we solve orally: b) 
- 
IN) 
- 
G) 
- 
- No. 000. Compare the numbers: a) and
Therefore, the function increases, then
Why ?
Increasing function and 
Therefore, the function is decreasing, then 
Both functions increase throughout their entire domain of definition, since they are exponential with a base of power greater than one.
What is the meaning behind it?
We build graphs:
Which function increases faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25">
Which function decreases faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25">
On the interval, which of the functions has greater value at a specific point?
D), https://pandia.ru/text/80/379/images/image068_0.gif" width="69" height="57 src=">. First, let's find out the scope of definition of these functions. Do they coincide?
Yes, the domain of these functions is all real numbers.
Name the scope of each of these functions.
The ranges of these functions coincide: all positive real numbers.
Determine the type of monotonicity of each function.
All three functions decrease throughout their entire domain of definition, since they are exponential with a base of powers less than one and greater than zero.
Which singular point does the graph of an exponential function exist?
What is the meaning behind it?
Whatever the basis of the degree of an exponential function, if the exponent contains 0, then the value of this function is 1.
We build graphs:
Let's analyze the graphs. How many points of intersection do the graphs of functions have?
Which function decreases faster when trying https://pandia.ru/text/80/379/images/image070.gif" width="41 height=57" height="57">
Which function increases faster when striving https://pandia.ru/text/80/379/images/image070.gif" width="41 height=57" height="57">
On the interval, which of the functions has greater value at a specific point?
On the interval, which of the functions has greater value at a specific point?
Why do exponential functions with different bases have only one intersection point?
Exponential functions are strictly monotonic throughout their entire domain of definition, so they can intersect only at one point.
The next task will focus on using this property. No. 000. Find the largest and smallest values of the given function on the given interval a) . Recall that a strictly monotonic function takes its minimum and maximum values at the ends of a given segment. And if the function is increasing, then its greatest value will be at the right end of the segment, and the smallest at the left end of the segment (demonstration on the poster, using the example of an exponential function). If the function is decreasing, then its largest value will be at the left end of the segment, and the smallest at the right end of the segment (demonstration on the poster, using the example of an exponential function). The function is increasing, because, therefore, the smallest value of the function will be at the point https://pandia.ru/text/80/379/images/image075_0.gif" width="145" height="29">. Points b ) 
, V) 
d) solve the notebooks yourself, we will check them orally.
Students solve the task in their notebooks
|
Decreasing function
|
Decreasing function
|
Increasing function
|
greatest value of the function on the segment
the smallest value of a function on a segment
the smallest value of a function on a segment
greatest value of the function on the segment- No. 000. Find the largest and smallest value of the given function on the given interval a) 
. This task is almost the same as the previous one. But what is given here is not a segment, but a ray. We know that the function is increasing, and it has neither the largest nor the smallest value on the entire number line https://pandia.ru/text/80/379/images/image063_0.gif" width="68" height ="20">, and tends to at , i.e. on the ray the function at tends to 0, but does not have its own lowest value, but it has the greatest value at the point 
. Points b) 
, V) 
, G) 
Solve the notebooks yourself, we will check them orally.
|
Properties of the Exponential Function |
y = , 0< a < 1 |
|
|
1. Function domain |
||
|
2. Function range |
||
|
3. Comparison intervals with unity |
for x > 0, > 1 |
for x > 0, 0< < 1 |
|
at x< 0, 0< < 1 |
at x< 0, > 1 |
|
|
4. Even, odd. |
The function is neither even nor odd (a function of general form). |
|
|
5. Monotony. |
monotonically increases on R |
monotonically decreases on R |
|
6. Extremes. |
The exponential function has no extrema. |
|
|
7. Asymptote |
The Ox axis is a horizontal asymptote. |
|
|
8. For any real values x and y; |
Examples:
Example No. 1. (To find the domain of definition of a function). What argument values are valid for functions:
Example No. 2. (To find the range of values of a function). The figure shows the graph of the function. Specify the domain of definition and range of values of the function:
Example No. 3. (To indicate the intervals of comparison with one). Compare each of the following powers with one:
Example No. 4. (To study the function for monotonicity). Compare real numbers m and n in size if:
Example No. 5. (To study the function for monotonicity). Draw a conclusion regarding base a if:
y(x) = 10x; f(x) = 6x; z(x) - 4x
How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?
Table. Conclusion:
Table. Conclusion:
One coordinate plane graphs of functions were constructed:
y(x) = (0,1)x; f(x) = (0.5)x; z(x) = (0.8)x.
How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?
Conclusion
In this course work on the topic “Exponential function” I reviewed its concept, basic properties and graphs.
The topic of exponential function, in general, is one of the frequently used in calculations and solving various problems.
The work provided examples and tasks of varying complexity and content.
The course work, in my opinion, was carried out within the framework of the methodology of teaching mathematics and can be used as a visual aid for full-time and part-time students.