Download presentation

Presentation is loading. Please wait.

Published byBelen Terrell Modified over 2 years ago

1
10.4: The Derivative

2
The average rate of change is the ratio of the change in y to the change in x The instantaneous rate of change of f at a is the limit of the average rates of change of f over shorter and shorter intervals around a Another name for the instantaneous rate of change is THE DERIVATIVE Note: Sometimes the word “instantaneous” is omitted

3
Think of the odometer and the speedometer in a car. The odometer measure the distance (in miles) that the car travels and the speedometer measure how fast (miles per hour) the car travels at a particular time. The speedometer will give you the derivative at a certain point.

4
Ex: Before I headed to Virginia, I looked at the odometer: 50,155. The trip started at 2:00 PM and ended at 8:30 PM. When I got to my parent-in-law’s house in VA, I checked my odometer: 50,517. Find the average speed? (50,517 - 50,155) / (8.5-2) = 362 / 6.5 = 55.7 mph The average speed is 55.7 mph; however, during my trip, I didn’t travel 55.7 mph all the time. Sometimes I went faster and sometimes I went slower. If I calculate the average speed at a smaller time interval (for example, during 1 second interval or smaller than that), I will have instantaneous average speed which is the derivative.

5
Given y = f (x), the average rate of change from x = a to x = a + h is The above expression is also called a difference quotient. It can be interpreted as the slope of a secant. f (a + h) – f (a) h

6
Application: The revenue generated by producing and selling calculators is given by R(x) = x (75 – 3x) for 0 x 20. What is the change in revenue if production changes from 9 to 12? R(12) – R(9) = $468 – $432 = $36. Increasing production from 9 to 12 will increase revenue by $36.

7
The revenue is R(x) = x (75 – 3x) for 0 x 20. What is the average rate of change in revenue (per unit change in x) if production changes from 9 to 12? To find the average rate of change we divide the change in revenue by the change in production: Thus the average change in revenue is $12 when production is increased from 9 to 12.

8
Example 1: given f(x)=x 2 A)Find the slope of the secant line for a =2 and h =1 Two ways to do this: *** Use slope formula We have (2, 4) and (3, 9) giving the above information *** Use different quotient TRY THE SAME PROBLEM WITH a=2, h=2. Do you get 6 for the answer?

9
Example 1- continue: given f(x)=x 2 B)Find and simplify the slope of the secant line for a = 2 and h is any nonzero number. *** Use different quotient

10
Example 1- continue: given f(x)=x 2 C)Find the limit of the expression in part B. D)Find the slope of the graph and the slope of the tangent line at a=2 The slope obtained from the limit of slopes of secant lines in part C is call the slope of the graph. Therefore, the slope of the graph and the slope of the tangent line at a=2 is 4 E)Find the equation of the tangent line at 2 Slope m=4, (2,4) y = mx + b 4 = 4(2) + b -4 = b Therefore the equation of the tangent line is y = 4x - 4

11
The Derivative For y = f (x), we define the derivative of f at x, denoted f ’ (x), to be if the limit exists. If f ’(a) exists, we call f differentiable at a. If f ’(x) exist for each x in the open interval (a, b), then f is said to be differentiable over (a, b).

12
Interpretations of the Derivative If f is a function, then f ’ is a new function with the following interpretations: ■ For each x in the domain of f ’, f ’ (x) is the slope of the line tangent to the graph of f at the point (x, f (x)). ■ For each x in the domain of f ’, f ’ (x) is the instantaneous rate of change of y = f (x) with respect to x. ■ If f (x) is the position of a moving object at time x, then v = f ’ (x) is the velocity of the object at that time.

13
Finding the Derivative To find f ‘ (x), we use a four-step process: Step 1. Find f (x + h) Step 2. Find f (x + h) – f (x) Step 3. Find Step 4. Find

14
Find the derivative of f (x) = x 2 – 3x Step 1. f (x + h) = (x + h) 2 – 3(x + h) = x 2 + 2xh + h 2 – 3x – 3h Step 2. Find f (x + h) – f (x) = x 2 + 2xh + h 2 – 3x – 3h – (x 2 – 3x) = 2xh + h 2 – 3h Step 3. Find Step 4. Find Example

15
continue Find the slope of the tangent to the graph of f (x) = x 2 – 3x at x = 0, x = 2, and x = 3. Solution: In example 2 we found the derivative of this function at x to be f ’ (x) = 2x – 3 Hence f ’ (0) = -3 f ’ (2) = 1, and f ’ (3) = 3 Graphing Calculator: Press: y= Type in the equation Graph Press: 2 nd then Trace Press 6 (dy/dx) Type in the number Enter

16
Practice: Find the derivatives

21
Example: application The total sales of a company (in millions of dollars) t months from now are given by Find S(12) and S’(12), and interpret. Use these results to estimate the total sales after 13 months and after 14 months. Answer: Use G.C: S’(12) = 0.125 In 12 months, the sales will be 4 million dollars In 12 months, the rate is increasing at 0.125 million dollars ($125,000) per month. After 13 months, the sales will be 4.125 (4+0.125) million dollars After 14 months, the sales will be 4.250 (4.125+0.125) million dollars

22
Nonexistence of the Derivative Some of the reasons why the derivative of a function may not exist at x = a are ■ The graph of f has a hole or break at x = a, or ■ The graph of f has a sharp corner at x = a, or ■ The graph of f has a vertical tangent at x = a.

Similar presentations

Presentation is loading. Please wait....

OK

Lecture 12 Average Rate of Change The Derivative.

Lecture 12 Average Rate of Change The Derivative.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on introduction to object-oriented programming polymorphism Ppt on file system in unix grep Ppt on spirit of unity in india Seminar ppt on cloud computing Ppt on life study of mathematician pascal Ppt on earth movements and major landforms in florida Ppt on aerobics dance Ppt on intelligent manufacturing software Doc convert to ppt online shopping Ppt on views in dbms tutorial