Differentiation of exponential and logarithmic functions. The antiderivative of the exponential function in the UNT tasks
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Differentiation of exponential and logarithmic functions
1. Number e. Function y = e x, its properties, graph, differentiation
Consider an indicative function y = ax, where a> 1. For different bases a we get different graphs (Fig. 232-234), but you can see that they all pass through the point (0; 1), they all have a horizontal asymptote y = 0 at , they all face downward convexity and, finally, they all have tangents at all their points. Let us draw, for example, a tangent to graphics function y = 2x at the point x = 0 (Fig. 232). If you make accurate constructions and measurements, you can make sure that this tangent forms an angle of 35 ° with the x-axis (approximately).
Now we draw a tangent line to the graph of the function y = 3 x also at the point x = 0 (Fig. 233). Here the angle between the tangent and the x-axis will be greater - 48 °. And for the exponential function y = 10 x in a similar
situation, we get an angle of 66.5 ° (Fig. 234).
So, if the base a of the exponential function y = ax gradually increases from 2 to 10, then the angle between the tangent to the graph of the function at the point x = 0 and the abscissa axis gradually increases from 35 ° to 66.5 °. It is logical to assume that there is a base a for which the corresponding angle is 45 °. This base should be between the numbers 2 and 3, since for the function y-2x the angle of interest to us is 35 °, which is less than 45 °, and for the function y = 3 x it is 48 °, which is already slightly more than 45 °. The base of interest to us is usually denoted by the letter e. It has been established that the number e is irrational, i.e. represents an infinite decimal non-periodic fraction:
e = 2.7182818284590 ...;
in practice, it is usually assumed that e = 2.7.
Comment(not very serious). It is clear that L.N. Tolstoy has nothing to do with the number e, nevertheless, in the notation of the number e, note that the number 1828 is repeated twice in a row - the year of birth of L.N. Tolstoy.
The graph of the function y = ex is shown in Fig. 235. This is an exponent that differs from other exponentials (graphs of exponential functions with other bases) in that the angle between the tangent to the graph at the point x = 0 and the abscissa axis is 45 °.
Properties of the function y = e x:
1)
2) is neither even nor odd;
3) increases;
4) not limited from above, limited from below;
5) has neither the highest nor the lowest values;
6) continuous;
7)
8) convex downward;
9) differentiable.
Go back to § 45, take a look at the list of properties of the exponential function y = ax for a> 1. You will find the same properties 1-8 (which is quite natural), and the ninth property associated with
differentiability of the function, we did not mention then. Let's discuss it now.
Let us derive a formula for finding the derivative y-ex. In this case, we will not use the usual algorithm that we developed in Section 32 and which we have successfully used more than once. In this algorithm, at the final stage, it is necessary to calculate the limit, and our knowledge of the theory of limits is still very, very limited. Therefore, we will rely on geometric prerequisites, considering, in particular, the very fact of the existence of a tangent to the graph of the exponential function beyond doubt (which is why we so confidently wrote down the ninth property in the above list of properties - the differentiability of the function y = ex).
1. Note that for the function y = f (x), where f (x) = ex, we already know the value of the derivative at the point x = 0: f / = tan45 ° = 1.
2. Introduce into consideration the function y = g (x), where g (x) -f (x-a), i.e. g (x) -ex "a. Fig. 236 shows the graph of the function y = g (x): it is obtained from the graph of the function y - fx) by shifting along the x-axis by | a | scale units. The tangent to the graph of the function y = g (x) at the point x-a is parallel to the tangent to the graph of the function y = f (x) at the point x -0 (see Fig. 236), which means that it forms an angle of 45 ° with the x-axis. Using the geometric meaning of the derivative, we can write that g (a) = tan45 °; = 1.
3. Let's return to the function y = f (x). We have:
4. We have established that for any value of a the relation is valid. Instead of the letter a, you can, of course, use the letter x; then we get
From this formula, the corresponding integration formula is obtained:
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Differentiation of exponential and logarithmic functions
Compiled by:
teacher of mathematics MOU SOSH №203 KHEC
Novosibirsk city
T.V. Vidutova
Number e. Function y = e x, its properties, graph, differentiation
1. Let's construct graphs for different bases: 1. y = 2 x 3. y = 10 x 2. y = 3 x (option 2) (option 1) "width =" 640 " Consider the exponential function y = a x, where a 1.
Let's build for different bases a graphs:
1. y = 2 x
3. y = 10 x
2. y = 3 x
(Option 2)
(Option 1)
1) All charts pass through the point (0; 1);
2) All graphs have a horizontal asymptote y = 0
at X ∞;
3) All of them are facing downward convexity;
4) They all have tangents at all their points.
Let's draw a tangent to the graph of the function y = 2 x at the point X= 0 and measure the angle that the tangent forms with the axis X
With the help of accurate plotting of the tangent lines to the graphs, you can see that if the base a exponential function y = a x the base gradually increases from 2 to 10, then the angle between the tangent to the graph of the function at the point X= 0 and the abscissa gradually increases from 35 'to 66.5'.
Therefore there is a reason a, for which the corresponding angle is 45 '. And this meaning a is between 2 and 3, because at a= 2 the angle is 35 ', for a= 3 it is equal to 48 '.
In the course of mathematical analysis, it was proved that this foundation exists, it is customary to denote it by the letter e.
Determined that e - an irrational number, that is, it is an infinite non-periodic decimal fraction:
e = 2, 7182818284590 ... ;
In practice, it is usually assumed that e ≈ 2,7.
Function graph and properties y = e x :
1) D (f) = (- ∞; + ∞);
3) increases;
4) not limited from above, limited from below
5) has neither the greatest nor the least
values;
6) continuous;
7) E (f) = (0; + ∞);
8) convex downward;
9) differentiable.
Function y = e x are called exhibitor .
In the course of mathematical analysis, it was proved that the function y = e x has a derivative at any point X :
(e x ) = e x
(e 5x ) "= 5e 5x
(e x-3 ) "= e x-3
(e -4x + 1 ) "= -4e -4x-1
Example 1 . Draw a tangent to the graph of the function at the point x = 1.
2) f () = f (1) = e
4) y = e + e (x-1); y = ex
Answer:
Example 2 .
x = 3.
Example 3 .
Examine function for extremum
x = 0 and x = -2
X= -2 - maximum point
X= 0 - minimum point
If the base of the logarithm is the number e, then they say that it is given natural logarithm ... For natural logarithms, a special notation is introduced ln (l is the logarithm, n is natural).
Graph and properties of the function y = ln x
Properties of the function y = ln x:
1) D (f) = (0; + ∞);
2) is neither even nor odd;
3) increases by (0; + ∞);
4) not limited;
5) has neither the highest nor the lowest values;
6) continuous;
7) E (f) = (- ∞; + ∞);
8) convex top;
9) differentiable.
0 the differentiation formula is valid "width =" 640 " In the course of mathematical analysis, it is proved that for any value x0 the differentiation formula is valid
Example 4:
Calculate the value of the derivative of a function at a point x = -1.
For instance:
Internet resources:
- http://egemaximum.ru/pokazatelnaya-funktsiya/
- http://or-gr2005.narod.ru/grafik/sod/gr-3.html
- http://ru.wikipedia.org/wiki/
- http://900igr.net/prezentatsii
- http://ppt4web.ru/algebra/proizvodnaja-pokazatelnojj-funkcii.html
Lesson outline
Subject: Algebra
Date: 2.04.13.
Grade: Grade 11
Teacher: Tyshibaeva N.Sh.
Topic: Differentiation of logarithmic and exponential functions. The antiderivative of the exponential function.
Target:
1) formulate formulas for derivatives of logarithmic and exponential functions; teach to find the antiderivative of the exponential function
2) develop memory, observation, logical thinking, mathematical speech of students, the ability to analyze and compare, develop a cognitive interest in the subject;
3) to educate the communicative culture of students, skills of collective activity, cooperation, mutual assistance.
Lesson type: explanation of new material and consolidation of the acquired knowledge, abilities and skills.
Equipment : cards, interactive whiteboard.
Technology: differentiated approach
During the classes :
1.Org. moment. (2min).
2. Solving the crossword puzzle (8min)
1. The French mathematician of the 17th century Pierre Fermat defined this line as "The straight line most closely adjacent to the curve in a small neighborhood of a point."
Tangent
2. The function, which is given by the formula y = a x.
Indicative
3. Function, which is given by the formula y = log a x.
Logarithmic
4. Derivative of displacement
Speed
5. What is the name of the function F (x) for the function f (x) if the condition F "(x) = f (x) is satisfied for any point from the interval I.
Antiderivative
6. What is the name of the relationship between X and Y, in which each element of X is associated with a single element of Y.
Function
7. If the function f (x) can be represented in the form f (x) = g (t (x)), then this function is called ...
Complex
Word vertical surname of French mathematician and mechanic
Lagrange
3.Explanation of the new material: (10 min)
The exponential function at any point of the domain of definition has a derivative and this derivative is found by the formula:
(.ln a in the formula, replace the number and on e, we get
(e x) "= e x_ formula derivative exponent
The logarithmic function has a derivative at any point in the domain of definition, and this derivative is found by the formula:
(log a x) "= in the formula, replace the number and on e, we get
Exponential function y =(a at any point in the domain of definition has an antiderivative and this antiderivative is found by the formula F (x) =+ C
4.Fixing new material (20min)
Mathematical dictation.
1.Write the formula for the derivative of the exponential function (a X )"
(a x) "= a x · ln a
2. Write down the formula for the derivative of the exponent. (e X )"
(e x) "= e x
3. Write down the formula for the derivative of the natural logarithm
4. Write down the formula for the derivative of the logarithmic function (log a x) "=?
(log a x) "=
5. Write down the general form of antiderivatives for the function f (x) = a X .
F (x) = + C
6. Write down the general view of the antiderivatives for the function:, x ≠ 0. F (x) = ln | x | + С
Working at the blackboard
№255,№256,№258,№259(2,4)
6.D / h # 257, # 261 (2min)
7. Lesson summary: (3min)
- What is the formula for the logarithmic function?
What is the formula for the exponential function?
What is the formula for the derivative of the logarithmic function?
What is the formula for the derivative of the exponential function
Lesson topic: “Differentiation of exponential and logarithmic functions. The antiderivative of the exponential function "in the UNT tasks
Target : develop students' skills in applying theoretical knowledge on the topic “Differentiation of exponential and logarithmic functions. The antiderivative of the exponential function "for solving the problems of the UNT.
Tasks
Educational: to systematize the theoretical knowledge of students, to consolidate the skills of solving problems on this topic.
Developing: develop memory, observation, logical thinking, mathematical speech of students, attention, self-assessment and self-control skills.
Educational: promote:
fostering a responsible attitude towards learning among students;
developing a sustained interest in mathematics;
creating a positive intrinsic motivation to study mathematics.
Teaching methods: verbal, visual, practical.
Forms of work: individual, frontal, in pairs.
During the classes
Epigraph: "The mind consists not only in knowledge, but also in the ability to apply knowledge in practice" Aristotle (slide 2)
I. Organizational moment.
II. Solving a crossword puzzle. (slide 3-21)
The French mathematician of the 17th century Pierre Fermat defined this line as "The straight line most closely adjacent to the curve in a small neighborhood of a point."
Tangent
The function that is given by the formula y = log a x.
Logarithmic
The function that is given by the formula y = a X.
Indicative
In mathematics, this concept is used to find the speed of movement of a material point and the slope of the tangent to the graph of a function at a given point.
Derivative
What is the name of the function F (x) for the function f (x) if the condition F "(x) = f (x) is satisfied for any point from the interval I.
Antiderivative
What is the name of the relationship between X and Y, in which each element of X is associated with a single element of Y.
Derivative of displacement
Speed
The function that is given by the formula y = e x.
Exhibitor
If the function f (x) can be represented as f (x) = g (t (x)), then this function is called ...
III. Mathematical dictation. (Slide 22)
1. Write down the formula for the derivative of the exponential function. ( a x) "= a x ln a
2. Write down the formula for the derivative of the exponent. (e x) "= e x
3. Write down the formula for the derivative of the natural logarithm. (ln x) "=
4. Write down the formula for the derivative of the logarithmic function. (log a x) "= 
5. Write down the general form of antiderivatives for the function f (x) = a X. F (x) = 
6. Write down the general form of antiderivatives for the function f (x) =, x ≠ 0. F (x) = ln | x | + C
Check the work (answers on slide 23).
IV. Solving UNT problems (simulator)
A) No. 1,2,3,6,10,36 on the board and in the notebook (slide 24)
B) Work in pairs No. 19.28 (simulator) (slide 25-26)
V. 1. Find mistakes: (slide 27)
1) f (x) = 5 e - 3x, f "(x) = - 3 e - 3x
2) f (x) = 17 2x, f "(x) = 17 2x ln17
3) f (x) = log 5
(7x + 1), f "(x) = 
4) f (x) = ln (9 - 4x), f "(x) = 
.
Vi. Student presentation.
Epigraph: "Knowledge is such a precious thing that it is not shameful to get it from any source" Thomas Aquinas (slide 28)
Vii. Household assignment No. 19.20 p. 116
VIII. Test (backup task) (slide 29-32)
IX. Lesson summary.
“If you want to participate in the big life, then fill your head with mathematics while you can. She will then provide you with great help throughout your life "M. Kalinin (slide 33)
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