- There is a talk this Thursday, at 3:05. I'll give you a GOHF card if you attend. I think that it should be really interesting: When Will We Run Out? Mathematical Models of the Depletion of Nonrenewable Resources
- Your implicit homework is returned.
We use the linear function representing the tangent line at a point to make certain approximations.
That linear function is called the linearization of $f$ at $a$:
- For example, suppose you need to compute $\sqrt{15.96}$. Now the
easy way is to use your calculator, but the cowboy way would be to use
the linearization! Let's have a go at that one.
- We know the square root of 16, which is close: our answer should be near 4.
- Draw a picture! The picture will demonstrate two different
types of quantities that we discuss in this section:
differentials $dy$, and increments $\Delta y$.
Differentials versus increments
The increment is the true change in the function value; the differential approximates the true change: One way to remember the difference:- Differentials $dy$ follow the direction suggested by the derivative
- Increments $\Delta y$ are actual changes in the function value.
$dy \approx \Delta y$
We want the increment, but may settle for the easily computed differential.
- We use the tangent line for our approximation, and, when we're done, we can ask "is our answer an over- or an under-estimate?"
- We know the square root of 16, which is close: our answer should be near 4.
- Examples
- #2, p. 187
- #5
- #32