1. DIFFERENTIALS
Section 3.9

2. When you are done with your homework, you should be able to…
Understand the concept of a tangent line approximation
Compare the value of the differential, dy, with the actual change in y,
Estimate a propagated error using a differential
Find the differential of a function using differentiation formulas

3. Aristotle Plato Pythagoras Archimedes
My mentor was exiled from Athens and committed suicide. I taught and clarified the Pythagorean philososphy of nature. I taught that atoms were in the shape of regular polyhedra. Who am I?
Aristotle
Plato
Pythagoras
Archimedes

4. TANGENT LINE APPROXIMATIONS (AKA LINEAR APPROXIMATION)
In the last section, we used Newton’s Method to use a tangent line to a graph to approximate the graph.
In this section we will examine other situations where the graph of a function can be approximated by a straight line.

5. What is the equation of the tangent line at
Both A and C

6. Find the equation of the tangent line T to the graph of

7. Find the equation of the tangent line T to the graph

8. Let’s examine what occurs when x approaches c.
1.9
1.99 2
2.01
2.1
1.6620
1.5151
1.5
1.4851
1.3605
1.65
1.515
1.485
1.35

9. DIFFERENTIALS When is small, is approximated by
When we use the approximation above, the quantity is usually denoted by dx, and is called the differential of x.
The expression is denoted by dy, and is called the differential of y, so we have

11. ERROR PROPAGATION
Estimation of errors propagated by physical measuring devices
Consider x representing the measured value of a variable and representing the exact value, then is the error in measurement. If the measured value x is used to compute another value , the difference between and is the propagated error.

12. DIFFERENTIAL FORMULAS
Let u and v be differentiable functions of x. Constant Multiple: Sum or Difference: Product: Quotient: