Linear Approximation and Differentials Lesson 4.8

Tangent Line Approximation Consider a tangent to a function at a point x = a Close to the point, the tangent line is an approximation for f(x) a f(a) y=f(x) The equation of the tangent line: y = f(a) + f ‘(a)(x – a)

Tangent Line Approximation We claim that This is called linearization of the function at the point a. Recall that when we zoom in on an interval of a function far enough, it looks like a line

New Look at dy = rise of tangent relative to  x = dx  y = change in y that occurs relative to  x = dx x  x = dx dy yy x +  x

New Look at We know that  then Recall that dy/dx is NOT a quotient  it is the notation for the derivative However … sometimes it is useful to use dy and dx as actual quantities

The Differential of y Consider Then we can say  this is called the differential of y  the notation is d(f(x)) = f ’(x) * dx  it is an approximation of the actual change of y for a small change of x

Animated Graphical View Note how the "del y" and the dy in the figure get closer and closer

Try It Out Note the rules for differentials Page 274 Find the differential of 3 – 5x 2 x e -2x

Differentials for Approximations Consider Use Then with x = 25, dx =.3 obtain approximation

Propagated Error Consider a rectangular box with a square base  Height is 2 times length of sides of base  Given that x = 3.5  You are able to measure with 3% accuracy What is the error propagated for the volume? x x 2x

Propagated Error We know that Then dy = 6x 2 dx = 6 * * = This is the approximate propagated error for the volume

Propagated Error The propagated error is the dy  sometimes called the df The relative error is The percentage of error  relative error * 100%

Assignment Lesson 4.8 Page 276 Exercises 1 – 45 odd