Tangent Line Approximations Consider a function f that is differentiable at c. The equation for the tangent line at the point (c,f(c)) is given by y – f(c) = f’(c)(x – c) y = f(c) + f’(c)(x – c) and is called the tangent line approximation of f at c. This is a linear function. It makes it really easy to evaluate. The nice part is when x -> c, the limit of y is f(c).

Tangent Line Approximations Tangent Line Approximations have become semi-obsolete with the invention of computers and calculators. Prior to modern technology, this was an amazing technique for finding derivative values of complex equations. For instance, it’s easy to find the value of 1 + sin(x) when x=0, but what about x = 0.01. (Without a calculator!!) Let’s take a look.

Tangent Line Approximations

Tangent Line Approximations

Differentials

Differentials

Differentials

Calculating Differentials It’s basically the same but the dx term was replaced because… du = u’ dx and dv = v’ dx

Calculating Differentials Even Leibniz, who developed this notation, did not fully understand his notation but everyone understood the application of his notation.

Calculating Differentials

Calculating Differentials