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Chapter 3 – Differentiation Rules 3.10 Linear Approximations and Differentials 1 3.10 Linear Approximations and Differentials Dr. Erickson Differentials 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 f '( x)dx dy is the dependent variable that depends on values of x and dx. 2 3.10 Linear Approximations and Differentials Dr. Erickson Differentials Let P x, f ( x) and Q x x, f ( x x) be points on the graph of f and let dx = x. The corresponding change in y is y f ( x x) f ( x) Propagated error Measured Value Exact Value Measurement error 3 3.10 Linear Approximations and Differentials Dr. Erickson NOTE 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 f ( x) f (a) f '(a)( x a) can be written as f (a dx) f (a) dy 4 3.10 Linear Approximations and Differentials Dr. Erickson Example 4 Find the differential of each function. s 1. y 1 2s 2. y eu cos u 5 3.10 Linear Approximations and Differentials Dr. Erickson Example 5 y cos x a) b) Find the differential dy. Evaluate dy for the given values of x and dx. x 6 3 dx 0.05 3.10 Linear Approximations and Differentials Dr. Erickson Example 6 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. 7 a) x = 1, x = 1 y x b) x = 0, x = 0.5 y ex 3.10 Linear Approximations and Differentials Dr. Erickson Example 7 Use a linear approximation (or differentials) to estimate the given number. 8 a) (2.001)5 b) e-0.015 c) tan44o 3.10 Linear Approximations and Differentials Dr. Erickson Example 8 The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. a. b. 9 Use differentials to estimate the maximum error in the calculated area of the disk. What is the relative error? What is the percentage error? 3.10 Linear Approximations and Differentials Dr. Erickson Example 9 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)? 10 3.10 Linear Approximations and Differentials Dr. Erickson Example 10 One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30o, with a possible error of 1o. a) b) 11 Use differentials to estimate the error in computing the length of the hypotenuse. What is the percentage error? 3.10 Linear Approximations and Differentials Dr. Erickson Example 11 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. 12 3.10 Linear Approximations and Differentials Dr. Erickson Example 12 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: F kR 4 (this is known as Poiseuille’s Law). A partially clogged artery can be expanded by an operation called angioplasty, in which a balloontipped 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? 13 3.10 Linear Approximations and Differentials Dr. Erickson