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

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

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

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

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

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.

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

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

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

Generally speaking ( is constant) ex

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

ex5. Find the derivative of function sol: or so

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

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:

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

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.

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

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

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;

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

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

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.

Class over 24