Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative a. Tangent.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative a. Tangent line to a curve, p.105, figures 3.1–3 Tangent line to a curve, p.105, figures 3.1–3 b. Definition – differentiable (3.1.1), p. 106 Definition – differentiable (3.1.1), p. 106 c. Equation of the tangent line, (3.1.2), p. 109 Equation of the tangent line, (3.1.2), p. 109 d. The derivative as a function, (3.1.3), p. xxx The derivative as a function, (3.1.3), p. xxx e. Tangent lines and normal lines, p. xxx, figure xxx Tangent lines and normal lines, p. xxx, figure xxx f. If f is differentiable at x, then f is continuous at x, p.111 If f is differentiable at x, then f is continuous at x, p.111 Differentiation Formulas a. Derivatives of sums, differences and scalar multiples, (3.2.3), (3.2.4), p. 115 Derivatives of sums, differences and scalar multiples, (3.2.3), (3.2.4), p. 115 b. The product rule, p. 117 The product rule, p. 117 c. The reciprocal rule, p. 119 The reciprocal rule, p. 119 d. Derivatives of powers and polynomials, (3.2.7), (3.2.8), pp. 117, 118 Derivatives of powers and polynomials, (3.2.7), (3.2.8), pp. 117, 118 e. The quotient rule, p. 121 The quotient rule, p. 121 Derivatives of higher Order a. The d/dx notation, p. 124, 125 The d/dx notation, p. 124, 125 b. Derivatives of higher order, p. 127 Derivatives of higher order, p. 127 The Derivative as a Rate of Change The Chain Rule a. Leibnitz form of the chain rule, p. 133 Leibnitz form of the chain rule, p. 133 b. The chain-rule theorem (3.5.6), p. 138 The chain-rule theorem (3.5.6), p. 138 Chapter 3: Differentiation Topics Differentiating the Trigonometric Functions a. Basic formulas, (3.6.1), (3.6.2), (3.6.3), (3.6.4), pp. 142, 143 Basic formulas, (3.6.1), (3.6.2), (3.6.3), (3.6.4), pp. 142, 143 b. The chain rule and the trig functions, (3.6.5), p. 144 The chain rule and the trig functions, (3.6.5), p. 144 c. My table of differentiation formulas My table of differentiation formulas Implicit differentiation; Rational Powers a. Example 1, p.147, Figures 1.71–2 Example 1, p.147, Figures 1.71–2 b. The derivative of rational powers, (3.7.1), p. 149 The derivative of rational powers, (3.7.1), p. 149 c. Chain-rule version, (3.7.2), p. 150 Chain-rule version, (3.7.2), p. 150

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 1, p. 106 Example 1 We begin with a linear function f(x) = mx + b. The graph of this function is the line y = mx + b, a line with constant slope m. We therefore expect f ’ (x) to be constantly m. Indeed it is: for h ≠ 0, and therefore f(x + h) – f(x) h = [m(x + h) + b] – [mx + b] b = mh h = m

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 2 Now we look at the squaring function f(x) = x 2. To find f ’ (x), we form the difference quotient and take the limit as h → 0. Since Therefore The slope of the graph changes with x. For x ＜ 0, the slope is negative and the curve tends down; at x = 0, the slope is 0 and the tangent line is horizontal; for x ＞ 0, the slope is positive and the curve tends up. f(x + h) – f(x) h = (x + h) 2 – x 2 h (Figure 3.1.2) (x + h) 2 – x 2 h = (x 2 + 2xh + h 2 ) – x 2 h = 2xh + h 2 h = 2x + h. Example 2, p. 106-107

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 3 Here we look for f ’ (x) for the square-root function Since f ’ (x) is a two-side limit, we can expect a derivative at most for x ＞ 0. We take x ＞ 0 and form the difference quotient We simplify this expression by multiplying both numerator and denominator by This gives us It follows that At each point of the graph to the right of the origin the slope is positive. As x increase, the slope diminishes and the graph flattens out. (Figure 3.1.3) Example 3, p. 107

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 4 Let’s differentiate the reciprocal function We begin by forming the difference quotient Now we simplify: It follows that The graph of the function consists of two curves. On each curve the slope, –1/x 2, is negative: large negative for x close to 0 (each curve steepens as x approaches 0 and tends toward the vertical) and small negative for x far from 0 (each curve flattens out as x moves away from 0 and tends toward the horizontal). (Figure 3.1.4) 1 x + h – 1x1x h = – x x(x + h) x + h x(x + h) h = –h x(x + h) h = –1 x(x + h). Example 4, p. 107-108

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 5 We take f(x) = 1 – x 2 and calculate f ’ (–2).. Example 5, p. 108

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 6 Let’s find f ’ (–3) and f ’ (1) given that f(x) =. x 2, x ≦ 1 2x – 1, x ＞ 1. Example 6, p. 109

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 7, p. 109 Example 7 We go back to the square-root function and write an equation for the tangent line at the point (4, 2). As we showed in Example 3, for x ＞ 0 Thus. The equation for the tangent line at the point (4, 2) can be written

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative Example 8 We differentiate the function f(x) = x 3 – 12x and seek the point of the graph where the tangent line is horizontal. Then we write an equation for the tangent line at the point of the graph where x = 3. First we calculate the difference quotient: Now we take the limit as h → 0: f(x + h) – f(x) h = [(x + h)] 3 – 12(x + h)] – [x 3 – 12x] h = x 3 + 3x 2 h + 3xh 2 + h 3 – 12x – 12h – x 3 + 12x h = 3x 2 + 3xh + h 2 – 12. Example 8, p. 109-110

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative The function has a horizontal tangent at the points (x, f(x)) where f ’(x) = 0. In this case f ’(x) = 0 iff 3x 2 – 12 = 0 iff x = ±2. The graph has a horizontal tangent at the points (–2, f(–2) ) = (–2, 16) and (2, f(2)) = (2, –16). The graph of and the horizontal tangents are shown in Figure 3.1.5. The point on the graph where x = 3 is the point (3, f(3)) = (3, –9). The slope at this point is f ’(3) = 15, and the equation of the tangent line at this point can be written y + 9 = 15(x – 3). Example 8, p. 109-110

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative If f is differentiable at x, then f is continuous at x, p. 111

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Derivatives of sums, differences and scalar multiples, (3.2.3), (3.2.4), p. 115, 116

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Derivatives of powers and polynomials, (3.2.7), (3.2.8), pp. 117, 118

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 1, p. 118 Example 1 Differentiate F(x) = (x 3 – 2x + 3)(4x 2 + 1) and find F ’(–1).

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 2, p. 118 Example 2 Differentiate F(x) = (ax + b)(cx + d), where a, b, c, d are constants.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 3, p. 119 Example 3 Suppose that g is differentiable at each x and that F(x) = (x 3 – 5x)g(x). Find F ’(2) given that g(2) = 3 and g’(2) = –1.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas The reciprocal rule, p. 119-120 5x25x2 6x6x 1212 Example 4 Differentiate f(x) = – and find f ’( ).

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 5, p. 120 Example 5 Differentiate f(x) =, where a, b, c are constants. 1 ax 2 + bx + c

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 6, p. 121 Example 6 Differentiate F(x) =. 6x 2 – 1 x 4 + 5x + 1

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 7, p. 121 Example 7 Find equations for the tangent and normal lines to the graph of f(x) = at the point (2, f(2)) = (2, –2). 3x 1 – 2x

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiation Formulas Example 8, p. 122 Example 8 Find the point on the graph of f(x) = where the tangent line is horizontal. 4x x 2 + 4

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 1, p. 125 Example 1 Find for y =. dy dx 3x – 1 5x + 2

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 2, p. 125 Example 2 Find for y = (x 3 + 1)(3x 5 + 2x – 1). dy dx

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 4, p. 126 Example 4 Find for u = x(x + 1)(x + 2).

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 5, p. 126 Example 5 Find dy/dx at x = 0 and x = 1 given that y =. x 2 x 2 – 4

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 6, p. 127 Example 6 If f(x) = x 4 – 3x –1 + 5, then f ’(x) = 4x 3 + 3x–2 and f ”(x) = 12x 2 – 6x –3.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Derivatives of Higher Order Example 8, p. 127-128 Example 8 Finally, we consider y = x –1. In the Leibniz notation. = –x –2, = 2x–3, = –6x –4, = 24x –5, …. On the basis of these calculations, we are led to the general result = (–1) n n!x –n – 1. [Recall that n! = n(n – 1)(n – 2)…3 ． 2 ． 1.] In Exercise 61 you are asked to prove this result. In the prime notation we have y’ = –x –2, y” = 2x –3, y’” = –6x –4, y (4) = 24x –5, …. In general y (n) = (–1) n n!x –n – 1. dy dx d 2 y dx 2 d 3 y dx 3 d 4 y dx 4 d n y dx n

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change The derivative as a rate of change, p. 130

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Example 1, p. 130 Example 1 The area of a square is given by the formula A = x 2 where x is the length of a side. As x changes, A changes. The rate of change of A with respect to x is the derivative = (x 2 ) = 2x. When x =, this rate of change is : the area is changing at half the rate of x. When x =, the rate of change of A with respect to x is 1: the area is changing at the same rate as x. When x = 1, the rate of change of A with respect to x is 2: the area is changing at twice the rate of x. In Figure 3.4.3 we have plotted A against x. The rate of change of A with respect to x at each of the indicated point appears as the slope as the slope of the tangent line. dA dx d dx 1414 1212 1212

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Example 2, p. 131 Example 2 An equilateral triangle of side x has area The rate of change of A with respect to x is the derivative When, the rate of change of A with respect to x is 3. In other words, when the side has length, the area is changing three time as fast as the length of the side.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Example 3, p. 131 Example 3 Set (a) Find the rate of change of y with respect to x at x = 2. (b) Find the value(s) of x at which the rate of change of y with respect to x is 0.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Example 4, p. 131-132 Example 4 Suppose that we have a right circular cylinder of changing dimensions. (Figure 3.4.4) When the base radius is r and the height is h, the cylinder has volume. If r remains constant while h changes, then V can be viewed as a function of h. The rate of change of V with respect to h is the derivative If h remains constant while r changes, then V can be viewed as a function of r. The rate of change of V with respect to r is the derivative dV dh = πr 2. dV dh = 2πrh.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative as a Rate of Change Example 4, p. 131-132 Suppose now that r changes but V is kept constant. How does h change with respect to r? To answer this, we express h in term of r and V Since V is held constant, h is now a function of r. There rate of change of h with respect to r is the derivative h = = r –2. Vπr2Vπr2 VπVπ dh dr = r –3 = – r –3 = –. 2Vπ2Vπ 2( π r 2 h) π 2hr2hr

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 1, p. 133 Example 1 Find dy/dx by the chain rule given that y = and u = x 2. u – 1 u + 2

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 3, p. 135 Example 3 Since We have d dx d dx d dx [1 + (2 + 3x) 5 ] 3 = 3[1 + (2 + 3x) 5 ] 2 [1 + (2 + 3x) 5 ]. d dx [1 + (2 + 3x) 5 ] 3 = 5(2 + 3x) 4 (2 + 3x) = 5(2 + 3x) 4 (3) = 15(2 + 3x) 4. d dx [1 + (2 + 3x) 5 ] 3 = 3[1 + (2 + 3x) 5 ] 2 [15(2 + 3x) 4 ] = 45(2 + 3x) 4 [1 + (2 + 3x) 5 ] 2.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 4, p. 135 Example 4 Calculate the derivative of f(x) = 2x 3 (x 2 – 3) 4.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 5, p. 135 Example 5 Find dy/ds given that y = 3u + 1, u = x –2, x = 1 – s.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 6, p. 136 Example 6 Find dy/dt at t = 9 given that

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 7, p. 136 Example 7 Gravel is being poured by a conveyor onto a conical pile at the constant rate of 60π cubic feet per minute. Frictional forces within the pile are such that the height is always two-thirds of the radius. How fast is the radius of the pile changing at the instant the radius is 5 feet?

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Basic formulas, (3.6.1), (3.6.2), Example 1, p. 142 Example 1 To differentiate f(x) = cos x sin x, we use the product rule: f ’(x) = cos x (sin x) + sin x (cos x) d dx d dx

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Basic formulas (3.6.3), (3.6.4), p.143

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 2, p.143 Example 2 Find f ’( π /4) for f(x) = x cot x.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 3, p.143 Example 3 Find

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 4, p.143 Example 4 Find an equation for the line tangent to the curve y = cos x at the point where x = π /3

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 5, p.144 Example 5 Set f(x) = x + 2 sin x. Find the numbers x in the open interval (0, 2 π ) at which (a) f ’(x) = 0, (b) f ’(x) ＞ 0, (c) f ’(x) ＜ 0.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions The chain rule and the trig functions, (3.6.5), p. 144

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 6, p. 144 Example 6 d dx (cos 2x) = –sin 2x (2x) = –2sin 2x. d dx

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 7, p. 144 Example 7 d dx [sec(x 2 + 1)] = sec(x 2 + 1)tan(x 2 + 1) (x 2 + 1) = 2x sec(x 2 + 1)tan(x 2 + 1). d dx

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Differentiating the Trigonometric Functions Example 9, p. 145 Example 9 Find (sin x°). d dx

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 1, p. 147, Figures 3.71–2

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 2, p. 147 Example 2 Assume that y is a differentiable function of x which satisfies the given equation. Use implicit differentiation to express dy/dx in terms of x and y. (a) 2x 2 y – y 3 + 1 = x + 2y. (b) cos(x – y) = (2x + 1) 3 y.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 3, p. 147-148, Figures 3.7.3 Example 3 Figure 3.7.3 show the curve 2x 3 + 2y 3 = 9xy and the tangent line at the point (1, 2). What is the slope of the tangent line at that point?

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 4, p. 148 Example 4 The function y = (4 + x 2 ) 1/3 satisfies differentiation y 3 – x 2 = 4. Use implicit differentiation to express d 2 y/dx 2 in terms of x and y.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers The derivative of rational powers, (3.7.1), p. 149

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Chain-rule version, (3.7.2), p. 150

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 5, p. 150 Example 5 d dx (a) [(1 + x 2 ) 1/5 ] = (1 + x 2 ) –4/5 (2x) = x(1 + x 2 ) –4/5. 1515 2525 d dx (b) [(1 + x 2 ) 2/3 ] = (1 – x 2 ) –1/3 (–2x) = – x(1 – x 2 ) –1/3. 2323 4343 d dx (c) [(1 – x 2 ) 1/4 ] = (1 – x 2 ) –3/4 (–2x) = – x(1 – x 2 ) –3/4. 1414 1212 The first statement holds for all real x, the second for all x ≠ ±1, and the third only for x ∊ (–1, 1).

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 6, p. 150 Example 6 The result holds for all x ＞ 0.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative a. Tangent.

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Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative a. Tangent.

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