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Introduction to Differential Equations

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Introduction to Differential Equations Derivatives – An Introduction Concepts of primary interest: Definition: first derivative Differential of a function Left-side and right-side limits Higher derivatives Mean value and Rolle’s theorem L’Hospital Rule for indeterminate forms Implicit differentiation Inverse functions Logarithmic Derivative Method for Products Extrema and the Determinant Test for functions of two variables Lagrange Undetermined Multipliers (*** needs serious edit ****) Sample calculations: 2 Derivative of x Chain Rule Product Rule Saddle-points and local extrema Derivative of the inverse function for Squared d(sinθ) /dθ = 1 for θ = 0 Applications Closest point of approach or CPA (nautical term) Tools of the Trade Derivatives of inverse functions Contact: [email protected] 11/1/2012 Plotting inverse functions ***ADD a table of the derivatives of common functions See Tipler; Lagrange Approximating Functions Many functions are complicated, and their evaluations require detailed, tortuous calculations. To avoid the stress, approximate representations are sometimes substituted for the actual functions of interest. For a continuous function f(x), it might be adequate to replace the function in a small interval around x0 by its value at that point f(x0). A better approximation for a function f(x) with a continuous derivative follows from the definition of that derivative. df fx()− fx ( ) = Limit 0 [DI.1] x0 xx→ dx 0 x− x0 df df Clearly, fx()≅+ fx ( ) ( x − x) where is the slope of the function at x0.. 00dx dx x0 x0 11/1/2012 Physics Handout Series.Tank: Derivatives D/DX- 2 Figure DI-1 This handout formalizes the definition of a derivative and illustrates a few basic properties and applications of the derivative. The derivative of the function at an general point x is to be defined as: The limit of the slope of the chord df() x fx()21− fx () = Limit xx, → x dx 12 (x21− x This relation requires that the limit exist and be the same for all limit paths with x1 and x2 approaching x. This condition means that the slope of the chord must converge to the same value no matter how x1 and x2 approach x. This includes arbitrary one-sided approaches dfx() fx (+∆ x ) − fx () fx (+∆ x ) − fx () = Limit = Limit [DI.2] dx ∆→xx00()x+∆ x − x ∆→ ∆x 11/1/2012 Physics Handout Series.Tank: Derivatives D/DX- 3 and the left-side (from smaller values) and right-side (from larger values) approaches which must exist and be equal. dfx() fx (+∆ x ) − fx () fx (+∆ x ) − fx () = Limit = Limit −+ [DI.3] dx ∆→xx00∆∆x ∆→ x left−− side right side At minimum, the function f(x) must be continuous at a point if its derivative exists at that point. As before, the derivative is useful for representing the change in the value of the function corresponding to an infinitesimal change in its argument. df df= f( x +− dx ) f () x ≅ dx [DI.4] dx The infinitesimal change df is called the differential of f(x). The function plotted to the right is smooth except in a neighborhood of x0. (The function has a cusp at x0.) The derivative of the function is not defined at that point. Which condition required for the derivative to be defined is not met at x0? Smoothness: A function is smooth if it is continuous and continuously differentiable. If the function is continuous and continuously differentiable through order n, it is n- smooth. Differentiating an n-smooth function yields a result that is (n-1)-smooth while integration yields an (n+1)-smooth result. Cauchy’s Mean Value Theorem: If a function is continuous and has a continuous 11/1/2012 Physics Handout Series.Tank: Derivatives D/DX- 4 derivative in an interval [a, b], then there exists at least one point c in that interval such that: df f() b− f () a = dxxc= b− a That is: For a continuously differentiable function, the derivative attains its average value over a closed interval somewhere in that interval. Rolle’s theorem is the special f(b) = f(a) case for which the derivative must vanish somewhere in the interval. Linear Operation: The derivative operation is linear. d dd [afx()+= bgx ()] a[ f () x] + b[ gx ()] dx dx dx Sample Calculation: Compute the derivative of x2. (We assume that x is the independent variable.) dx()2 ( x+∆ x )22 − x 2x∆ x + ( ∆ x )2 =Limit =Limit =Limit[22 x +∆ x] = x [DI.5] dx ∆→xxx000()x+∆ x − x ∆→ ∆x ∆→ Vital Rule: The Chain Rule (Know this one by heart!) Start: F(x) = f(u(x)) , a function of a function of x. dfux(()) fux ((+∆ x )) − fux (()) fux((+∆ x )) − fux (()) ux (+∆ x ) − ux () = Limit = Limit dx ∆→xx00∆x ∆→ ux(+∆ x ) − ux () ∆x df(()) u x f ( u+∆ u ) − f () u u( x +∆ x ) − u () x df du = Limit * Limit = [DI.6] dx ∆→xx00()u+∆ u − u ∆→ ∆x du dx The derivative of a function f of a function u(x) is the derivative of f with respect to its argument times the derivative of its argument with respect to x. The rate of change of f(u(x)) with respect to x is the rate of change of f(u) with respect to u times the rate of change of u(x) with respect to x. Given the importance of the Chain Rule, it is to be df(()) u x stated one more time. is the rate of change in the value of f(u(x)) with respect dx 11/1/2012 Physics Handout Series.Tank: Derivatives D/DX- 5 df() u to changes in the value of x which is the rate of change of the function f(u) with du respect to changes in the value of its argument multiplied by the rate of change of the df(()) u x df du argument u with respect to changes in the value of x. = dx du dx Sample Calculation: The Chain Rule f(u) = eu ; u(x) = ln(x). Examine: df(()) u x df du = . dx du dx u df u d[ln(x)] -1 f(u) = e ⇒ /du = e ; u(x) = ln(x) ⇒ /dx = x . Hence df(()) u x ln( x )) 1 1 =ex = =1 dx x x The result is expected as e ln(x) = x. Sample Calculation: The Product Rule Start: F(x) = f(x) g(x) d( f () xgx ()) f ( x+∆ xgx ) ( +∆ x ) − f () xgx () = Limit dx ∆→x 0 ∆x f( x+∆ xgx ) () − f () xgx () f ( x+∆ xgx ) ( +∆ x ) − f ( x +∆ xgx ) () = Limit + ∆→x 0 ∆∆xx f( x+∆ x ) − f () x gx(+∆ x ) − gx () =Limit g() x+ Limit[ f ( x +∆ x )] Limit ∆→x0∆∆xx ∆→xx00∆→ d( f () xgx ()) d ( f ()) x d( gx ()) ⇒=+gx() f () x [DI.7] dx dx dx The rule can be understood in terms of a rectangle representing the product f(x+∆x) g(x+∆x) – f(x g(x) = (f + ∆f) (g +∆g) – f g = f ∆g + ∆f g + ∆f ∆g Full Area = (f + ∆f) (g +∆g) The increase in the product is f ∆g + ∆f g + ∆f ∆g where the piece ∆f ∆g is second order in small ∆g ∆f (∆f ∆g) things. The limit of f /∆x + /∆x g + /∆x dg df df becomes [ f /dx + /dx g + /dx (dg)] which 11/1/2012 Physics Handout Series.Tank: Derivatives D/DX- 6 dg df approaches f /dx + /dx g as dx and dg go to zero. Used: g(x) is a continuous function. ∆(fg) = f ∆g + ∆f g + ∆f ∆g ≈ f ∆g + ∆f g Logarithmic Derivative Method for Products It can be tedious to apply the product rule to f(x), a product of several factors. One can take the derivative of the natural log of f(x). d 1 df df d (ln[fx ( )]) = . It follows that: = fx( ) (ln[ fx ( )]) . dx f() x dx dx dx df x2 x +1 Exercise: Compute forfx ( ) = using the product and quotient rules. dx (x2+ 3) 2/3 Repeat using the logarithmic derivative approach. Higher Order Derivatives: A second derivative of f(x) w.r.t. x is the first derivative of the first derivative of f(x) and so on. Partial Derivative: A function of several variables requires some generalization of the concept of derivative. Consider a function of three variables: f(x,y,t). The partial derivative is introduced to represent the rate of change of the value of the function when only one variable is incremented with respect to the variable incremented. To designate that only one argument of the list is to be incremented, the ‘straight-back’ derivative symbol is replaced by the ‘curly-back’ derivative symbol. More formally, the symbol ∂ is known as Jacobi’s delta. It is pronounced as ‘curly dee’ or partial. The definition of the partial derivative of f(x,y,t) with respect to x is: 11/1/2012 Physics Handout Series.Tank: Derivatives D/DX- 7 ∂fxyt(,,) fx (+∆ xyt ,,) − fxyt (,,) = Limit [DI.8] ∂ ∆→x 0 +∆ − x y, t fixed ()x xx For our purposes the partial derivative with respect to x is defined just as the derivative with respect to x with the understanding that the other arguments are to be treated as constants. This definition leads to the partial differential w.r.t. x which is ∂f dx and the total differential of f(x,y,t): ∂x ∂∂∂fff df=++ dx dy dt ∂∂∂xyt The Total Derivative: Given a function of several variables such as f(x,y,t), the total rate of change of the with respect to one variable is needed when the other arguments have an implicit dependence on that variable. That is: for f(x(t),y(t), t), df∂∂∂ f dx f dy f =++ [DI.9] dt∂∂∂ x dt y dt t Note that this expression includes all the variation in the value of f(x,y,t) when t varies.
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