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### Lesson 3[1]

Course: M 1314, Fall 2009
School: U. Houston
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Word Count: 924

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3 M1314 lesson Math 1314 Lesson 3 The Derivative The Limit Definition of the Derivative 1 We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specific point. We could attempt to answer this question by graphing the function and its tangent line at the point of interest. However, with many functions, we'd get an...

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3 M1314 lesson Math 1314 Lesson 3 The Derivative The Limit Definition of the Derivative 1 We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specific point. We could attempt to answer this question by graphing the function and its tangent line at the point of interest. However, with many functions, we'd get an approximation at best. We need a way to find the slope of the tangent line analytically for every problem that will be exact every time. We can draw a secant line across the curve, then take the coordinates of the two points on the curve, P and Q, and use the slope formula to approximate the slope of the tangent line. Consider this function: M1314 lesson 3 2 Now suppose we move point Q closer to point P. When we do this, we'll get a better approximation of the slope of the tangent line. Suppose we move point Q even closer to point P. We get an even better approximation. We are letting the distance between P and Q get smaller and smaller. What does this sound like? M1314 lesson 3 3 Now let's give these two points names. We'll express them as ordered pairs. Now we'll apply the slope formula to these two points. This expression is called a difference quotient. The last thing that we want to do is to let the distance between P and Q get arbitrarily small, so we'll take a limit. This gives us the definition of the slope of the tangent line. Definition: The slope of the tangent line to the graph of f at the point P ( x, f ( x)) is given by f ( x + h) - f ( x) lim h 0 h provided the limit exists. The difference quotient gives us the average rate of change. We find the instantaneous rate of change when we take the limit of the difference quotient. The derivative of f with respect to x is the function f ' (read "f prime") defined by f ( x + h) - f ( x ) f ' ( x) = lim . The domain of f ' ( x) is the set of all x for which the limit exists. h 0 h M1314 lesson 3 4 We can use the derivative of a function to solve many types of problems. Once we have completed our study of the derivative, we will be able to solve problems, such as these Core Problems: Core Problem 2: A ball is thrown straight up from the top of a building that is 64 feet high with an initial velocity of 96 feet per second, so that its height (in feet) after t seconds is given by s (t ) = -16t 2 + 96t + 64. 1. Find the average velocity of the ball over the time intervals [2, 3], [2, 2.5], [2, 2.1]. 2. Find the instantaneous velocity of the ball when t = 2. 3. Compare your results questions for (1) and (2). 4. When does the ball hit the ground? What is its velocity at the moment when its hits the ground? Core Problem 3: The oxygen content of a pond t days after organic waste has been dumped into t 2 - 4t + 4 the pond is given by f (t ) = 100 2 t + 4 , t > 0 , percent of the normal level. Sketch the graph of the function and interpret your results. Core Problem 4: A tennis racket manufacturer finds that the total daily cost of manufacturing x rackets per day is given by the function C ( x) = 400 + 4 x + 0.0001x 2 . The rackets can be sold at a price of p dollars, where p = 10 - 0.0004 x. Assume that all rackets that are manufactured can be sold. Find the daily level of production that will yield a maximum profit. The Four-Step Process for Finding the Derivative Now that we know what the derivative is, we need to be able to find the derivative of a function. We'll use an algebraic process to do so. We'll use a Four-Step Process to find the derivative. The steps are as follows: 1. Find f ( x + h). 2. Find f ( x + h) - f ( x). 3. Form the difference quotient f ( x + h) - f ( x ) . h h 0 4. Find the limit of the difference quotient as h gets close to 0: lim f ( x + h) - f ( x ) . h f ( x + h) - f ( x) h So, f ' ( x) = lim h 0 M1314 lesson 3 5 Example 1: Find the rule for the slope of the tangent line for the function f ( x) = 3 x + 2. Step 1: Step 2: Step 3: Step 4: Example 2: Use the four-step process to find the derivative of f ( x) = x 2 . Step 1: Step 2: Step 3: Step 4: M1314 lesson 3 1...

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U. Houston - M - 1314
M1314lesson 11 Math 1314 Lesson 11 Applications of the Second Derivative Concavity1Earlier in the course, we saw that the second derivative is the rate of change of the first derivative. The second derivative can tell us if the rate of change o
U. Houston - M - 1314
M1314Lesson 19 Math 1314 Lesson 19 The Fundamental Theorem of Calculus1In the last lesson, we approximated the area under a curve by drawing rectangles, computing the area of each rectangle and then adding up their areas. We saw that the actual
U. Houston - M - 1314
M1314Lesson 4 Math 1314 Lesson 4 Basic Rules of Differentiation1We can use the limit definition of the derivative to find the derivative of every function, but it isn't always convenient. Fortunately, there are some rules for finding derivative
U. Houston - M - 1314
M1314Lesson 12 Math 1314 Lesson 12 Curve Sketching1One of our objectives in this part of the course is to be able to graph functions. In this lesson, we'll add to some tools we already have to be able to sketch an accurate graph of each functio
U. Houston - M - 1314
M 1314Lesson 20 Math 1314 Lesson 20 Evaluating Definite Integrals1We will sometimes need these properties when computing definite integrals. Properties of Definite Integrals Suppose f and g are integrable functions. Then: 1. 2. 3. 4. 5. a
U. Houston - M - 1314
M1314lesson 5 Math 1314 Lesson 5 The Product and Quotient Rules1In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x) f
U. Houston - M - 1314
M1314Lesson 13 Math 1314 Lesson 13 Absolute Extrema1In earlier sections, you learned how to find relative (local) extrema. These points were the high points and low points relative to the other points around them. In this section, you will lear
U. Houston - M - 1314
M1314lesson 21 Math 1314 Lesson 21 Area Between Two Curves1Two advertising agencies are competing for a major client. The rate of change of the client's revenues using Agency A's ad campaign is approximated by f(x) below. The rate of change of
U. Houston - M - 1314
M 1314lesson 6 Math 1314 Lesson 6 The Chain Rule1In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1: Decompose h( x) = 3 x 2 - 5 x + 6 into functions f (x) and g (x) such that h( x) = (
U. Houston - M - 1314
M 1314Lesson 14 Math 1314 Lesson 14 Optimization1Now youll work some problems where the objective is to optimize a function. That means you want to make it as large as possible or as small as possible depending on the problem. The first task is
U. Houston - M - 1314
M1314lesson 22 Math 1314 Lesson 22 Functions of Several Variables1So far, we have looked at functions of a single variable. In this section, we will consider functions of more than one variable. You are already familiar with some examples of th
U. Houston - M - 1314
M1314Lesson 7 Math 1314 Lesson 7 Higher Order Derivatives1Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second
U. Houston - M - 1314
M1314lesson 15 Math 1314 Lesson 15 Exponential Functions as Mathematical Models1In this lesson, we will look at a few applications involving exponential functions. Well first consider some word problems having to do with money. Next, well consi
U. Houston - M - 1314
M1314lesson 23 Math 1314 Lesson 23 Partial Derivatives1When we are asked to find the derivative of a function of a single variable, f (x), we know exactly what to do. However, when we have a function of two variables, there is some ambiguity. W
U. Houston - M - 1314
M 1314lesson 8 Math 1314 Lesson 8 Some Applications of the Derivative Equations of Tangent Lines1The first applications of the derivative involve finding the slope of the tangent line and writing equations of tangent lines. Example 1: Find the
U. Houston - MATH - 3303
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U. Houston - MATH - 3303
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U. Houston - MATH - 3303
Homework Module 4 3303 Name: email address: phone number:Who helped me:Who I helped:This is a 55 point assignment. Homework rules: Front side only. Keep the questions and your answers in order. If you send it pdf, send it in a single scanned
U. Houston - M - 1300
Math 1300 1. Homework is due before class begins. a. True b. FalsePopper 012. I must bubble in _ on popper scantrons or I will get a zero for that grade. a. Section number b. Assignment number c. Grading ID d. Form A e. All of the above 3. All te
U. Houston - M - 1300
Math 1300 1. Classify the number -2.5. a. Real b. Real, Rational c. Real, Rational, Integer d. Real, Irrational e. None of the above 2. Fill in the appropriate symbol: -9 _ -5. a. &gt; b. &lt; c. = d. None of the above 3. Add: -3 + (-5) a. -8 b. -2 c. 2 d.
U. Houston - M - 1300
Math 1300 1. Find the GCD of 28 and 42. a. 7 b. 14 c. 28 d. 84 e. None of the above 2. Find the LCM of 12 and 28. a. 4 b. 12 c. 28 d. 84 e. None of the above 3. Perform the given operation:a. b. c. d. e. 7 17 12 30 53 30 67 30 None of the above 3 6
U. Houston - M - 1300
Math 1300 1. Find the GCD of 28 and 42. a. 7 b. 14 c. 28 d. 84 e. None of the above 2. Find the LCM of 12 and 28. a. 4 b. 12 c. 28 d. 84 e. None of the above 3. Perform the given operation:a. b. c. d. e. 7 17 12 30 53 30 67 30 None of the above 3 6
U. Houston - M - 1300
Math 1300 1. Simplify: m4 m2. a. m2 b. m6 c. m8 d. m42 e. None of the above 2. Simplify: a. b. c.d. e.Popper 04x 2 y -2 x3x5 y2 x y2 1 5 2 x y 1 xy 2 None of the above3. Simplify 24x 2 a. 2x 6 b. 4x 6 c. 4x 3 d. 12x e. None of the above4.
U. Houston - M - 1300
Math 1300 1. Simplify: m4 m2. a. m2 b. m6 c. m8 d. m42 e. None of the above 2. Simplify: a. b. c.d. e.Popper 04x 2 y -2 x3x5 y2 x y2 1 5 2 x y 1 xy 2 None of the above3. Simplify 24x 2 a. 2x 6 b. 4x 6 c. 4x 3 d. 12x e. None of the above4.
U. Houston - M - 1300
Math 1300 1. Simplify: (2 + 4) 2 5 a. 31 b. 15 c. 7 d. 1 e. None of the above 2. Simplify: 2 3 + 15 3 a. 7 b. 11 c. 12 d. 16 e. None of the above c+b ? 3. If a = 2, b = -1, and c = 13, what is cb 6 a. 7 36 b. 49 49 c. 36 12 d. 14 e. None of the
U. Houston - M - 1300
Math 1300 1. Solve for x: 6x 13 = -5 a. -3 4 b. 3 c. 3 d. 2 e. None of the above2. Solve for x: 2(x + 3) = 6 a. -6 3 b. 2 c. 0 d. 3 e. None of the above 3. Write x &lt; -3 in interval notation. a. (-, -3) b. (-, -3] c. [-3, ) d. (-3, ) e. None of the
U. Houston - M - 1300
Math 1300 1. Solve for x: 6x 13 = -5 a. -3 4 b. 3 c. 3 d. 2 e. None of the above2. Solve for x: 2(x + 3) = 6 a. -6 3 b. 2 c. 0 d. 3 e. None of the above 3. Write x &lt; -3 in interval notation. a. (-, -3) b. (-, -3] c. [-3, ) d. (-3, ) e. None of the
U. Houston - M - 1300
Math 1300 1. Solve for x: (x + 3) = 6 a. 0 b. 3 c. 9 d. 15 e. None of the above 2. Solve for x: 2(x 5) = 3(x 2) 4 a. - 5 14 b. - 5 c. -4 d. -14 e. None of the above3. Write x 8 in interval notation. a. (-, 8) b. (-, 8] c. [8, ) d. (8, ) e. None o
U. Houston - M - 1300
Math 1300 1. Solve for x: (x + 3) = 6 a. 0 b. 3 c. 9 d. 15 e. None of the above 2. Solve for x: 2(x 5) = 3(x 2) 4 a. - 5 14 b. - 5 c. -4 d. -14 e. None of the above3. Write x 8 in interval notation. a. (-, 8) b. (-, 8] c. [8, ) d. (8, ) e. None o
U. Houston - M - 1300
1. a. b. c. d. e.Simplify the expression 16 - 8(6 + 2(-1) 3 ) -60 -28 -56 -24 None of the above1 x&lt;6 62. Solve -a. b. c. d. e.(-, -1) (-, -36) (-1, ) (-36, ) None of the abovex 2 ( y 4 ) -1 3. Simplify z -3 z3 a. 2 4 x yb.x2 y4 z3 1 c.
U. Houston - M - 1300
1. a. b. c. d. e.Simplify the expression 16 8(6 + 2(1) 3 ) -60 -28 -56 -24 None of the above1 x&lt;6 62. Solve a. b. c. d. e.(-, -1) (-, -36) (-1, ) (-36, ) None of the above( x 2 y 4 ) 1 3. Simplify z 3 z3 a. 2 4 x y b.x2 y4 z3 1 c. 2 4 3
U. Houston - M - 1300
Math 1300Popper 09 02/20/09Solve each of the following: 1. -3(4 + 5x) &lt; -2(7 x) a. (-, 2/17) b. (2/17, ) c. (-, 2/17] d. [2/17, ) e. None of the above 2. -9 &lt; 2x 3 &lt; 13 a. (-3, 8) b. (-6, 5) c. (-, -3) u (8, ) d. (-, -6) u (5, ) e. None of the
U. Houston - M - 1300
Math 1300Popper 09 02/20/09Solve each of the following: 1. -3(4 + 5x) &lt; -2(7 x) a. (-, 2/17) b. (2/17, ) c. (-, 2/17] d. [2/17, ) e. None of the above 2. -9 &lt; 2x 3 &lt; 13 a. (-3, 8) b. (-6, 5) c. (-, -3) u (8, ) d. (-, -6) u (5, ) e. None of the
U. Houston - M - 1300
Math 1300Popper 10 02/23/091. Solve: -4 3 7x 17 a. (, -2] u [1, ) b. (-, 1] u [-2, ) c. [-1, 2] d. [-2, 1] e. None of the above 2. The line y = 3 is a _ line. a. Horizontal b. Vertical c. Diagonal d. None of the above 3. The point (-3, 4) lies
U. Houston - M - 1300
Math 1300Popper 10 02/23/091. Solve: -4 3 7x 17 a. (, -2] u [1, ) b. (-, 1] u [-2, ) c. [-1, 2] d. [-2, 1] e. None of the above 2. The line y = 3 is a _ line. a. Horizontal b. Vertical c. Diagonal d. None of the above 3. The point (-3, 4) lies
U. Houston - M - 1300
Popper 11 1300 2/25/09 1. a. b. c. e. 2. a. b. c. e. 3. a. b. c. d. e. 4. a. b. c. d. e. Which type of line is y = 3? horizontal vertical diagonal None of the above Which type of line is x = 5? horizontal vertical diagonal None of the above What is t
U. Houston - M - 1300
Popper 11 1300 2/25/09 1. a. b. c. e. 2. a. b. c. e. 3. a. b. c. d. e. 4. a. b. c. d. e. Which type of line is y = 3? horizontal vertical diagonal None of the above Which type of line is x = 5? horizontal vertical diagonal None of the above What is t
U. Houston - M - 1300
Math 1300 1. What is the distance between the points (7, 4) and (-2, -2)? A. 15 B. 3 13 C. 29 D. 7 E. None of the abovePopper 12 02/27/092. What's the midpoint between the points (-, -3) and (, -3)? A. (-, -3) B. (-, 0) C. (0, -3) D. (0, 0) E. No
U. Houston - M - 1300
Math 1300 1. What is the distance between the points (7, 4) and (-2, -2)? A. 15 B. 3 13 C. 29 D. 7 E. None of the abovePopper 12 02/27/092. What's the midpoint between the points (-, -3) and (, -3)? A. (-, -3) B. (-, 0) C. (0, -3) D. (0, 0) E. No
U. Houston - M - 1300
Math 1300Popper 13 03/02/091. What is the slope of the line connecting the points (7, 4) and (-2, -2)? A. 5 2 B. 3 2 C. 2 3 D. 2 5 E. None of the above 2. What's the slope of the line connecting the points (-, -3) and (, -3)? A. -12 B. 0 C. 12 D.
U. Houston - M - 1300
Math 1300Popper 14 03/05/091. What is the slope of the line connecting the points (0, 5) and (7, -2)? A. 1 B. -1 C. -3/4 D. 3/7 E. None of the above. 2. What's the slope of the line connecting the points (6, -3) and (4, 7)? A. 1 B. -1 C. -5 D. -2
U. Houston - M - 1300
Math 1300Popper 14 03/05/091. What is the slope of the line connecting the points (0, 5) and (7, -2)? A. 1 B. -1 C. -3/4 D. 3/7 E. None of the above. 2. What's the slope of the line connecting the points (6, -3) and (4, 7)? A. 1 B. -1 C. -5 D. -2
U. Houston - M - 1300
Math 1300 For questions 1 4, use y = 5x + 8. 1. In what form of a line is this equation? A. Slope-Intercept B. General C. Point-Slope D. Cannot be determined E. None of the above. 2. What's the slope of this line? A. 1 B. 5 C. 8 D. 1/5 E. None of th
U. Houston - M - 1300
Math 1300 For questions 1 4, use 4x 3y = 7. 1. In what form of a line is this equation? A. Slope-Intercept B. General C. Point-Slope D. Cannot be determined E. None of the above. 2. What's the slope of this line? A. 4 B. -4 C. 3/4 D. 4/3 E. None of
U. Houston - M - 1300
Math 1300 For questions 1 4, use 4x 3y = 7. 1. In what form of a line is this equation? A. Slope-Intercept B. General C. Point-Slope D. Cannot be determined E. None of the above. 2. What's the slope of this line? A. 4 B. -4 C. 3/4 D. 4/3 E. None of
Sanford-Brown Institute - CS - 195
CS195-5, Lecture 4: Derivations and notes by Greg Shakhnarovich gregory@csSlide 5What is the relationship between the Euclidean distance x - and the argument of the exponent? Let us start with writing out the expression for the distance between t
Sanford-Brown Institute - CS - 195
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U. Houston - MATH - 1314
Math 1314 Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18th century. 1. How can we find the line tangent to a curve at a given point on
U. Houston - MATH - 1314
Math 1314 Lesson 2 One-Sided Limits and Continuity One-Sided Limits Sometimes we are only interested in the behavior of a function when we look from one side and not from the other. Example 1: Consider the function f ( x) =x x . Find lim f ( x).x0
U. Houston - MATH - 1314
Math 1314 Lesson 3 The Derivative The Limit Definition of the Derivative We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specific point.We need a
U. Houston - MATH - 1314
Math 1314 Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isn't always convenient. Fortunately, there are some rules for finding derivatives which will make th
U. Houston - MATH - 1314
Math 1314 Lesson 5 The Product and Quotient Rules In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x) f ' ( x ) dxExample
U. Houston - MATH - 1314
Math 1314 Lesson 6 The Chain Rule In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1: Decompose h( x) = 3 x 2 5 x + 6 into functions f (x) and g (x) such that h( x) = ( f g )( x).()4
U. Houston - MATH - 1314
Math 1314 Lesson 7 Higher Order Derivatives Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second derivative, and is
U. Houston - MATH - 1314
Math 1314 Lesson 8 Some Applications of the Derivative Equations of Tangent Lines The first applications of the derivative involve finding the slope of the tangent line and writing equations of tangent lines. Example 1: Find the slope of the line tan
U. Houston - MATH - 1314
Math 1314 Lesson 9 Marginal Functions in Economics Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level. Sometimes the business owner will want to know how much it costs to produce one more
U. Houston - MATH - 1314
Math 1314 Lesson 10 Applications of the First Derivative Determining the Intervals on Which a Function is Increasing or Decreasing From the Graph of f Definition: A function is increasing on an interval (a, b) if, for any two numbers x1 and x 2 in (a
U. Houston - MATH - 1314
Math 1314 Lesson 11 Applications of the Second Derivative Concavity Earlier in the course, we saw that the second derivative is the rate of change of the first derivative. The second derivative can tell us if the rate of change of the function is inc
U. Houston - MATH - 1314
Math 1314 Lesson 12 Curve Sketching One of our objectives in this part of the course is to be able to graph functions. In this lesson, we'll add to some tools we already have to be able to sketch an accurate graph of each function. From prerequisite
U. Houston - MATH - 1314
Math 1314 Lesson 13 Absolute Extrema In earlier sections, you learned how to find relative (local) extrema. These points were the high points and low points relative to the other points around them. In this section, you will learn how to find absolut
U. Houston - MATH - 1314
Math 1314 Lesson 14 Optimization Now you'll work some problems where the objective is to optimize a function. That means you want to make it as large as possible or as small as possible depending on the problem. The first task is to write a function