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3.10 Linear Approximations and Differentials 1 Dr. Erickson

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If y = f(x), where f is a differentiable function, then the differential dx is an independent variable (can be given the value of any real number). The differential dy is defined in terms of dx by dy is the dependent variable that depends on values of x and dx. 3.10 Linear Approximations and Differentials 2Dr. Erickson

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Let and be points on the graph of f and let dx = x. The corresponding change in y is 3.10 Linear Approximations and Differentials 3 Propagated error Exact Value Measured Value Measurement error Dr. Erickson

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The approximation y dy becomes better as x becomes smaller. For more complicated functions, it may be impossible to compute y exactly. In such cases, the approximation by differentials is useful. The linear approximation can be written as 3.10 Linear Approximations and Differentials 4Dr. Erickson

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Find the differential of each function. 3.10 Linear Approximations and Differentials 5Dr. Erickson

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a) Find the differential dy. b) Evaluate dy for the given values of x and dx. 3.10 Linear Approximations and Differentials 6Dr. Erickson

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Compute y and dy for the given values of x and dx= x. Sketch a diagram showing the line segments with lengths dx, dy, and y. a) x = 1, x = 1 b) x = 0, x = 0.5 3.10 Linear Approximations and Differentials 7Dr. Erickson

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Use a linear approximation (or differentials) to estimate the given number. a) (2.001) 5 b) e -0.015 c) tan44 o 3.10 Linear Approximations and Differentials 8Dr. Erickson

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The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. a. Use differentials to estimate the maximum error in the calculated area of the disk. b. What is the relative error? What is the percentage error? 3.10 Linear Approximations and Differentials 9Dr. Erickson

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a. Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness r. b. What is the error involved in using the formula from part (a)? 3.10 Linear Approximations and Differentials 10Dr. Erickson

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One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30 o, with a possible error of 1 o. a) Use differentials to estimate the error in computing the length of the hypotenuse. b) What is the percentage error? 3.10 Linear Approximations and Differentials 11Dr. Erickson

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If current I passes through a resistor with resistance R, Ohm’s Law states that the voltage drop is V=RI. If V is constant and R is measured with a certain error, use differentials to show that the relative error in calculating I is approximately the same (in magnitude) as the relative error in R. 3.10 Linear Approximations and Differentials 12Dr. Erickson

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When blood flows along a blood vessel, the flux F (volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel: (this is known as Poiseuille’s Law). A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon- tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in F is about four times the relative change in R. How will a 5% increase in the radius affect the flow of blood? 3.10 Linear Approximations and Differentials 13Dr. Erickson

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