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**Unformatted text preview: **L21 – Linear Approximations and Diﬀerentials Linear or Tangent Line Approximations Tangent to y = f (x) at ( a, f (a) ) L(x) = The linearization of f at a 1 Suppose a cup of coﬀee which is 200◦ F is placed in a 75◦ F room. Suppose that it takes 10 minutes to reach 150◦ F and 20 minutes to reach 127◦ F. Let T (t) represent the temperature at t minutes. Use a linear approximation to predict the temperature at 30 minutes. 2 Find the linearization of f (x) = √ √ 8.95 and 9.07 . √ x + 5 at x = 4, and use it to approximate 3 Def – Let y = f (x) be a diﬀerentiable function. The diﬀerential dx is an independent variable (it can be any real number). The diﬀerential dy is deﬁned by Find dy if y = 3x2 − x4 . Evaluate dy when x = 3 and dx = .01 . 4 Geometrically f (a + ∆x) = f (a + ∆x) ≈ 5 Compare ∆y and dy if y = f (x) = x3 + x and x changes from 2 to 2.01. 6 Use diﬀerentials to approximate √ 3 7.8 . Actual value 1.98319248 . . . 7 Approximate the error in using 4 inches to calculate the volume of a cube if the measurement might be oﬀ by 0.2 inches. Relative error: ∆V V Percentage error: 8 ...

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