Presentation is loading. Please wait.

Presentation is loading. Please wait.

Friday, September 18, 2015MAT 145. Friday, September 18, 2015MAT 145.

Published byJeffry Cook Modified over 9 years ago

Similar presentations

Presentation on theme: "Friday, September 18, 2015MAT 145. Friday, September 18, 2015MAT 145."— Presentation transcript:

1 Friday, September 18, 2015MAT 145

2 Friday, September 18, 2015MAT 145

3 Friday, September 18, 2015MAT 145

4 Friday, September 18, 2015MAT 145

5 Friday, September 18, 2015MAT 145

6 Friday, September 18, 2015MAT 145 Here is a graph of a function f. At which values is f discontinuous? Why?

7 Friday, September 18, 2015MAT 145

8 Friday, September 18, 2015MAT 145 Removable Discontinuity Infinite Discontinuity Jump Discontinuity Removable Discontinuity

9 Friday, September 18, 2015MAT 145

10 Friday, September 18, 2015MAT 145 Match the expression in the left column (A thru F) with its correct description in the right column (1 thru 6). Explain your determination of each correct match!

11 Friday, September 18, 2015MAT 145 Match the expression in the left column (A thru F) with its correct description in the right column (1 thru 6). Explain your determination of each correct match!

12 Friday, September 18, 2015MAT 145

13 Friday, September 18, 2015MAT 145 A B C D E

14 Friday, September 18, 2015MAT 145 Calculate the slope at x = −2. Calculate the slope at x = −1. Calculate the slope at x = 0. Calculate the slope at x = a.

15 Friday, September 18, 2015MAT 145 Calculate the slope of f(x) = x 2 at x = a.

16 Friday, September 18, 2015MAT 145 We call this slope calculation the derivative of f at x = a.

17 Friday, September 18, 2015MAT 145

18 Friday, September 18, 2015MAT 145

19 Friday, September 18, 2015MAT 145

20 Friday, September 18, 2015MAT 145 The value f ’(a) is called: the derivative of f at x = a, the instantaneous rate of change of f at x = a, the slope of f at x = a, and the slope of the tangent line to f at x = a.

21 Friday, September 18, 2015MAT 145 The derivative in action! S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100?

22 Friday, September 18, 2015MAT 145 The derivative in action! C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5?

23 Friday, September 18, 2015MAT 145 The derivative in action! V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π?

24 Friday, September 18, 2015MAT 145 Can we create a derivative function f that will be true for any x value where a derivative exists?

25 Friday, September 18, 2015MAT 145

26 Friday, September 18, 2015MAT 145

27 Friday, September 18, 2015MAT 145 The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of f at x is the limit of the value of the slope function as the secant lines get closer and closer to being a tangent line.

28 Friday, September 18, 2015MAT 145 Calculate the derivative function, f ’(x), for f(x) = x 2. Use the limit definition of the derivative.

29 Friday, September 18, 2015MAT 145

30 Friday, September 18, 2015MAT 145

31 Friday, September 18, 2015MAT 145

32 Friday, September 18, 2015MAT 145

33 Friday, September 18, 2015MAT 145

34 Friday, September 18, 2015MAT 145

Download ppt "Friday, September 18, 2015MAT 145. Friday, September 18, 2015MAT 145."

Similar presentations

Warm Up A particle moves vertically(in inches)along the x-axis according to the position equation x(t) = t4 – 18t2 + 7t – 4, where t represents seconds.

Warm Up A particle moves vertically(in inches)along the x-axis according to the position equation x(t) = t4 – 18t2 + 7t – 4, where t represents seconds.
The Derivative.

The Derivative.
2.1 Derivatives and Rates of Change. The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of.

2.1 Derivatives and Rates of Change. The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of.
Rate of change and tangent lines

Rate of change and tangent lines
THE DERIVATIVE AND THE TANGENT LINE PROBLEM

THE DERIVATIVE AND THE TANGENT LINE PROBLEM
Wednesday, January 28, 2015MAT 145. Wednesday, January 28, 2015MAT 145 Which of the following... ?True or False? Explain!

Wednesday, January 28, 2015MAT 145. Wednesday, January 28, 2015MAT 145 Which of the following... ?True or False? Explain!
CHAPTER 2 THE DERIVATIVE.

CHAPTER 2 THE DERIVATIVE.
The Derivative. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine.

The Derivative. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine.
1.4 – Differentiation Using Limits of Difference Quotients

1.4 – Differentiation Using Limits of Difference Quotients
1 Instantaneous Rate of Change  What is Instantaneous Rate of Change?  We need to shift our thinking from “average rate of change” to “instantaneous.

1 Instantaneous Rate of Change  What is Instantaneous Rate of Change?  We need to shift our thinking from “average rate of change” to “instantaneous.
Moving from Average Rate of Change (AROC) to Instantaneous Rate of Change (IROC) Today you will use the average rate of change to find the instantaneous.

Moving from Average Rate of Change (AROC) to Instantaneous Rate of Change (IROC) Today you will use the average rate of change to find the instantaneous.
Derivative at a point. Average Rate of Change of A Continuous Function on a Closed Interval.

Derivative at a point. Average Rate of Change of A Continuous Function on a Closed Interval.
2.1 The Derivative and the Tangent Line Problem

2.1 The Derivative and the Tangent Line Problem
3.2 Continuity JMerrill, 2009 Review 3.1 Find: Direct substitution causes division by zero. Factoring is not possible, so what are you going to do?

3.2 Continuity JMerrill, 2009 Review 3.1 Find: Direct substitution causes division by zero. Factoring is not possible, so what are you going to do?
An introduction to limits Limits in calculus : This section gives some examples of how to use algebraic techniques to compute limits. these In cludethe.

An introduction to limits Limits in calculus : This section gives some examples of how to use algebraic techniques to compute limits. these In cludethe.
Wdnesday, September 16, 2015MAT 145. Wdnesday, September 16, 2015MAT 145.

Wdnesday, September 16, 2015MAT 145. Wdnesday, September 16, 2015MAT 145.
Unit 1 Limits. Slide 2 1.1 Limits Limit – Assume that a function f(x) is defined for all x near c (in some open interval containing c) but not necessarily.

Unit 1 Limits. Slide Limits Limit – Assume that a function f(x) is defined for all x near c (in some open interval containing c) but not necessarily.
1.6 – Tangent Lines and Slopes Slope of Secant Line Slope of Tangent Line Equation of Tangent Line Equation of Normal Line Slope of Tangent = 0 3.1.

1.6 – Tangent Lines and Slopes Slope of Secant Line Slope of Tangent Line Equation of Tangent Line Equation of Normal Line Slope of Tangent =
2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.

2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.
2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.

2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.

Similar presentations

Ads by Google