Presentation on the topic exponential function around us. The exponential function and its application. Exploring a new topic
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MAOU "Sladkovskaya secondary school" Exponential function, its properties and graph Grade 10
A function of the form y \u003d a x, where a is a given number, a > 0, a ≠ 1, x is a variable, is called exponential.
An exponential function has the following properties: O.O.F: the set R of all real numbers; Mn.zn.: the set of all positive numbers; The exponential function y \u003d a x is increasing on the set of all real numbers if a> 1, and decreasing if 0
Graphs of the function y \u003d 2 x and y \u003d (½) x 1. The graph of the function y \u003d 2 x passes through the point (0; 1) and is located above the Ox axis. a>1 D(y): х є R E(y): y > 0 Increases over the entire domain of definition. 2. The graph of the function y= also passes through the point (0; 1) and is located above the Ox axis. 0
Using the ascending and descending properties of an exponential function, you can compare numbers and solve exponential inequalities. Compare: a) 5 3 and 5 5 ; b) 4 7 and 4 3 ; c) 0.2 2 and 0.2 6 ; d) 0.9 2 and 0.9. Solve: a) 2 x >1; b) 13 x + 1 0.7; d) 0.04 x a b or a x 1, then x>b (x
Solve graphically the equations: 1) 3 x \u003d 4-x, 2) 0.5 x \u003d x + 3.
If you remove a boiling kettle from the fire, then at first it cools down quickly, and then the cooling goes much more slowly, this phenomenon is described by the formula T \u003d (T 1 - T 0) e - kt + T 1 Application of the exponential function in life, science and technology
The growth of wood occurs according to the law: A - change in the amount of wood over time; A 0 - initial amount of wood; t - time, k, a - some constants. Air pressure decreases with height according to the law: P - pressure at height h, P0 - pressure at sea level, and - some constant.
Population growth The change in the number of people in the country over a short period of time is described by the formula, where N 0 is the number of people at time t=0, N is the number of people at time t, a is a constant.
The law of organic reproduction: under favorable conditions (lack of enemies, a large number of food) living organisms would reproduce according to the law of exponential function. For example: one housefly can produce 8 x 10 14 offspring in a summer. Their weight would be several million tons (and the weight of the offspring of a pair of flies would exceed the weight of our planet), they would take up a huge space, and if you line them up in a chain, then its length will be greater than the distance from the Earth to the Sun. But since, in addition to flies, there are many other animals and plants, many of which are natural enemies of flies, their number does not reach the above values.
When a radioactive substance decays, its amount decreases, after a while half of the original substance remains. This period of time t 0 is called the half-life. The general formula for this process is: m \u003d m 0 (1/2) -t / t 0, where m 0 is the initial mass of the substance. The longer the half-life, the slower the decay of the substance. This phenomenon is used to determine the age of archaeological finds. Radium, for example, decays according to the law: M = M 0 e -kt. Using this formula, scientists calculated the age of the Earth (radium decays in about the time equal to the age of the Earth).
On the topic: methodological developments, presentations and notes
The use of integration in the educational process as a way to develop analytical and creative abilities....
Exponential function. A function of the form y \u003d a x, where a is a given number, a> 0, and 1, x is a variable, is called exponential. 0, and 1, x is a variable, is called exponential."> 0, and 1, x is a variable, is called exponential."> 0, and 1, x is a variable, is called exponential." title="Exponential function A function of the form y \u003d a x, where a is a given number, a\u003e 0, and 1, x is a variable, is called exponential."> title="Exponential function. A function of the form y \u003d a x, where a is a given number, a> 0, and 1, x is a variable, is called exponential."> !}
The exponential function has the following properties: 1.D(y): the set R of all real numbers; 2.E(y): the set of all positive numbers; 3. The exponential function y \u003d a x is increasing on the set of all real numbers if a> 1, and decreasing if 0 1, and decrease"> 1, and decrease if 0"> 1, and decrease" title=" An exponential function has the following properties: 1.D(y): the set R of all real numbers; 2.E( y): the set of all positive numbers 3. The exponential function y \u003d a x is increasing on the set of all real numbers, if a > 1, and decreasing"> title="The exponential function has the following properties: 1.D(y): the set R of all real numbers; 2.E(y): the set of all positive numbers; 3. The exponential function y \u003d a x is increasing on the set of all real numbers, if a> 1, and decreasing"> !}
1 D(y): х є R Е(y): y>0 Increases over the entire domain of definition. 2. The graph of the function y= also passes through the point (0;1) and is located above oc "title=" The graphs of the function y=2 x and y=(½) x 1. The graph of the function y=2 x passes through point (0;1) and is located above the axis Ox. a>1 D(y): х є R E(y): y>0 Increases over the entire domain of definition 2. The graph of the function y= also passes through the point (0; 1) and is located above" class="link_thumb"> 6 !} Graphs of the function y \u003d 2 x and y \u003d (½) x 1. The graph of the function y \u003d 2 x passes through the point (0; 1) and is located above the Ox axis. a>1 D(y): х є R E(y): y>0 Increases over the entire domain of definition. 2. The graph of the function y= also passes through the point (0; 1) and is located above the Ox axis. 0 1 D(y): х є R Е(y): y>0 Increases over the entire domain of definition. 2. The graph of the function y \u003d also passes through the point (0; 1) and is located above os "\u003e 1 D (y): x є R E (y): y\u003e 0 Increases over the entire domain of definition. 2. The graph of the function y \u003d also passes through the point (0; 1) and is located above the Ox axis. 2. The graph of the function y= also passes through the point (0;1) and is located above oc "title=" The graphs of the function y=2 x and y=(½) x 1. The graph of the function y=2 x passes through point (0;1) and is located above the axis Ox. a>1 D(y): х є R E(y): y>0 Increases over the entire domain of definition 2. The graph of the function y= also passes through the point (0; 1) and is located above"> title="Graphs of the function y \u003d 2 x and y \u003d (½) x 1. The graph of the function y \u003d 2 x passes through the point (0; 1) and is located above the Ox axis. a>1 D(y): х є R E(y): y>0 Increases over the entire domain of definition. 2. The graph of the function y \u003d also passes through the point (0; 1) and is located above the axis"> !}
exponential equations. Equations in which the unknown is in the exponent are called exponential. Solutions: 1. By the property of the degree; 2. Taking the common factor out of brackets; 3. Division of both parts of the equation by the same expression, which takes on a value other than zero for all real values of x; 4. Way of grouping; 5. Reducing the equation to a quadratic one; 6.Graphic.. For example:
1; b) 13 x + 1 0.7; d) 0.04 x a and "title=" Using the properties of increasing and decreasing exponential functions, you can compare numbers and solve exponential inequalities. 1. Compare: a) 5 3 and 5 5 ; b) 4 7 and 4 3 ; c) 0.2 2 and 0.2 6 ; d) 0.9 2 and 0.9. 2. Solve: a) 2 x > 1; b) 13 x + 1 0.7; d) 0.04 x a b i" class="link_thumb"> 8 !} Using the increase and decrease properties of an exponential function, you can compare numbers and solve exponential inequalities. 1. Compare: a) 5 3 and 5 5; b) 4 7 and 4 3 ; c) 0.2 2 and 0.2 6 ; d) 0.9 2 and 0.9. 2. Solve: a) 2 x > 1; b) 13 x + 1 0.7; d) 0.04 x a b or a x 1, then x>b (x 1; b) 13 x + 1 0.7; d) 0.04 x a b i "> 1; b) 13 x + 1 0.7; d) 0.04 x a b or a x 1, then x> c (x"> 1; b) 13 x +1 0.7; d) 0.04 x a and "title=" Using the properties of increasing and decreasing exponential functions, you can compare numbers and solve exponential inequalities. 1. Compare: a) 5 3 and 5 5 ; b) 4 7 and 4 3 ; c) 0.2 2 and 0.2 6 ; d) 0.9 2 and 0.9. 2. Solve: a) 2 x > 1; b) 13 x + 1 0.7; d) 0.04 x a b i"> title="Using the increase and decrease properties of an exponential function, you can compare numbers and solve exponential inequalities. 1. Compare: a) 5 3 and 5 5; b) 4 7 and 4 3 ; c) 0.2 2 and 0.2 6 ; d) 0.9 2 and 0.9. 2. Solve: a) 2 x > 1; b) 13 x + 1 0.7; d) 0.04 x a b i"> !}
Methods for solving exponential inequalities. 1. By the nature of the degree; 2. Taking the common factor out of brackets; 3. Reduction to square; 4. Graphic. Some exponential inequalities by replacing a x \u003d t are reduced to quadratic inequalities, which are solved, given that t>0. x y 0. x y ">
Where a is a given number, a>o, Function graph, x N consists of points with abscissas 1,2,3… lying on some curve - it is called the Exponent o, The graph of the function, x N consists of points with abscissas 1,2,3 ..., lying on some curve - it is called the Exponent "> o, The graph of the function, x N consists of points with abscissas 1,2,3 ..., lying on some curve - it is called the Exponent "> o, The graph of the function, x N consists of points with abscissas 1,2,3 ... lying on some curve - it is called the Exponent" title = " Where a is a given number, a>o, The graph of the function, x N consists of points with abscissas 1,2,3 ..., lying on some curve - it is called the Exponent"> title="Where a is a given number, a> o, The graph of the function, x N consists of points with abscissas 1,2,3 ... lying on some curve - it is called the Exponent"> !}
A good household example! Everyone must have noticed that if you remove a boiling kettle from the fire, then at first it cools quickly, and then the cooling goes much more slowly. The fact is that the rate of cooling is proportional to the difference between the temperature of the kettle and the temperature environment. The smaller this difference becomes, the slower the kettle cools. If at first the temperature of the kettle was To, and the temperature of the air T1, then after t seconds the temperature T of the kettle will be expressed by the formula: Everyone probably noticed that if you remove the boiling kettle from the fire, then at first it cools quickly, and then cooling goes much more slowly. The fact is that the cooling rate is proportional to the difference between the temperature of the kettle and the ambient temperature. The smaller this difference becomes, the slower the kettle cools. If at first the temperature of the kettle was To, and the air temperature T1, then after t seconds the temperature T of the kettle will be expressed by the formula: T=(T1-T0)e-kt+T1, T=(T1-T0)e-kt+T1, where k - a number depending on the shape of the kettle, the material from which it is made, and the amount of water that is in it. where k is a number depending on the shape of the teapot, the material from which it is made, and the amount of water that is in it.
When bodies fall in airless space, their speed continuously increases. When bodies fall in the air, the falling speed also increases, but cannot exceed a certain value. When bodies fall in the air, the falling speed also increases, but cannot exceed a certain value.
Consider the problem of a parachutist falling. If we assume that the force of air resistance is proportional to the speed of the fall of the parachutist, i.e. that F=kv, then after t seconds the falling speed will be equal to: v=mg/k(1-e-kt/m), where m is the parachutist's mass. After a certain period of time, e-kt/m will become a very small number, and the fall will become almost uniform. The proportionality factor k depends on the size of the parachute. This formula is suitable not only for studying the fall of a skydiver, but also for studying the fall of a drop of rainwater, a feather, etc. Consider the problem of a parachutist falling. If we assume that the force of air resistance is proportional to the speed of the fall of the parachutist, i.e. that F=kv, then after t seconds the falling speed will be equal to: v=mg/k(1-e-kt/m), where m is the parachutist's mass. After a certain period of time, e-kt/m will become a very small number, and the fall will become almost uniform. The proportionality factor k depends on the size of the parachute. This formula is suitable not only for studying the fall of a skydiver, but also for studying the fall of a drop of rainwater, a feather, etc.
Many difficult mathematical problems have to be solved in the theory of interplanetary travel. One of them is the problem of determining the mass of fuel required to give the rocket the desired speed v. This mass M depends on the mass m of the rocket itself (without fuel) and on the velocity v0 at which the combustion products flow out of the rocket engine. Many difficult mathematical problems have to be solved in the theory of interplanetary travel. One of them is the problem of determining the mass of fuel required to give the rocket the desired speed v. This mass M depends on the mass m of the rocket itself (without fuel) and on the velocity v0 at which the combustion products flow out of the rocket engine.
If we do not take into account air resistance and Earth's attraction, then the mass of fuel is determined by the formula: M=m(ev/v0-1) (formula of K.E. Tsialkovsky). For example, in order to give a rocket with a mass of 1.5 tons a speed of 8000 m / s, it is necessary to take approximately 80 tons of fuel at a gas outflow velocity of 2000 m / s. If we do not take into account air resistance and Earth's attraction, then the mass of fuel is determined by the formula: M=m(ev/v0-1) (formula of K.E. Tsialkovsky). For example, in order to give a rocket with a mass of 1.5 tons a speed of 8000 m / s, it is necessary to take approximately 80 tons of fuel at a gas outflow velocity of 2000 m / s.
If, during the oscillations of a pendulum, a weight swinging on a spring, air resistance is not neglected, then the amplitude of the oscillations becomes less and less, the oscillations die out. Deviations of a point making damped oscillations is expressed by the formula: s=Ae-ktsin(?t+?). Since the factor e-kt decreases over time, the swing becomes smaller and smaller. If, during the oscillations of a pendulum, a weight swinging on a spring, air resistance is not neglected, then the amplitude of the oscillations becomes less and less, the oscillations die out. Deviations of a point making damped oscillations is expressed by the formula: s=Ae-ktsin(?t+?). Since the factor e-kt decreases over time, the swing becomes smaller and smaller.
As the radioactive material decays, its quantity decreases. After a while, half of the original amount of the substance remains. This period of time to is called the half-life. In general, after t years, the mass m of the substance will be equal to: m=m0(1/2)t/t0, where m0 is the initial mass of the substance. The longer the half-life, the slower the decay of the substance. As the radioactive material decays, its quantity decreases. After a while, half of the original amount of the substance remains. This period of time to is called the half-life. In general, after t years, the mass m of the substance will be equal to: m=m0(1/2)t/t0, where m0 is the initial mass of the substance. The longer the half-life, the slower the decay of the substance. The phenomenon of radioactive decay is used to determine the age of archaeological finds, for example, the approximate age of the Earth, about 5.5 billion years, is determined to maintain a standard of time. The phenomenon of radioactive decay is used to determine the age of archaeological finds, for example, the approximate age of the Earth, about 5.5 billion years, is determined to maintain a standard of time.
Problem: The half-life of plutonium is 140 days. How much plutonium will be left after 10 years if its initial mass is 8 g? m = ? Answer: 1, (d).
Here are some of the Nobel Laureates who have received an award for research in physics using the exponential function: Here are some of the Nobel Laureates who have received an award for research in physics using the exponential function: Pierre Curie d. Pierre Curie d. Richardson Owen d. Richardson Owen d. Igor Tamm d. Igor Tamm d. Alvarez Luis d. Alvarez Luis d. Alfven Hannes d. Alfven Hannes d. Wilson Robert Woodrow d. Wilson Robert Woodrow d.
She never ceases to amaze us! The exponential function is also used in solving some problems of navigation, for example, e-x function used in tasks requiring the application of the binomial law (repetition of experiments), Poisson's law (rare events), Rayleigh's law (the length of a random vector). The exponential function is also used in solving some problems of navigation, for example, the e-x function is used in problems that require the application of the binomial law (repetition of experiments), Poisson's law (rare events), Rayleigh's law (the length of a random vector). Application of the logarithmic function in biology. In a nutrient medium, the bacterium Escherichia coli divides every minute. It is clear that the total number of bacteria doubles every minute. If at the beginning of the process there was one bacterium, then after x minutes their number (N) will become equal to 2 x, i.e. N(x) = 2 x.
Properties of the function Let's analyze according to the scheme: Let's analyze according to the scheme: 1. domain of the function 1. domain of the function 2. set of function values 2. set of function values 3. zeros of the function 3. zeros of the function 4. intervals of constant sign of the function 4. intervals of constant sign of the function 5. function even or odd 5. function even or odd 6. function monotonicity 6. function monotonicity 7. maximum and minimum values 7. maximum and minimum values 8. function periodicity 8. function periodicity 9. function boundedness 9. function boundedness
0 for x R. 5) The function is neither even nor "title=" An exponential function, its graph and properties y x 1 o 1) The domain of definition is the set of all real numbers (D(y)=R). 2) The set of values is the set of all positive numbers (E(y)=R +). 3) There are no zeros. 4) y>0 for x R. 5) The function is neither even nor" class="link_thumb"> 10 !} An exponential function, its graph and properties y x 1 o 1) The domain of definition is the set of all real numbers (D(y)=R). 2) The set of values is the set of all positive numbers (E(y)=R +). 3) There are no zeros. 4) y>0 for x R. 5) The function is neither even nor odd. 6) The function is monotonic: it increases on R for a>1 and decreases on R for 0 0 for x R. 5) The function is neither even, nor "> 0 for x R. 5) The function is neither even nor odd. 6) The function is monotonic: it increases on R for a> 1 and decreases on R for 0"> 0 for x R. 5) The function is neither even nor "title=" An exponential function, its graph and properties y x 1 o 1) The domain of definition is the set of all real numbers (D(y)=R). 2) The set of values is the set of all positive numbers (E(y)=R +). 3) There are no zeros. 4) y>0 for x R. 5) The function is neither even nor"> title="An exponential function, its graph and properties y x 1 o 1) The domain of definition is the set of all real numbers (D(y)=R). 2) The set of values is the set of all positive numbers (E(y)=R +). 3) There are no zeros. 4) y>0 for x R. 5) The function is neither even nor"> !}
The growth of wood occurs according to the law, where: A - change in the amount of wood over time; A 0 - initial amount of wood; t is time, k, a are some constants. The growth of wood occurs according to the law, where: A - change in the amount of wood over time; A 0 - initial amount of wood; t is time, k, a are some constants. t 0 t0t0 t1t1 t2t2 t3t3 tntn А A0A0 A1A1 A2A2 A3A3 AnAn
The temperature of the kettle changes according to the law, where: T is the change in the temperature of the kettle with time; T 0 - boiling point of water; t is time, k, a are some constants. The temperature of the kettle changes according to the law, where: T is the change in the temperature of the kettle with time; T 0 - boiling point of water; t is time, k, a are some constants. t 0 t0t0 t1t1 t2t2 t3t3 tntn T T0T0 T1T1 T2T2 T3T3
Radioactive decay occurs according to the law, where: Radioactive decay occurs according to the law, where: N is the number of undecayed atoms at any time t; N 0 - initial number of atoms (at time t=0); t-time; N is the number of undecayed atoms at any time t; N 0 - initial number of atoms (at time t=0); t-time; T is the half-life. T is the half-life. t 0 t 1 t 2 N N3N3 N4N4 t4t4 N0N0 t3t3 N2N2 N1N1
C An essential property of the processes of organic and change in quantities is that for equal periods of time the value of a quantity changes in the same ratio. Growth of wood Change in temperature of the kettle Change in air pressure Processes of organic change in quantities include: Radioactive decay
Compare the numbers 1.3 34 and 1.3 40. Example 1. Compare the numbers 1.3 34 and 1.3 40. General solution method. 1. Present the numbers as a power with the same base (if necessary) 1.3 34 and 1. Find out whether the exponential function is increasing or decreasing a = 1.3; a>1, the next exponential function increases. a=1.3; a>1, the next exponential function increases. 3. Compare exponents (or function arguments) 34 1, the next exponential function increases. a=1.3; a>1, the next exponential function increases. 3. Compare exponents (or function arguments) 34">
Solve graphically the equation 3 x = 4-x. Example 2. Solve graphically the equation 3 x \u003d 4-x. Solution. We use the functional-graphical method for solving equations: we construct graphs of the functions y=3 x and y=4-x in one coordinate system. graphs of functions y=3x and y=4x. Note that they have one common point (1;3). So the equation has only one root x=1. Answer: 1 Answer: 1 y \u003d 4-x
4th. Example 3. Solve graphically the inequality 3 x > 4-x. Solution. y=4-x We use the functional-graphical method for solving inequalities: 1. Construct in one system 1. Construct graphs of functions "title=" Graphically solve the inequality 3 x > 4-x in one coordinate system. Example 3. Solve graphically the inequality 3 x > 4-x. Solution y=4-x We use the functional-graphical method for solving inequalities: 1. Construct in one system 1. Construct graphs of functions in one coordinate system" class="link_thumb"> 24 !} Solve graphically the inequality 3 x > 4 x. Example 3. Solve graphically the inequality 3 x > 4-x. Solution. y=4-x We use the functional-graphical method for solving inequalities: 1. We construct in one system 1. We construct in one coordinate system the graphs of the coordinate functions of the graphs of the functions y=3 x and y=4-x. 2. Select the part of the graph of the function y=3 x, located above (because the > sign) of the graph of the function y=4-x. 3. Mark on the x-axis the part that corresponds to the selected part of the graph (otherwise: project the selected part of the graph onto the x-axis). 4. Write the answer as an interval: Answer: (1;). Answer: (1;). 4th. Example 3. Solve graphically the inequality 3 x > 4-x. Solution. y \u003d 4-x We use the functional-graphical method for solving inequalities: 1. Construct in one system 1. Construct graphs of functions "\u003e 4-x in one coordinate system. Example 3. Solve graphically the inequality 3 x > 4-x. Solution. y =4-x We use the functional-graphical method for solving inequalities: 1. Construct in one system 1. Construct in one coordinate system the graphs of the coordinate functions of the graphs of the functions y=3 x and y=4-x 2. Select part of the graph of the function y=3 x, located above (because the > sign) of the graph of the function y=4-x 3. Mark on the x-axis the part that corresponds to the selected part of the graph (otherwise: project the selected part of the graph onto the x-axis) 4. Write down the answer as an interval: Answer: (1;). Answer: (1;)."> 4-x. Example 3. Solve graphically the inequality 3 x > 4-x. Solution. y=4-x We use the functional-graphical method for solving inequalities: 1. Construct in one system 1. Construct graphs of functions "title=" Graphically solve the inequality 3 x > 4-x in one coordinate system. Example 3. Solve graphically the inequality 3 x > 4-x. Solution y=4-x We use the functional-graphical method for solving inequalities: 1. Construct in one system 1. Construct graphs of functions in one coordinate system"> title="Solve graphically the inequality 3 x > 4 x. Example 3. Solve graphically the inequality 3 x > 4-x. Solution. y=4-x We use the functional-graphical method for solving inequalities: 1. Construct in one system 1. Construct graphs of functions in one coordinate system"> !}
Solve graphically inequalities: 1) 2 x >1; 2) 2 x 1; 2) 2 x "> 1; 2) 2 x "> 1; 2) 2 x "title=" Solve graphically inequalities: 1) 2 x >1; 2) 2 x"> title="Solve graphically inequalities: 1) 2 x >1; 2) 2 x"> !}
Independent work(test) 1. Specify the exponential function: 1. Specify the exponential function: 1) y=x 3 ; 2) y \u003d x 5/3; 3) y \u003d 3 x + 1; 4) y \u003d 3 x + 1. 1) y \u003d x 3; 2) y \u003d x 5/3; 3) y \u003d 3 x + 1; 4) y \u003d 3 x + 1. 1) y \u003d x 2; 2) y \u003d x -1; 3) y \u003d -4 + 2 x; 4) y=0.32 x. 1) y \u003d x 2; 2) y \u003d x -1; 3) y \u003d -4 + 2 x; 4) y=0.32 x. 2. Indicate a function that increases on the entire domain of definition: 2. Specify a function that increases on the entire domain of definition: 1) y = (2/3) -x; 2) y=2-x; 3) y \u003d (4/5) x; 4) y \u003d 0.9 x. 1) y \u003d (2/3) -x; 2) y=2-x; 3) y \u003d (4/5) x; 4) y \u003d 0.9 x. 1) y \u003d (2/3) x; 2) y=7.5 x; 3) y \u003d (3/5) x; 4) y \u003d 0.1 x. 1) y \u003d (2/3) x; 2) y=7.5 x; 3) y \u003d (3/5) x; 4) y \u003d 0.1 x. 3. Indicate a function that decreases over the entire domain of definition: 3. Specify a function that decreases over the entire domain of definition: 1) y = (3/11) -x; 2) y=0.4 x; 3) y \u003d (10/7) x; 4) y \u003d 1.5 x. 1) y \u003d (2/17) -x; 2) y=5.4 x; 3) y = 0.7 x; 4) y \u003d 3 x. 4. Indicate the set of values of the function y=3 -2 x -8: 4. Indicate the set of values of the function y=2 x+1 +16: 5. Indicate the smallest of these numbers: 5. Indicate the smallest of these numbers: 1) 3 - 1/3; 2) 27 -1/3; 3) (1/3) -1/3; 4) 1 -1/3. 1) 3 -1/3; 2) 27 -1/3; 3) (1/3) -1/3; 4) 1 -1/3. 5. Indicate the largest of these numbers: 1) 5 -1/2; 2) 25 -1/2; 3) (1/5) -1/2; 4) 1 -1/2. 1) 5 -1/2; 2) 25 -1/2; 3) (1/5) -1/2; 4) 1 -1/2. 6. Find out graphically how many roots the equation 2 x \u003d x -1/3 (1/3) x \u003d x 1/2 has 6. Find out graphically how many roots the equation 2 x \u003d x -1/3 (1/3) has x \u003d x 1/2 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots.
1. Specify the exponential function: 1) y=x 3; 2) y=x 5/3; 3) y=3 x+1; 4) y \u003d 3 x + 1. 1) y \u003d x 3; 2) y=x 5/3; 3) y=3 x+1; 4) y=3 x Indicate a function that is increasing over the entire domain of definition: 2. Indicate a function that is increasing over the entire domain of definition: 1) y = (2/3)-x; 2) y=2-x; 3) y \u003d (4/5) x; 4) y \u003d 0.9 x. 1) y \u003d (2/3) -x; 2) y=2-x; 3) y \u003d (4/5) x; 4) y \u003d 0.9 x. 3. Indicate a function decreasing over the entire domain of definition: 3. Indicate a function decreasing over the entire domain of definition: 1) y = (3/11)-x; 2) y=0.4 x; 3) y \u003d (10/7) x; 4) y \u003d 1.5 x. 1) y \u003d (3/11) -x; 2) y=0.4 x; 3) y \u003d (10/7) x; 4) y \u003d 1.5 x. 4. Indicate the set of values of the function y=3-2 x-8: 4. Indicate the set of values of the function y=3-2 x-8: 5. Indicate the smallest of these numbers: 5. Indicate the smallest of these numbers: 1) 3- 1/3; 2) 27-1/3; 3) (1/3)-1/3; 4) 1-1/3. 1) 3-1/3; 2) 27-1/3; 3) (1/3)-1/3; 4) 1-1/3. 6. Find out graphically how many roots the equation 2 x=x- 1/3 has 6. Find out graphically how many roots the equation 2 x=x- 1/3 has 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. Verification work Select exponential functions, which: Select exponential functions, which: I option - decrease on the domain of definition; Option I - decrease on the domain of definition; II option - increase on the domain of definition. II option - increase on the domain of definition.
The presentation "Exponential function, its properties and graph" clearly presents the educational material on this topic. During the presentation, the properties of the exponential function, its behavior in the coordinate system are considered in detail, examples of solving problems using the properties of the function, equations and inequalities are considered, important theorems on the topic are studied. With the help of the presentation, the teacher can increase the effectiveness of the math lesson. A vivid presentation of the material helps to keep the attention of students on the study of the topic, animation effects help to more clearly demonstrate the solutions to problems. For faster memorization of concepts, properties and features of the solution, color highlighting is used.
The demonstration begins with examples of the exponential function y=3x with various exponents - positive and negative integers, common fractions and decimals. For each indicator, the value of the function is calculated. Next, a graph is built for the same function. On slide 2, a table is built filled with the coordinates of the points belonging to the graph of the function y \u003d 3 x. According to these points on the coordinate plane, the corresponding graph is built. Next to the graph, similar graphs are built y \u003d 2 x, y \u003d 5 x and y \u003d 7 x. Each function is highlighted in different colors. The graphs of these functions are made in the same colors. Obviously, as the base of the degree of the exponential function grows, the graph becomes steeper and more pressed against the y-axis. The same slide describes the properties of the exponential function. It is noted that the domain of definition is the real line (-∞;+∞), the function is not even or odd, the function increases over all domains of definition and does not have the largest or smallest value. The exponential function is bounded from below, but not bounded from above, continuous on the domain of definition and convex downwards. The range of values of the function belongs to the interval (0;+∞).
Slide 4 presents a study of the function y \u003d (1/3) x. The graph of the function is built. To do this, the table is filled with the coordinates of the points belonging to the graph of the function. Based on these points, a graph is built on a rectangular coordinate system. The properties of the function are described next. It is noted that the domain of definition is the entire numerical axis. This function is not odd or even, decreasing over the entire domain of definition, has no maximum, the smallest values. The function y=(1/3) x is bounded from below and unbounded from above, is continuous in the domain of definition, and has a downward convexity. The range of values is the positive semiaxis (0;+∞).
Using the given example of the function y=(1/3) x, one can single out the properties of an exponential function with a positive base less than one and refine the idea of its graph. Slide 5 shows a general view of such a function y \u003d (1 / a) x, where 0
Slide 6 compares the graphs of the functions y=(1/3)x and y=3x. It can be seen that these graphs are symmetrical about the y-axis. To make the comparison more visual, the graphs are colored in colors that highlight the function formulas. The following is the definition of an exponential function. On slide 7, a definition is highlighted in the box, which indicates that a function of the form y \u003d a x, where positive a, not equal to 1, is called exponential. Further, using the table, an exponential function is compared with a base greater than 1 and positive less than 1. Obviously, almost all properties of the function are similar, only a function with a base greater than a is increasing, and with a base less than 1, decreasing. The following is an example solution. In example 1, you need to solve the equation 3 x \u003d 9. The equation is solved graphically - a graph of the function y \u003d 3 x and a graph of the function y \u003d 9 are built. The point of intersection of these graphs is M (2; 9). Accordingly, the solution to the equation is the value x=2. Slide 10 describes the solution to the equation 5 x =1/25. Similarly to the previous example, the solution of the equation is determined graphically. The construction of graphs of functions y=5 x and y=1/25 is demonstrated. The point of intersection of these graphs is the point E (-2; 1/25), which means that the solution to the equation x \u003d -2. Next, it is proposed to consider the solution of the inequality 3 x<27. Решение выполняется графически - определяется точка пересечения графиков у=3 х и у=27. Затем на плоскости координат хорошо видно, при каких значениях аргумента значения функции у=3 х будут меньшими 27 - это промежуток (-∞;3). Аналогично выполняется решение задания, в котором нужно найти множество решений неравенства (1/4) х <16. На координатной плоскости строятся графики функций, соответствующих правой и левой части неравенства и сравниваются значения. Очевидно, что решением неравенства является промежуток (-2;+∞).