1. 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, 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…

5. Definition of Differentials Let y = f(x) represent a function that is differentiable in an open interval containing x. Since then Therefore dx is called the differential of x and is equal to  x. dy is called the differential of y and is approximately equal to  y as long as  x is close to zero.

6. Example 3 Find both  y and dy and compare. y = 1 – 2x 2 at x = 1 when  x = dx = -.1 Solution:  y = f(x +  x) – f(x) = f(.9) – f(1) = -.62 – (-1) =.38 dy = f’(x)dx = (-4)(-.1) =.4 ** Note that  y ≈ dy

7. Example 4 The radius of a ball bearing is measured to be.7 inches. If the measurement is correct to within.01 inch, estimate the propagated error in the volume V of the ball bearing. **Propagated error means the resulting change, or error, in measurement

8. Example 4 Continued To decide whether the propagated error is small or large, it is best looked at relative to the measurement being calculated. Find the relative error in volume of the ball bearing. ** Relative error is dy/y, or in this case dV/V Find the percent error, (dV/V) * 100.