1. Section 3.9 - Differentials Tangent Line Approximations A tangent line approximation is a process that involves using the tangent line to approximate a value. Newton used this method to approximate zeros of functions. Another use of this method is to approximate the value of a function.

2. Approximating the Value of a Function To approximate the value of f at x = c, f(c): 1.Choose a convenient value of x close to c, call this value x 1. 2.Write the equation of the tangent line to the function at x 1. y – f(x 1 ) = f ’(x 1 )( x - x 1 ) 3.Substitute the value of c in for x and solve for y. 4.f(c) is approximately equal to this value of y.

3. Example 1 If f(x) = x^2 + 3, use a tangent line approximation to find f(1.2). 1.Since c = 1.2, choose x 1 = 1, which is close to 1.2 2.Since f(1) = 4 and f’(x) = 2x gives f’(1) = 2, then eq. of tangent line is y – 4 = 2(x – 1) 3.Substitute 1.2 in for x and y = 4 + 2(.2) = 4.4 4.Therefore, since 1 is close to 1.2, f(1.2) is close to 4.4 5.Check this with the actual value f(1.2) = 4.44

4. Example 2 Approximate sqrt(18) using the tangent line approximation method (also called using differentials). 1.Choose x 1 = 16 and let f(x) = sqrt(x), then f (x 1 ) = 4 and f’(x 1 ) = 1/(2 * sqrt(16)) = 1/8 2.Eq. of Tangent: y – 4 = (1/8)(x – 16) 3.When x = 18, y = 4 + (1/8)(2) = 4 + ¼ = 4.25 4.Therefore sqrt(18) is approximately 4.25 5.Check with sqrt(18) on your calculator: 4.243…

6. Definition of 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 non-zero real number. The differential of y (denoted by dy) is dy = f’(x)dx. x (the change in x) and dx are equivalent. We say that dx = x y (the change in y) can be approximated using dy. We say that y is approximately equal to dy. Therefore, y ≈ f’(x)dx y