Differentials
- Page ID
- 219415
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Differentials (Definitions) Recall that the derivative is defined by \( \displaystyle \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \) If we drop the limit and assume that Dx is small we have: \( \displaystyle \frac{dy}{dx} \approx \frac{\Delta y}{\Delta x} \) we can rearrange this equation to get:
Applications Suppose that a die is manufactured so that each side is .5 inches \( \pm \).01 inches. Then its volume is \(V = x^3 \) So that \( V' = 3x^3 = 3(0.5)^2 = .75 \) and \( \Delta y = (0.75)(0.01) = 0.0075 \) cu inches So that the volume of the die is in the range \((.5)^3 \pm 0.0075 = 0.125 \pm .0075 \) or between .1175 and .1375 cubic inches.
Example We can use differentials to approximate \( \sqrt{1.01} \) We have \( f(x) = x^{\frac{1}{2}} \) Since \( f(1 + \Delta x) - f(1) \approx f '(1) \Delta x \) We have \( f(1 + \Delta x) \approx f'(1) \Delta x + f(1) \) \( f(1) = 1 \) \( f'(1) = \frac{1}{2} \) \( \Delta x = .01 \) we have \( f(1 + \Delta x) \approx \frac{1}{2}(.01) + 1 = 1.005 \) (The true value is 1.00499)
Exercise: A spherical bowl is full of jellybeans. You count that there are \( 25 \pm 1 \) beans that line up from the center to the edge. Give an approximate error of the number of jelly beans in the jar for this estimate.
Example: The level of sound in decibels is equal to \( V = \frac{5}{ r^3} \) Where r is the distance from the source to the ear. If a listener stands 10 feet \( \pm \) 0.5 feet for optimal listening, how much variation will there be in the sound? What is the relative and percent error.
Solution \( V' = -15r^{-4} = \frac{-15}{10,000} = -0.0015 \) \(V' \Delta v = (-0.0015)(.5) = -0.00075 \) \( V = 0.005 \pm -0.00075 \) We have a percent error of Percent Error = \( \frac{0.00075}{0.005}\cdot 100% = 15% \) Back to the Applications home page
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