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Understanding Differentials and Their Applications in Calculus

This text explores the concept of differentials in calculus, particularly focusing on the relationship between the differential of a function ( y = f(x) ) and its derivative. It illustrates how to calculate dy for a specific function, ( y = x^2 ), at a given point and examines the concept of error estimation using real-world problems, such as measuring the radius of a ball bearing. The text emphasizes the practical applications of differentials in estimating changes and understanding propagated errors in measurements.

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Understanding Differentials and Their Applications in Calculus

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  1. 3.9 Differentials Let y = f(x) represent a function that is differentiable in an open interval containing x. The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is given by dy = f’(x) dx

  2. dy x

  3. Comparing Let y = x2. Find dy when x = 1 and dx = 0.01. Compare this value to when x = 1 and = 0.01. dy = f’(x) dx dy = 2x dx dy = 2(1)(.01) = .02 = f(1.01) – f(1) = 1.012 – 12 = .0201 dy = 0.02 (1, 1)

  4. Estimation of error. The radius of a ball bearing is measured to be .7 inch. If the measurement is correct to within .01 inch, estimate the propagated error in the Volume of the ball bearing. r = .7 and r = .7 relative error is propagated error Percentage error

  5. Finding differentials Function Derivative Differential dy = 2x dx y = x2 dy = 2cos x dx y = 2sin x y = x cos x y = sin 3x

  6. Let x = 100 and and dy = f’(x) dx

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