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# Note for MATH 263A with Professor Holston at Ohio-section 2.2 notes

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###### Class Notes

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Date Created: 02/06/15

The Derivative as 2 Function Chris Hoiston The Derivative as a Function Chris Holston The Derivative as 2 Function Tangent Lines Chris Holston Preliminaries The following concept is a way of extending the notion of slopes to all functions not just linear ones The Derivative as 2 Function Tangent Lines cm Holston Preliminaries The following concept is a way of extending the notion of slopes to all functions not just linear ones Definition The tangent line ofa function fx at a real number a is the line crossing through the point 37 fa whose slope is given by The Derivative as 2 Function Tangent Lines cm Holston Preliminaries The following concept is a way of extending the notion of slopes to all functions not just linear ones Definition The tangent line ofa function fx at a real number a is the line crossing through the point 37 fa whose slope is given by fX fa m lim provided this limit exists The Derivative as a Function Chris Holston Preliminaries The following concept is a way of extending the notion of slopes to all functions not just linear ones Definition The tangent line ofa function fx at a real number a is the line crossing through the point 37 fa whose slope is given by m lim w provided this limit exists The limit above will obviously always be indeterminate of the form The Derivative as a Function Tangent Lines cm Holston Preliminaries The following concept is a way of extending the notion of slopes to all functions not just linear ones Definition The tangent line ofa function fx at a real number a is the line crossing through the point 37 fa whose slope is given by m lim w provided this limit exists The limit above will obviously always be indeterminate of O the form 7 m is called the instantaneous rate of change of f at a We will almost always be using the equivalent expression 2 a h 7 f a m m M H0 h The Derivative as 2 Function Definition of derivative Chris Hoiston Preiiminzries Definition Given a function fx we define the derivative of f with respect to X to be fx h 7 fx fX Iiilli h The Derivative as 2 Function Definition of derivative cm Hoiston Preiiminzries Definition Given a function fx we define the derivative of f with respect to X to be fx h 7 fx 7 f X 7 210 h Given a constant a if f a exists we say that f is differentiable at a The Derivative as a Function Definition of derivative cm Hoiston Preliminaries Definition Given a function fx we define the derivative of f with respect to X to be fx h 7 fx 7 f X 7 210 h Given a constant a if f a exists we say that f is differentiable at a We say if is differentiable on an interval I is f is differentiable at every 3 in I The Derivative as a Function Definition of derivative cm Hoiston Preliminaries Definition Given a function fx we define the derivative of f with respect to X to be 7 X h 7 7 X f X 7 210 h Given a constant a if f a exists we say that f is differentiable at a We say if is differentiable on an interval I is f is differentiable at every 3 in I We say if is differentiable if f is differentiable at every real number The Derivative as 2 Function Differentiability Chris Hoiston Preiiminzries If f is differentiable at a then f is continuous at a The Derivative as 2 Function Differentiability cm Hoiston Preiiminzries If f is differentiable at a then f is continuous at a In fact differentiability requires notjust continuity but also continuity of f The Derivative as a Function Differentiability cm Holston Preliminaries If f is differentiable at a then f is continuous at a In fact differentiability requires notjust continuity but also continuity of 2quot There are 3 ways to tell graphically if a function is not differentiable at a The Derivative as a Function Differentlablllty cm Holston Preliminaries If f is differentiable at a then f is continuous at a In fact differentiability requires notjust continuity but also continuity of 2quot There are 3 ways to tell graphically if a function is not differentiable at a gt 1 f has a discontinuity at a The Derivative as a Function Differentiability cm Holston Preliminaries If f is differentiable at a then f is continuous at a In fact differentiability requires notjust continuity but also continuity of 2quot There are 3 ways to tell graphically if a function is not differentiable at a gt 1 f has a discontinuity at a gt 2 f has a sharp turn or corner The Derivative as a Function Differentiability cm Holston Preliminaries If f is differentiable at a then f is continuous at a In fact differentiability requires notjust continuity but also continuity of 2quot There are 3 ways to tell graphically if a function is not differentiable at a gt 1 f has a discontinuity at a gt 2 f has a sharp turn or corner gt 3 the tangent line is vertical The Derivative as 2 Function Notation Chris Holston Preliminaries The following are all notations for the derivative of a function y fx The Derivative as 2 Function Notation Chris Holston Preliminaries The following are all notations for the derivative of a function y fx f x y lm Dxmxn The Derivative as a Function Notation cm Holston Preliminaries The following are all notations for the derivative of a function y fx dy d 7 7 i i f x a y 7 dx dxmxn Dxmxn We can take the derivative of the derivative function ie the second derivative and continue to do so any number of times The Derivative as a Function cm Holston Preliminaries Notation The following are all notations for the derivative of a function y fx dy d 7 7 i i f x a y 7 dx dxmxn Dxmxn We can take the derivative of the derivative function ie the second derivative and continue to do so any number of times dzy The second derIvatIve Is notated f X y W and In general the nth derivative is notated The Derivative as a Function cm Holston Preliminaries Notation The following are all notations for the derivative of a function y fx dy d 7 7 i i f x a y 7 dx dxmxn Dxmxn We can take the derivative of the derivative function ie the second derivative and continue to do so any number of times dzy The second derIvatIve Is notated f X y W and In general the nth derivative is notated d y n n X y W The Derivative as a Function Chris Hoiston Exerci 595 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative fx 15X2 7 x 3 7 The Derivative as a Function Chris Hoiston Exerci 595 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative fx 15X2 7 x 3 7 f x 3x71 The Derivative as a Function Chris Hoiston Exerci 595 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative fx 15X2 7 x 3 7 f x 3x 7 1 Domf R The Derivative as a Function Chris Hoiston Exerci 595 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative fx 15X2 7 x 3 7 f x 3x 7 1 Domf R Domf R The Derivative as a Function Chris Hoiston Exerci 595 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative X X173x The Derivative as a Function Exerci 595 Chris Hoiston 22 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative x 0 7 17 3x fx i 1 7 3x2 The Derivative as a Function Exerci 595 Chris Hoiston 22 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative fx 137 10 W W Domf m The Derivative as a Function Exerci 595 Chris Hoiston 22 Find the derivative of the function using the de nition of derivative State the domain of the function and the domain of its derivative 0 1 10 W 1 7 3x2 Domf R Domf R wigt

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