1. Ch. 5 – Applications of Derivatives
5.5 – Linearization and Differentials

2. All differentiable functions are locally linear
They look like straight lines when you zoom in enough
If we look at 2 points that are really close on a curve, we can approximate the slope of those points using a straight line:
L(x) defines the linearization of f at a. Linearizations can be used to approximate values of irrational numbers.
(a, f(a))
(x, y) a x

3. Ex: Find the linearization of f(x) = cosx at x = π/2
Ex: Find the linearization of f(x) = cosx at x = π/2. Then use that to approximate cos(1.75).
We can use linearization because 1.75 is close to π/2.
Basically, we’re using point-slope form, so we need 2 things:
Slope:
Point:
Put it all together!
To approximate cos(1.75), plug in 1.75 for x in the above equation.

4. Special Linearization for Binomials:
This only works when x is very small!
Ex: Find a polynomial that approximates the following functions for x≈0 .

5. Differentials
We can consider derivatives as fractions (even though they’re not fractions)
This notation allows us to view the change in y (dy) as a function of the change in x (dx)
Ex: Find the differential dy of y = x5 + 37x. Then evaluate at x = 1, dx = .01.
Take the derivative, but put a dy when you take a derivative of a y, and put a dx when you take a derivative of an x.

6. Differentials
Ex: Find the differential dy of x + y = xy. Then evaluate at x = 2, dx = .01.
Take the derivative implicitly!
Plug x into the original problem to find y = 2 when x = 2...

7. Ex: About how accurately should we measure the radius r of a sphere to calculate its surface area S within 1% of its true value?
We are given , and we want to find .

8. Newton’s Method
To find solutions to the equation f(x)=0, guess a solution near the solution you desire.
Use that guessed x-value as xn, then enter it into the formula:
Take the new value xn+1 and enter it back into the equation. Repeat this process until you have converged on the solution.

9. Newton’s Method
Ex: Find the solution to x3 + 3x + 1 = 0 using Newton’s Method.
Let’s guess x = 0 to start.
Evaluate this expression, then enter the following into your calculator:
Press enter until you converge on the same value each time. This value is the solution... ≈