1. Advanced Mathematics Ⅰ
主讲教师：卢学飞
西安石油大学理学院 1

2. Chapter 3 The Derivative
Founder of calculus
British mathematician Newton
German mathematician Leibniz
Derivative: Describe how fast the function changes 2

3. Section 3.2 The Definition of Derivative
一、Rectilinear motion and tangent lines
二、The Definition of Derivative
三、The Geometric Meaning of Derivatives
四、the relationship between differentiability and
continuity
五、one-sided derivatives

4. The position function of the particle
一、 Examples
1. Rectilinear motion
The position function of the particle
The average velocity over a time interval
The average velocity over a time interval
The instantaneous velocity at time t0

5. 2. The slope of tangent line
The tangent line of at M
Limit position M T of secant line
(if )
The slope of tangent line MT
The slope of secant line MN

6. Two examples in common:
Instantaneous velocity
The slope of tangent line
Two examples in common:
The demand is the limit of the ratio about function increment to the independent variable increment.

7. 二、The Definition of Derivative
definition : A function
is said to be differentiable at If
exists,
The limit is called the derivative of at x0
the derivative of f(x) at x0 or

8. The position of particle
The instantaneous velocity
The slope of tangent line

10. ex2. Find the derivative of constant function
sol: so
ex3. Find the derivative of
sol:

11. Generally speaking
( is constant) ex

12. ex4
Find the derivative of function
sol: let so
We also have

13. ex5. Find the derivative of function
sol: or so

14. ex6. show that
isn’t differentiable at x = 0.
pro:
is not exist , so is not differentiable at x0
ex7. if
exist, find
sol: original limit

15. 三、 The Geometric Meaning of Derivatives
The tangent slope of f(x) at ( x0,y0) if
The curve is increasing if
The curve is decreasing if
The tangent line is parallel with x-axis if
The tangent line is perpendicular with y-axis
Equation of tangent line:
Equation of normal line:

16. 四、 the relationship between differentiability and continuity
3.2.1 Theorem. If a function f(x) is differentiable at x0, the f(x) is
continuous at x0

17. is continuous at x = 0 but not differentiability.
But f(x) is continuous at x needn’t to be differentiable
ex:
is continuous at x = 0 but not differentiability.

18. 五、 one-sided derivative
Right –hand derivatives
Left –hand derivatives
ex,
at x = 0

19. 3.2.2 Theorem. If a function f(x) is differentiable at x0, if and
only if
also so
exists

20. summary 1. Essence of derivative: The limit of increment; 2.
3. The Geometric Meaning of Derivatives: the slope of tangent line
4. the relationship between differentiability and continuity;
5. Some primary formula
not continuous not differentiable.
6. Determination of differentiability
The definition of derivative;
Left-hand derivative equal right-hand derivative;

21. Homework
P139
3 -6, 9-13,21-22

22. Alternate question
find
1. If
exist, and
sol: since so

23. 2. if
is continuous at x=0, and
exist，
Show that is differentiable
sol：since
exist，
We have
And f(x) is continuous at x=0, so
We get so
is differentiable at.

24. Class over 24