1. Question
Find the derivative of
Sol.

2. Question
Find

3. Question For what values of the function
(1) is continuous at 0, (2) is differentiable at 0,
(3) is continuous at 0.
Sol. (1) (2)
(3)

4. Hyperbolic functions
Definition of the hyperbolic functions

5. Hyperbolic functions
Derivatives of hyperbolic functions

6. Differentials
The differential of the independent variable x, denoted by dx, is the increment
If y=f(x) is a differentiable function, the
differential of the function, denoted by dy, is
Generally, the differential dy is not equal to the
increment

7. Linear approximations
An equation of the tangent line to y=f(x) at the point (a,f(a)) is
If the graph of f is smooth, we can approximate the curve using the tangent line in the neighborhood of a, i.e.,
This is called the linear approximation of f at a.

8. Example
Find the linear approximation of at a=1 and use it to approximate the numbers and Are these approximations overestimates or underestimates?
Sol.
They are overestimates because the tangent line lies above the
curve.

9. Error estimation of linear approximation
Suppose f is differentiable at a, then its linear approximation at a, Consider the error
It can be easily proved by the definition of derivative that
which means that the error is much smaller then

10. Error estimation of linear approximation
Some notations: increment of x at
corresponding increment of y at
We proved

11. Example
Note that only when x is close to a, the approximation of f(x) by L(x) is accurate enough.
Ex. Find an approximate value of
Sol. Since we use the linear
approximation So

12. Question
Question. Find an approximation of
Sol.

13. Differentials In the linear approximation
we call the differential of f at
and denote Letting f(x)=x, we see
That is, the increment of x is the same as its differential.
but generally not the same
The formula justifies the notation
The linear approximation is using to approximate

14. Example Ex. Compare the values of and if and x changes from 2 to 2.01.
Sol.
When |x| is small, we have the following approximation,
which can be derived by replacing increment by differential.

15. Homework 7
Section 3.11: 24, 32, 36, 39
Review exercises: 14, 26, 53, 100, 101, 102, 103, 105