Part IV_01 derivative 导数.ppt

1、Part IV THE DERIVATIVE,The Derivative Differentiability and Continuity Differentiations of Some Important Functions Basic Rules of Differentiation,An Intuitive Example,Consider the maglev example from Section 2.4. The position of the maglev is a function of time given by s = f (t) = 4t2 (0 t 30) whe

2、re s is measured in feet and t in seconds. Its graph is:,An Intuitive Example,The graph rises slowly at first but more rapidly over time. This suggests the steepness of f (t) is related to the speed of the maglev, which also increases over time. If so, we might be able to find the speed of the magle

3、v at any given time by finding the steepness of f at that time. But how do we find the steepness of a point in a curve?,Slopes of Lines and of Curves,The slope of a line is a measure of its “steepness”: for any two points on the line, it is the ratio of the “rise” (difference in the y - coordinates)

4、 to the “run” (difference in the x -coordinates), i.e. slope = y / x .,Slopes of Lines and of Curves,Steep lines have large slopes , flatter lines have smaller slopes. Decreasing lines (which go “downhill” as x increases towards the right) have negative slopes.,The slope of a line doesnt depend on t

5、he pair of points on the line used to calculate it; all pairs of points on the same line will give the same slope.,Slopes of Lines and of Curves,For curves that arent lines, the idea of a single overall slope is not very useful. Intuitively, the steepness of a typical curve is different at different

6、 places on the curve, so an appropriate definition of slope for the curve should somehow reflect this variable steepness.,Slopes of Lines and of Curves,The slope at a point of a curve is given by the slope of the tangent to the curve at that point: Suppose we want to find the slope at point A. The t

7、angent line has the same slope as the curve does at point A.,A,To define the tangent line to a curve C at a point A on the curve, we should consider a straight line that passes through A and another point P on C distinct from A.,Slopes of Lines and of Curves,P,A,x,y,C,As the point P is allowed to mo

8、ve toward A along the curve, the secant line rotates about the fixed point A and approaches a limiting position (a fixed line) which is the tangent line to the curve C at the point A.,We can show the process more precisely as follows:,Slopes of Lines and of Curves,We can show the process more precis

9、ely as follows:,Slopes of Lines and of Curves,We can show the process more precisely as follows:,Slopes of Lines and of Curves,We can show the process more precisely as follows:,Slopes of Lines and of Curves,We can show the process more precisely as follows:,Slopes of Lines and of Curves,We can show

10、 the process more precisely as follows:,Slopes of Lines and of Curves,In general, we can express the slope of the secant as follows:,Slopes of Lines and of Curves,Thus, as h approaches zero, the slope of the secant approaches the slope of the tangent to the curve at that point:,Slopes of Lines and o

11、f Curves,Thus, as h approaches zero, the slope of the secant approaches the slope of the tangent to the curve at that point. Expressed in limits notation: The slope of the tangent line to the graph of f at the point P (x, f (x) is given by if it exists.,Slopes of Lines and of Curves,Rates of Change,

12、We can see that measuring the slope of the tangent line to a graph is mathematically equivalent to finding the rate of change of f at x. Definition: The ratio of the change in the output value and change in the input value of a function is called as rate of change. For examples, velocity is the rate

13、 of change in distance with respect to time; rate of change of velocity is known as acceleration; slope of a line or linear function represents the rate of change.,Average Rates of Change,The number f (x + h) f (x) measures the change in y that corresponds to a change h in x. Then the difference quo

14、tient measures the average rate of change of y with respect to x over the interval x, x + h. In the maglev example, if y measures the position the train at time x, then the quotient give the average velocity of the train over the time interval x, x + h.,Average Rates of Change,The average rate of ch

15、ange of f over the interval x, x + h or slope of the secant line to the graph of f through the points (x, f (x) and (x + h, f (x + h) is,Instantaneous Rates of Change,By taking the limit of the difference quotient as h goes to zero, evaluating we obtain the rate of change of f at x. This is known as

16、 the instantaneous rate of change of f at x (as opposed to the average rate of change). In the maglev example, if y measures the position of a train at time x, then the limit gives the velocity of the train at time x.,Instantaneous Rates of Change,The instantaneous rate of change of f at x or slope

17、of the tangent line to the graph of f at (x, f (x) is This limit is called the derivative of f at x .,The Derivative of a Function,The derivative of a function f with respect to x is the function f (read “f prime”). The domain of f is the set of all x where the limit exists.,Thus, the derivative of

18、function f is a function f that gives the slope of the tangent to the line to the graph of f at any point (x, f (x) and also the rate of change of f at x.,The Derivative of a Function,Four Step Process for Finding f (x) 1. Compute f (x + h). 2. Form the difference f (x + h) f (x). 3. Form the quotie

19、nt 4. Compute,Example 1,Find the slope of the tangent line to the graph f (x) = 3x + 5 at any point (x, f (x). Solution: The required slope is given by the derivative of f at x. To find the derivative, we use the four-step process: Step 1. f (x + h) = 3(x + h) + 5 = 3x + 3h + 5. Step 2. f (x + h) f

20、(x) = 3x + 3h + 5 (3x + 5) = 3h. Step 3. Step 4.,Example 2(a),Find the slope of the tangent line to the graph f (x) = x2 at any point (x, f (x). Solution: The required slope is given by the derivative of f at x. To find the derivative, we use the four-step process: Step 1. f (x + h) = (x + h)2 = x2

21、+ 2xh + h2. Step 2. f (x + h) f (x) = x2 + 2xh + h2 x2 = h(2x + h). Step 3. Step 4.,Example 2(b),Find the slope of the tangent line to the graph f (x) = x2 at any point (x, f (x). The slope of the tangent line is given by f (x) = 2x. Now, find and interpret f (2). Solution: f (2) = 2(2) = 4. This means that, at the point (2, 4) the slope of the tangent line to the graph is 4.,Exercises:,Textbook P146 10,12,14,16,18,20,22,Differentiability