DIFFERENTIALS Section 3.9.

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Presentation transcript:

DIFFERENTIALS Section 3.9

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

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

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.

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

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

Find the equation of the tangent line T to the graph

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

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

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.

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

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