Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3...

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Exponential Functions Section 1.3

Transcript of Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3...

Page 1: Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x

Exponential Functions

Section 1.3

Page 2: Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x

Exponential Functions

f(x) = ax

Example:

y1 = 2x

y2 = 3x

y3 = 5x

For what values of x is 2x<3x<5x?

For what values of x is 2x>3x>5x?

For what values of x is 2x=3x=5x?

Page 3: Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x

Important Concepts• Exponent rules• Half-life: the amount of time it takes for half

of a radioactive substance to change from its original state to a nonradioactive state by emitting energy in the form of radiation

• Exponential growth and decay

y = kax

(Growth if a > 1, decay if 0 < x < 1)

Page 4: Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x

Important Concepts• e: the number that the function given below

approaches as x approaches infinity

• e is commonly used with models of exponential growth or decay

• Example: Interest compounded continuously

y = Pert

f (x) 11

x

x

Page 5: Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x

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Page 6: Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x

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Section 4.3 Logarithmic Functions and Graphs. Flashback Consider the graph of the exponential function y = f(x) = 3 x. Is f(x) one-to-one? Does f(x) have.

Section 4.3 Logarithmic Functions and Graphs. Flashback Consider the graph of the exponential function y = f(x) = 3 x. Is f(x) one-to-one? Does f(x) have.

3.5 EXPONENTIAL AND LOGARITHMIC MODELS Introduction · Section 3.5 Exponential and Logarithmic Models 255 ... y = 1 + log x y x −1 1 −1 −2 2 1 y = 1 + ln x y x y −11 ... 25.7

3.5 EXPONENTIAL AND LOGARITHMIC MODELS Introduction · Section 3.5 Exponential and Logarithmic Models 255 ... y = 1 + log x y x −1 1 −1 −2 2 1 y = 1 + ln x y x y −11 ... 25.7

8.1 Exploring Exponential Models. What is an exponential equation? An exponential equation has the general form y=ab x.

8.1 Exploring Exponential Models. What is an exponential equation? An exponential equation has the general form y=ab x.

Exponential functions y=a x

Exponential functions y=a x

Section 5.2: Derivative of the General Exponential Function, y = b x

Section 5.2: Derivative of the General Exponential Function, y = b x

Liceo Scientifico Isaac Newton Roma School Year 2011-2012 Maths course Exponential function X Y O (0,1)

Liceo Scientifico Isaac Newton Roma School Year 2011-2012 Maths course Exponential function X Y O (0,1)

MATLAB Basics€¦ · MATLAB Basics Math functions † Numeruous math & trigonometric funktions sqrt(x) square root rem(x;y) rest after division x=yexp(x) exponential function round(x)

MATLAB Basics€¦ · MATLAB Basics Math functions † Numeruous math & trigonometric funktions sqrt(x) square root rem(x;y) rest after division x=yexp(x) exponential function round(x)

Exponential Growth and Decay Functions. What is an exponential function? An exponential function has the form: y = ab x Where a is NOT equal to 0 and.

Exponential Growth and Decay Functions. What is an exponential function? An exponential function has the form: y = ab x Where a is NOT equal to 0 and.

Exponential Functions. An exponential function is a function where the variable is an exponent. Examples: f(x) = 3 x g(x) = 5000(1.02) x h(x) = (¾) x+2.

Exponential Functions. An exponential function is a function where the variable is an exponent. Examples: f(x) = 3 x g(x) = 5000(1.02) x h(x) = (¾) x+2.

Lesson Objective: Draw graphs of exponential functions of the form y = ka x and understand ideas of exponential growth and decay.

Lesson Objective: Draw graphs of exponential functions of the form y = ka x and understand ideas of exponential growth and decay.

GUI with Matlab - Columbia University - Electricalmarios/matlab/GUIwithMatlab.pdf · legend('Sine Wave', 'Decaying Exponential') x=0:.1:2*pi; y=sin(x); plot(x,y) grid on hold on plot(x,

GUI with Matlab - Columbia University - Electricalmarios/matlab/GUIwithMatlab.pdf · legend('Sine Wave', 'Decaying Exponential') x=0:.1:2*pi; y=sin(x); plot(x,y) grid on hold on plot(x,

Section 9.2 Exponential Functions Evaluating Rational & Irrational Exponents Graphing Exponential Functions f(x) = a x Equations with x and y Interchanged.

Section 9.2 Exponential Functions Evaluating Rational & Irrational Exponents Graphing Exponential Functions f(x) = a x Equations with x and y Interchanged.

Applications of the Exponential and Natural Logarithm ... I n Chapter 4, we introduced the exponential function y = ex and the natural logarithm function y = ln x, and we studied their

Applications of the Exponential and Natural Logarithm ... I n Chapter 4, we introduced the exponential function y = ex and the natural logarithm function y = ln x, and we studied their

Derivatives of Exponential and Logarithmic Functions...Find the derivative of y = 3e x2+3. 10/23 Chain rule for exponential functions d dx eg(x) = eg(x) g0(x) Example Find the derivative

Derivatives of Exponential and Logarithmic Functions...Find the derivative of y = 3e x2+3. 10/23 Chain rule for exponential functions d dx eg(x) = eg(x) g0(x) Example Find the derivative

Section 5.2: Derivative of the General Exponential Function, y = b x Dana, Dina, Isabella.

Section 5.2: Derivative of the General Exponential Function, y = b x Dana, Dina, Isabella.

Exponential family by representation theory · Motivation G=H-method Examples and related work Exponential family Background Exponential family X: manifold, R(X): the set of all Radon

Exponential family by representation theory · Motivation G=H-method Examples and related work Exponential family Background Exponential family X: manifold, R(X): the set of all Radon

Applications of Exponential Functions in Daily LifeApplications of Exponential Functions in Daily Life Introduction An exponential function is a function in the form y ≠= a x , where

Applications of Exponential Functions in Daily LifeApplications of Exponential Functions in Daily Life Introduction An exponential function is a function in the form y ≠= a x , where

Exponential functions - haesemathematics.com · B GRAPHS OF EXPONENTIAL FUNCTIONS Example 1 Self Tutor A EVALUATING EXPONENTIAL FUNCTIONS y 1 1-3 -2 -1 2 3 2 4 6 8 y=2x x

Exponential functions - haesemathematics.com · B GRAPHS OF EXPONENTIAL FUNCTIONS Example 1 Self Tutor A EVALUATING EXPONENTIAL FUNCTIONS y 1 1-3 -2 -1 2 3 2 4 6 8 y=2x x

Exponential and Logarithmic Functionsasmith4math.weebly.com/uploads/2/3/6/9/23693402/chapter_7_pgs_486... · 5. 2x 2 ( x 3 ) (x) 6. 3 y -1 ( 5 x 2 y ) 7. a 8 _ a 2 ... exponential

Exponential and Logarithmic Functionsasmith4math.weebly.com/uploads/2/3/6/9/23693402/chapter_7_pgs_486... · 5. 2x 2 ( x 3 ) (x) 6. 3 y -1 ( 5 x 2 y ) 7. a 8 _ a 2 ... exponential

GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0.

GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0.