Calculus I

Lecture

Section s

1

1.1, 1.2

Objective Distance, midpoint formula, translation of points Graphs and intercepts of functions, standard form of equations of circles

Assignment 1.1: 3, 9, 11, 15, 21, 23, 25, 27 1.2: 3, 5, 13, 17, 19, 41, 45, 55, 57, 74 b,c

Understanding Goals: 1. To understand the rectangular coordinate system. 2. To understand how to find the distance between two points and midpoints of line segments connecting two points. 3. To understand how to translate points in a coordinate plane. 4. To understand how to sketch graphs of equations by hand. 5. To understand how to find the x- and y- intercepts of equations and points of intersection of two graphs. 6. To understand how to complete the square and write the standard forms of equation of circles. 7. To understand the use of mathematical models and to use it to solve real-life problems. Cartesian Plane (rectangular coordinate system) is formed by x-axis and y-axis. The point of intersection of these two axes is the origin. Example 1: Plot the points (2,1) , (0, 0) , (0,5) and (1, 2) .

The Distance Formula The distance d between the points ( x1 , y1 ) and ( x2 , y2 ) in the plane is d  ( x2  x1 ) 2  ( y2  y1 ) 2 .

1

Example 2: Use the Distance Formula to show that the points (2,5), (2, 2) , and (6, 2) are vertices of a right triangle.

The Midpoint Formula The midpoint of the segment joining the points ( x1 , y1 ) and ( x2 , y2 ) is  x1  x2 y1  y2  ,  2 2  

Midpoint  

Example 3: Find the midpoint of the line segment joining the points (5, 3) and (9,3) .

2

Example 4: Find the vertices of the parallelogram [ (1, 0), (3, 2), (3, 6), (1, 4) ] after it has been translated 1 unit to the left and 2 units down.

x-intercept is in the form of (a,0) if it is a solution point of the equation. y-intercept is in the form of (0,b) if it is a solution point of the equation.

Example 1: Find the x- and y- intercepts of the graph of the equation y  x 2  4 x  5 .

3

Example 2: Find the center and radius of the circle 4 x 2  4 y 2  20 x  16 y  37  0 .

A point of intersection of two graphs is an ordered pair that is a solution point of both graphs. To find the points analytically, set the two y-values or two x-values equal to each other and solve the equation. Example 3: Find the points of intersection (if any) of the graphs of the equations.

4

a) y  x , y  x ;

b) x 2  y  4, 2 x  y  1

5

Graphs of six basic algebraic equations:

6

Lecture

Section s

2

1.3

Objective Assignment slope-intercept, point-slope form of a 1.3: 7, 9, 19, 23, 39, 45, linear equation 49, 53, 65, 69, equations of parallel and perpendicular lines

Understanding Goals: 1. To understand the slope-intercept form of a linear equation and how to use it to sketch graphs and to write equations of a line. 2. To understand how to find slopes of lines passing through two points. 3. To understand how to find equations of parallel and perpendicular lines. 4. To understand how to use linear equations to model and solve real-life problems.

Slope of a line:

Example 1: Sketch the graph of y  2 x  1 .

7

Example 2: Find the slope of the line passing through each pair of points. a) (1, 2) and (2, 2) .

b) (0, 4) and (1, 1) .

8

Example 3: Find the equation of the line that has a slope of 2 and passes through the point (1, 1) .

Summary:

9

Example 4: Write the equations of the lines through the point (6, 4) (a) parallel to the line 3 x  4 y  7 . (b) perpendicular to the line 3 x  4 y  7 .

 7 3 ,   8 4

Example 5: Write the equations of the lines through the point  (a) parallel to the line 5 x  3 y  0

10

(b) perpendicular to the line 5 x  3 y  0 .

11

Lecture

Section s

3

1.4

Objectives

Assignment 1.4: 1-7 odds, 17-29 odds, 37, 39, 41, 47-57 odds

Domains, ranges of functions Evaluate functions, inverse functions Understanding Goals: 1. To understand the definition of functions and know how to find the domains and ranges of functions. 2. To be able to determine whether a relation between two variables is a function or not. 3. To be able to evaluate functions and combine functions to create other functions. To understand the definition of inverse functions and to find inverse functions algebraically

Example 1: Which of the equations below define y as a function of x? Explain your answer. a) x  y  4

b) x 2  y 2  4

c) x 2  y  4

12

d) x  y 2  4

Vertical line test: If every vertical line intersects the graph of an equation at most once, then the equation defines y as a function of x.

Example 2: Find the domain and range of each function. (a) y  x  1

 1  x,

(b) y   

x  1,

x 1 x 1

A function is one-to-one if to each value of the dependent variable in the range there corresponds exactly one value of the independent variable.

13

Horizontal line test: If every horizontal line intersects the graph of the function at most once, then the function is one-to-one. Note: The graph of a one-to-one function must satisfy both the vertical line test and the horizontal line test.

Function notation: f ( x) - read as f of x. Example 3: Let f ( x )  x 2  2 x  1 , find (a) f ( x Vx)

(b)

f ( x Vx)  f ( x) Vx

Example 4: Let f ( x )  2 x  3 and g ( x)  x 2  1 , find a) f ( g ( x))

b) g ( f ( x ))

14

The graphs of f and f 1 are mirror images of each other with respect to the line y x.

Example 5: Find the inversion function of f ( x)  2 x  1 .

15

Lecture Sections Objective 4-5 1.5 Limits of functions

Assignment 1.5: 3, 5, 13, 15, 17, 19, 23-57 eoo.

Understanding Goals: 1. To understand what limit is. 2. To be able to find limits of functions graphically and numerically. 3. To use the properties of limits to evaluate limits of functions. 4. To use different analytic techniques to evaluate limits of functions. 5. To understand one-sided limits to be able to evaluate it. 6. To be able to recognized unbounded behavior of functions.

Example 1: Find the limits of the following functions both numerically and graphically as x  1 . a)

x3  1 x 1

x 3  1 ( x  1)( x 2  x  1) x3  1 Question: We know that , is it true that   x2  x  1? x 1 x 1 x 1 x  1? What is the behavior of the graph of this function near

16

x

0.9

f(x)

2.71

0.99 2.970 1

| x 1| x 1 0.9 0.99

0.999

0.9999

1

1.0001

1.001

1.01

1.1

2.997

2.997001

?

3.0003

3.003

3.0301

3.31

0.999

0.9999

1

1.0001

1.001

1.01

1.1

b) f ( x )  x f(x)

17

x2  1 c) f ( x)  x 1 x f(x)

0.9

0.99

0.999

0.9999

1

1.0001

1.001

1.01

1.1

18

19

Example 2: Find the following limits: x 2  3 x  11) (a) lim( x 2

x2 1 x 1 x  1

(b) lim

(c) lim x 0

x 1 1 x

2  h (d) lim  h 0

2

4

h

(e) xlim 4

(f) lim x 1

x2  5x  4 x 2  3x  4

x 1 x x

Techniques for evaluating limits: 1. Direct substitution 2. Cancellation 3. Rationalization

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f ( x)  L Limit from the right One-sided limits: xlim c  lim f ( x)  L Limit from the left

x c 

Example 3: Find the limit as x  0 from the left and the limit as x  0 from the right x for the function f ( x)  . x

21

Example 4: Find the limit of f ( x ) as x  1 . x 1  4  x, f ( x)   2 x 1  4x  x ,

Example5: Find the limit (if possible) lim x 1

1 . x 1

5 . x 2 x  2

Example 6: Find the limit (if possible) lim

22

Example7: Given the following graph,

Compute each of the following, a). f  4  b). lim f  x  e) f  1 i) f  6 

x 4

f  x f). xlim 1

f  x j). xlim  6

f  x c) xlim 4 f  x g) xlim 1

f  x k). xlim  6

f  x d). xlim 4

f  x h). lim x 1 f  x l) lim x 6

23

Lecture Sections Objective 6 1.6 Continuity of functions

Assignment 1.6: 1-37 eoo. 39, 43, 45, 53,55

Understanding Goals: 1. To understand the meaning of “a function is continuous at a point or on a closed interval” and know how to determine it. 2. To understand how the greatest integer function is defined and how to use it to model and solve real-life problem. 3. To understand how to use compound interest model to solve real-life problems.

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Example 1: Discuss the continuity of each function: (a) f ( x)  x 2  2 x  3

(b) f ( x) 

1 x

(c) f ( x) 

1 x 1 2





f  g ( x)   f lim g ( x) . If f ( x) is continuous then, lim x a x a

25

 removable 

 gap  nonremovable  infinity  

Two types of discontinuity 

26

Example 2: Discuss the continuity of each function: (a) f ( x)  2  x

 5 x

(b) g ( x)  

2  x 1

1  x  2 2 x3

 x2

if x  1

2

if x  1

(c) f ( x)  

 x

(d) The greatest integer function § x ¨ at x  1 .

§ x ¨  greatest integer n such that n  x.

27

Example 3: Determine where the function f ( x) 

2x  1 is discontinuous. x  2 x  15 2

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Lecture Sections 6

2.1

Objectives Tangent line, limit definition of derivative Differentiability and continuity

Assignment 2.1: 5, 9, 15-47 eoo. 49-55 odds, 68-71

Understanding Goals: 1. To understand how to identify tangent lines to a graph at a point. 2. To understand how to use the limit definition to find the slopes of graphs at points. 3. To understand the limit definition of the derivative and use it to find derivatives of functions. 4. To understand the relationship between differentiability and continuity.

For a line, the slope (rate of change) is the same at every point on the line. How about for graphs other than lines?

Question: How can we determine the rate at which a graph rises or falls at a single point?

29

The tangent line to the graph of a function at a point is the line that best approximates the graph at that point. How do we define a tangent line then?

Example 1: Find the equation of the tangent line to the graph of f ( x)  x 2 at the point  2, 4  .

30

Example 2: Find the slope of the graph of f ( x)  3x  1 .

31

Notation:

dy y f ( x Vx)  f ( x )  lim  lim  f '( x ) V x  0 V x  0 dx x x

Example 3: Find the derivative of

f ( x)  3 x 2  2 x .

32

Example 4: Find the derivative of

f ( x)  2 x 2  x  1 .

Example 5: Find the derivative of y with respect to t for the 1

function y  t .

differentiability  continuity ? continuity  differentiability ?

33

34

Lecture Sections 7

2.2

Objectives

Assignment 2.2: 3, 5, 9, 13-53 eoo. 55, 57

Some Rules for Differentiation

Understanding Goals: 1. To understand how to use the Constant, Power, Constant Multiple, Sum and Difference Rules to find the derivatives of functions. 2. To understand how to use derivative to solve real-life problems.

Example 1: Find the derivate of the following functions: (a)

d 2 dx

(c)

y  1,

dy  dx

(b)

f ( x)  0 , f '( x ) 

(d)

f (t ) 

1 , f '(t )  2

Example 2: Find the derivate of the following functions: 35

(a)

f ( x)  x 5 , f '( x ) 

(c)

y t,

dy  dt

(d)

(b)

f ( x) 

A  x2 ,

dA  dx

Example 3: If possible, find the slopes of the graph of when x  3, 0, 1 and 2.

1 x5

,

f '( x ) 

f ( x)  3 x

Example 4: Differentiate each function. (a)

y  33 x

36

3x 2 4

(b)

f ( x) 

(c)

y

1 3x 2

(d)

y

1 (3 x) 2

Example 5: Find the slope of the graph of

f ( x)  x 3  4 x  2

at

x 1.

Example 6: Find an equation of the tangent line to the graph of 1 g ( x)   x 4  3 x 3  2 x 2

at the point



3  1,   . 2 

37

Lecture Sections

Objective

Assignment 38

8

2.3

Rates of Change

2.3: 3-9 odds, 13, 15, 19, 21, 27, 33(c. d. e)

Understanding Goals: 1. To understand the difference between average rates of change and instantaneous rates of change. 2. To understand how to find the average rates of change of functions over intervals. 3. To understand how to find the instantaneous rates of change of functions at points. 4. To understand how to find the marginal revenues, costs and profits for products. Two applications of derivatives: slope and rate of change. Real-life applications of rate of change: velocity, acceleration, population growth rates, production rate…

39

Example 1: The concentration C (in milligrams per milliliter) of a drug in a patient’s blood stream is monitored over 10-minute intervals, where t is measured in minutes, as shown in the table. Find the average rate of change over each interval. (a) [0, 10] (b) [20, 50] (c) [60, 80] t C

0 0

10 3

20 11

30 35

40 71

50 89

60 111

70 103

80 73

The average velocity is the average change in a function over a given time interval. If we let s (t ) represent the position of a particle at time t, then the average velocity over the time interval  t1 , t2  is calculated as: Average velocity =

s s(t2 )  s (t1 )  t t2  t1

The instantaneous velocity of s (t ) is found by determining the limiting value of the average velocity as the time interval, t2  t1  t approaches 0. Instantaneous velocity = lim t 0

s s(t )  s (t1 )  lim 2 t t 0 t2  t1

40

Example 2: A man standing on a platform that is 200 meters above the ground drops a baseball. Let s (t )  4.9t 2  200 be the Law of Motion for the path of the baseball. (a) Sketch the function by plotting the points at t  0, 2,3, 4 , and 6 seconds. t s

0

2

3

4

6

(b) Find the average velocity between 2 and 3 seconds, and between 3 and 4 seconds.

(c)

Estimate the instantaneous velocity at 3 seconds.

41

(d)

Find the average velocity between t and t  h seconds.

(e)

Determine the instantaneous velocity as at 3 seconds as h goes to zero.

The general position function for a free-falling object, neglecting air resistance is h  16t 2  v0t  h0 h : height (in feet) t : time (in seconds) where v0 : initial velocity (in feet per second) h0 : initial height (in feet) Example 3: At time t  0 , a diver jumps from a diving board that is 32 feet high. Because the diver’s initial velocity is 16 feet per seconds, his position function is h  16t 2  16t  32 . (a) When is the diver hit the water? (b) What is the diver’s velocity at impact?

42

4 3 2 Example 4: Find the instantaneous rate of change of V  r    r and A  r    r 3

Rate of Change in Economics: Marginals Total profit = Total revenue – total cost  P  R  C

Example 5: Find the marginal cost for producing x units. C  100 9  3 x





Example 6: The revenue (in dollars) from renting x apartments can be modeled by

43

R  2 x (900  32 x  x 2 ) (a) Find the additional revenue when the number of rentals is increased from 14 to 15.

(b) Find the marginal revenue when x  14 .

Example 7: The position of an object at any time t (in hours) is given by, s  t   2t 3  21t 2  60t  10 Determine when the object is moving to the right and when the object is moving to the left.

44

45

Lecture 9

Sections 2.4

Objectives The Product and Quotient Rule

Assignment 2.4: 1-37 eoo. 39, 43, 45,47

Understanding Goals: 1. To understand how to use the Product Rule and Quotient Rule to find the derivatives of functions. 2. To be able to simplify derivatives. 3. To understand the meaning of derivative and use it to answer questions about reallife situations.

Note:

d  f ( x) g ( x)  f '( x) g '( x) dx

Example 1: Differentiate the following functions: (a) y  ( x 2  2)( x  5)

 1   1( x  1)  x 

(b) f ( x)  

46

 1  2   x 

5 (c) y  ( x  3 x) 

Note:

d  f ( x)  f '( x)    dx  g ( x)  g '( x )

Example 2: Differentiate the following functions: (a) y 

x 1 2x  3

1 (b) x f ( x)  x5 3

47

(c) y 

3(3 x  2 x 2 ) 7x

 x5   x 1 

Example 3: Find the equation of the tangent line to the graph of g ( x)  ( x  2)  when x  0 .

Example 4: As blood moves from the heart through the major arteries out to the capillaries and back through the veins, the systolic pressure continuously drops. Consider a person whose blood pressure P (in millimeters of mercury) is given by 25t 2  125 P , 0  t  10 t2 1 where t is measured in seconds. At what rate is the blood pressure changing 5 seconds after blood leaves the heart?

48

Example 5: It is possible for one medication to reduce the effectiveness of other medicines taken simultaneously. If a patient taking an antibiotic also takes a kaolin-pectin medication for diarrhea, the antibiotic adsorbs to the surface of the kaolin particles and passes out of the body. The amount of antibiotic available to the body is reduced. Suppose the adsorption coefficient for the amount of antibiotic adsorbed to the kaolin 16C particles is given by: A(C )  where C represents the resulting concentration of k  2C antibiotic in the blood plasma (mg/dL), A represents the adsorption coefficient of antibiotic (mg of antibiotic/g of kaolin-pectin mixture), and k is a constant. Find an equation that represents the rate of change of the adsorption coefficient.

z

49

Lecture

Sections

Objectives

10

2.5

The Chain Rule

Assignment 2.5: 1-39 eoo. 45-63 eoo. 67, 69, 73

Understanding Goals: 1. 2. 3. 4. 5.

To understand where the Chain Rule is used and to find derivatives using it. To understand how to use the General Power Rule to find derivatives. To be able to write derivatives in simplified form. To understand the application of derivatives in real-life situations. To understand how to use the differentiation rules to differentiate algebraic functions.

Example 1: Write each function as the composition of two functions: (a) y 

1 x 1

(b) y  x 2  2 x  5

Example 2: Find the derivative of y respect to x and simplify your result, where y  u 1  u , u  x2 .

50





5

Example 3: Find the derivative of y  x 3  x 2  1 .





4

Example 4: Find the derivative of y  3 x  2 x 2 .

51

2

 3x  1  . 2  x 3

Example 5: Find the derivative of f ( x )  

x 2 using x 2 (a) the Quotient Rule and (b) the Chain Rule respectively. Example 6: Find the derivative of h  x  

Example 7: Find the tangent line to the graph of y  3 ( x 2  4) 2 when x  2 .

52

53

54

Lecture 11

Sections 2.6

Objectives Higher-Order Derivatives

Assignment 2.6: 1-39 eoo. 43, 47, 51-57

Understanding Goals: 1. To understand the definition of higher-order derivatives. 2. To understand how to use the position functions to determine the velocity and acceleration of moving objects. Notation for Higher-Order Derivatives First derivative:

y ',

f '( x ),

dy , dx

d  f ( x)  , dx

Dx  y 

Second derivative:

y ",

f "( x),

d2y , dx 2

d2  f ( x)  , dx 2

Dx 2  y 

Third derivative:

y '",

f '"( x),

d3y , dx3

d3  f ( x) , dx 3

Dx 3  y 

Fourth derivative:

y (4) ,

f (4) ( x),

d4y , dx 4

d4  f ( x)  , dx 4

Dx 4  y 

y (n) ,

f ( n ) ( x ),

dny , dx n

dn  f ( x)  , dx n

Dx n  y 

. . . nth derivative:

The nth-order derivative of an n th-degree polynomial function f ( x)  an x n  an 1 x n 1   a1 x  a0 (n) is the constant function f ( x )  n !an . Each derivative of order higher than n is the zero function.

Example 1: Find the fourth derivative of f ( x)  x 4  2 x 3 .

55

Example 2: Find the value of f (4) (2) if f ( x) 

1 . x

56

Position function s(t ) 

s (t )

differentiate differentiate       Velocity function v(t )        Acceleration function a(t )

ds  v (t ) dt

d 2s  v '(t )  a(t ) dt 2

Example 3: A ball is thrown into the air from the top of a 160-foot cliff. The initial velocity of the ball is 48 feet per second, which implies that the position function is s  16t 2  48t  160 where the time t is measured in seconds. Find the height, the velocity, and the acceleration of the ball when t  3.

57

2 Example 4: Find the second order derivative of f  x  | x  4 | .

58

Lecture 12

Section 2.7

Objectives Implicit Differentiation

Assignment 2.7: 1-39 eoo.

Understanding Goals: 1. 2. 3. 4.

Understand the difference between explicit form and implicit form of functions. Understand how to find derivatives explicitly. Understand how to find derivatives implicitly. Understand how to use implicit differentiation to answer questions about real-life situation

Example 1: Use two ways to find y ' for xy  1 . a) Solve for y b) Without Solving for y

The process that we used in the second solution to the previous example is called implicit differentiation. In the previous example we were able to just solve for y and avoid implicit differentiation. However, that won’t always be the case. For example it is difficult to express y as a function of x explicitly if x 3  y 3  2 y  1 . Example 2: Differentiating with respect to x: (a) 3x 2 (b) 2 y 3

(c) x + 2 y

(d) xy 2

59

Example 3: Use implicit differentiation to find y ' : (a) x 2  y 2  1

(b) x 2  xy  y 5  3

60

(c)

x  1  y 3  2 xy

5

2

(d) x 6  y 3  2

61

Example 4: Find the tangent line to the graph given by x 2 ( x 2  y 2 )  y 2 at the point 

2 2 ,  .  2 2  

62

Lecture 13

Section 2.8

Objectives Related Rates

Assignment 2.8: 1-19 odds, 23

Understanding Goals: 1. Understand relationship between variables in a function. 2. Understand how to use derivatives to solve related-rate problems. Related rates: the rates of two of more related variables that are changing with respect to time. Related rates problems: Find and solve a relation between different rates of changes.

63

Example 1: A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 ft/second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

64

Example 2: Air is being pumped into a spherical balloon at a rate of 4.5 ft 3 / min . Find the rate of change of radius when the radius is 2 feet.

Example 3: A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 ft 3 / min , find the rate of change of the depth of water when the water is 8 feet deep.

65

Example 4: A tumor is modeled as being roughly spherical, with radius R. If the radius of the tumor is currently R = 0.54 cm and is increasing at the rate of 0.13 cm per month, 4 R 3 what is the corresponding rate of change of the volume V  ? 3

Example 5: A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat. The winch pulls the rope at a rate of 4 feet per second. Find the speed of the boat when 13 feet of rope is out. What happens to the speed of the boat as it gets closer and closer to the dock?

66

Lecture 14

Section 3.1

Objectives Increasing and Decreasing Functions

Assignment 3.1: 3, 7, 9-33 odd, 37

Understanding Goals: 1. Understand the definition of increasing and decreasing functions and know how to test it. 2. Understand what critical number(s) of a function is and know how to find it. 3. Understand how to use critical number(s) to find the open intervals on which a function is increasing or decreasing. 4. Understand how to use increasing and decreasing functions to model and solve real-life problems.

67

b g

Example 1: Show that the function f ( x )  x 4 is decreasing on the open interval ,0 and increasing on the open interval 0,  .

bg

At what x-values can f ' ( x ) change signs?

68

3 Example 2: Find the open intervals on which f ( x )  x 

3 2 x is increasing or decreasing. 2

Interval Test Value Sign of f'(x) Conclusion

69

Example 3: Find the open intervals on which the function f ( x ) 

x4  1 is increasing or x2

decreasing.

Example 4: Find the critical numbers and the open intervals on which the function 1 f ( x )  x 4  2 x 2 is increasing or decreasing. 4

70

Example 5: Find the critical numbers and the open intervals on which the function   x 3  1, x0 y is increasing or decreasing. 2   x  2 x, x  0

Interval Test Value Sign of f'(x) Conclusion

71

Lecture 15

Section Objectives Assignment 3.2 Extrema and the First-Derivative Test 3.2: 1, 5-11 odd, 19-29 odd, 35, 39

Understanding Goals: 1. Understand the difference between relative extrema and absolute extrema. 2. Understand how to use the First-Derivative Test to find the relative extrema of functions. 3. Understand how to find absolute extrema of continuous functions on a closed interval. 4. Understand how to find minimum and maximum values of real-life models and interpret the results in context. A function has a relative extremum at points where the function changes from increasing to decreasing or vice versa.

72

For a continuous function, the relative extrema must occur at critical numbers of the function.

73

74

Example 1: Find all relative extrema of the function f ( x )  2 x 3  3x 2  36 x  14 .

Interval Test Value Sign of f'(x) Conclusion

2

Example 2: Find all relative extrema of the function f ( x )  2 x  3x 3 .

75

76

t2 Example 3: Find the minimum and maximum values of g (t )  2 on the interval t 3  1,1 .

77

Example 4: Coughing force the trachea (windpipe) to contract, which affects the velocity v of the air passing through the trachea. The velocity of the air during coughing is v  k ( R  r )r 2 , 0r R where k is constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air velocity?

Example 5: Poiseuille’s Law asserts that the speed of blood that is r centimeters from the 2 2 central axis of an artery of radius R is S (r )  c R  r , where c is a positive constant. Where is the speed of the blood greatest?





78

Lecture Section 16 3.3

Objectives Concavity and The Second-derivative Test

Assignment 3.3: 1, 7, 9-17 odd, 23-33 odd, 47, 65

Understanding Goals: 1. Understand the definition of concavity and points of inflection. 2. Understand how to determine the intervals on which the graphs of functions are concave upward or downward. 3. Understand how to find the points of inflection of the graphs of the function and how to use the second-derivative test to find the relative extrema of functions. 4. Understand the concept of diminishing returns in economics and know how to find it in an input-output model.

79

Example 1: Determine the open intervals on which the graph of f ( x)  upward or downward.

1 is concave x 1 2

80

Interval Test Value Sign of f''(x) Conclusion

Example 2: Determine the open intervals on which the graph of f ( x)  upward or downward.

x2  1 is concave x2  4

81

Example 3: Determine the points of inflection and discuss the concavity of the graph of f ( x)  x 3 ( x  4) .

82

Question: Are all points where the second derivative is zero points of inflection?

Example 4 :The graph below represents f  x  . On what intervals is f '  x  positive? Negative? Are there any points where f ''  x   0 ?

Example 5 : The graph below represents f '  x  . On what intervals is f  x  increasing? Decreasing? Concave up? Concave down?

83

Example 6: Find the relative extrema of f ( x)  3 x 5  5 x 3 .

84

Lecture Section 17 3.4

Objectives Optimization Problems

Assignment 3.4: 1-23 odd, 27, 33

Understanding Goals: 1. Understand how to write a primary and secondary equation. 2. Understand how to use the first- and second-derivative test to solve real-life optimization problems. Primary equation: an equation gives a formula for the quantity to be optimized. Secondary equation: an equation relates the independent variables of the primary equation, which is used to reduce the primary equation to one with a single independent variable.

Example 1: Find two numbers whose difference is 100 and whose product is a minimum.

85

Example 2: We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.

86

Example 3: We are going to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10/ ft 2 and the material used to build the sides cost $6/ ft 2 . If the box must have a volume of 50 ft 3 , determine the dimensions that will minimize the cost to build the box.

87

Example 4: A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equalsized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume?

88

Example 5: Determine the point(s) on y  x 2  1 that are closest to (0, 2).

89

Lecture Section 18 3.6

Objectives Asymptotes

Assignment 3.6: 1-35 eoo, 39, 41, 53, 61, 63

Understanding Goals: 1. Understand how vertical asymptotes of functions are defined and know how to find them and find infinite limits. 2. Understand how horizontal asymptotes of functions are defined and know how to find them and find limits at infinity. 3. Understand how to use asymptotes to answer real-life situations.

Example 1: Find each limit. a) lim

1 x 1

x 1

b) lim

1 x 1

x 1

c) xlim 1

1 ( x  1) 2

x 1

x 1

lim

1 x 1

lim

1 x 1

lim

1 ( x  1) 2

x 1

90

d) xlim 1

1 ( x  1) 2

lim

x 1

1 ( x  1) 2

Example 2: Find the vertical asymptotes of the graph of a) f ( x ) 

x 1 x2  x

b) f ( x ) 

x2  4x  5 x2  1

91

1 1  0, r  0 and lim r  0, r  0 r x  x x  x

Recall that lim

Example 3: Find the limit 1   2 2  a) lim  x  x  

1   5 3  b) xlim   x  

92

Example 4: Find the horizontal asymptote of the graph of each function: a) y 

5 x  7 4x2  3

5 x 2  7 b) y  4 x2  3

c) y 

5 x 3  7 4x2  3

93

Example 5: Find vertical and horizontal asymptotes of following functions: x2  5x  6 a) g  x   x2  9

b) f  x  

9 x6  x x3  1

ttt

Example 6: Sketch the graph of the equation y  asymptotes as sketching aids.

x 3 . Use intercepts, extrema, and x2

94

95

Lecture Section Objectives 19 3.7 Summary of Curve Sketching

Assignment 3.7: 1, 5, 7, 9, 13, 19, 33, 37, 43-51 odd

Understanding Goals: 1. To be able to analyze the graph of functions using previous knowledge of function. 2. To be able to recognize the graphs of simple polynomial functions. So far, we have studied several concepts that are useful in analyzing the graph of a function.  x-intercept and y-intercept  Symmetry  Domain and range  Continuity  Vertical asymptotes  Differentiability  Relative extrema  Concavity  Points of inflection  Horizontal asymptotes

96

Summary of Simple Polynomial Graphs

97

Example 1: Analyze and sketch the graph of f ( x)  x 3  3x 2  9 x  5 .

Domain, Range, end Behavior Symmetry x-intercept y-intercept Vertical asymptotes Horizontal asymptotes First order derivative Second order derivative Relative extrema points of inflection Test intervals

;;

98

2x2  5x  5 Example 2: Analyze and sketch the graph of f ( x)  . x2

Domain, Range, end Behavior Symmetry x-intercept y-intercept Vertical asymptotes Horizontal asymptotes First order derivative Second order derivative Relative extrema points of inflection Test intervals

Interval

f(x)

f'(x)

f"(x)

Characteristic of Graph 99

100

`

Example 3: Analyze and sketch the graph of f ( x) 

2( x 2  9) . x2  4

Domain, Range, end Behavior Symmetry x-intercept y-intercept Vertical asymptotes Horizontal asymptotes First order derivative Second order derivative Relative extrema points of inflection Test intervals

Interval

f(x)

f'(x)

f"(x)

Characteristic of Graph

101

5

4

Example 4: Analyze and sketch the graph of f ( x)  2 x 3  5 x 3 .

Domain, Range, end Behavior Symmetry x-intercept y-intercept Vertical asymptotes Horizontal asymptotes First order derivative Second order derivative Relative extrema points of inflection Test intervals

Interval

f(x)

f'(x)

f"(x)

Characteristic of Graph

102

Lecture Section Objectives 20 3.8 Differentials and Marginal Analysis

Assignment 3.8: 1-15 odd, 21, 23, 26, 42

Understanding Goals: 1. Understand the definition of differentials of functions and when to use it. 2. Understand how to use differentials to approximate changes and how to use it to solve real-life problems.

103

Example 1: Let y  x 2 . Find dy when x  1 and dx  0.01 . Compare this value with y for x  1 and x  0.01 .

104

Example 2: A state game commission introduces 50 deer into newly acquired state game 10(5  3t ) lands. The population N of the herd can be modeled by N  where t is the 1  0.04t time in years. Use differentials to approximate the change in the herd size from t  5 to t  6.

105

Example 3: Find the differential dy of the given functions. a) y  x 2

2

b) y  3x 3

c) y  x x  1

d) y  x 

1 x

e) y  x 2  1

106

f ( x  x)  f ( x)  y

Example 4: The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the propagated error in the volume V of the ball bearing.

The relative error in the volume is defined to be the ratio of dV to V .

107

Example 5: Use differentials to approximate

3

26 .

108

Lecture Section 21

4.1-4.3

Objectives Exponential Functions Natural Exponential Functions Derivatives of Exponential Functions

Assignment 4.1: 1-33 eoo, 35, 39, 41b 4.2: 1-21 eoo, 39-45 odd 4.3: 1-29 eoo, 39, 41

Understanding Goals: 1. Understand the properties of exponents and know how to use it to evaluate and simplify exponential expressions.rl 2. To be able to sketch the graphs of exponential functions. 3. Understand how to evaluate and graph functions involving the natural exponential function. 4. To be able to solve compound interest and present value problems. 5. To be able to differentiate natural exponential functions. 6. Understand how to use calculus to analyze the graphs of functions that involve the natural exponential function. 7. Explore the normal probability density function.

109

Example 1: After t years, the initial mass of 16 grams of a radioactive element whose t

30 half-life is 30 years is given by y  16  1  , t  0 .  2 a) How much of the initial mass remains after 50 years?

b) After how many years 5 grams of the element will be remaining?

110

Example 2: A bacterial culture is growing according to the logistic growth model 1.25 y , t0 1  0.25e 0.4t Where y is the culture weight (in grams) and t is the time (in hours). Find the weight of the culture after 0 hours, 1 hour, and 10 hours. What is the limit of the model as t increases without bound?

111

Example 3: How much should be deposited in an account paying 7.8% interest compounded monthly in order to have a balance of $21,154.03 fours years from now?

112

Example 4: The demand function for a product is modeled by 3   p  10, 000  1  0.001 x  3  xe   Find the price of product if the quantity demanded is (a) x  1000 units and (b) x  1500 units. What is the limit of the price as x increases without bound?

Example 5: Find the slope of the tangent lines to f ( x)  e x at the point (0,1) and (1, e) .

113

Example 6: Differentiate each function. (a) f ( x)  e5 x

2

(b) f ( x)  e 2 x

(c) f ( x)  6e x

3

(d) f ( x)  e  x (e) f ( x)  xe x

(f) f ( x) 

ex x

(g) f ( x)  xe x  e x

(h) f ( x) 

e x  e x 2

114

Example 7: Show that the graph of the normal probability density function 2 1 2x f ( x)  e 2 has points of inflection at x  1 .

115

Lecture Section 22-23

4.4-4.6

Objectives Logarithmic Function Derivatives of Logarithmic Functions Exponential Growth and Decay

Assignment 4.4: 1-17 odd, 21, 23-67 eoo. 71 4.5: 1-57 eoo. 61, 63, 67, 69 4.6: 1-21 odd, 31, 39, 41

Understanding Goals: 1. To be able to sketch the graphs of natural logarithmic function. 2. To be able to use properties of logarithms to simplify, expand and condense logarithmic expressions. 3. Understand how to use inverse properties of exponential and logarithmic functions to solve exponential and logarithmic equations. 4. Use properties of natural logarithms to solve real-life problems. 5. Understand how to differentiate natural logarithmic functions. 6. Understand how to use calculus to analyze the graphs of functions that involve the natural logarithmic functions. 7. To be able to evaluate logarithmic expressions involving other bases. 8. Understand how to differentiate exponential and logarithmic functions involving other bases. 9. Understand how to use exponential growth and decay to model real-life situations.

116

117

Example 1: Simplify each expression: a) ln e 2

b) ln e

e

c) eln 2 x

d) eln( x

2

1)

118

Example 2: Use the properties of logarithms to rewrite each expression as a sum, difference or multiple of logarithms. (Assume x > 0 and y > 0) a) ln

5 4

b) ln x 3  1

c) ln

xy 2

3 d) ln  x ( x  1) 

Example 3: Use the properties of logarithms to rewrite each expression as the logarithm of a single quantity. (Assume x > 0 and y > 0) a) ln 2 x  3ln y

b) 2 ln( x  1)  5ln x

119

Example 4: Solve each equation. a) e x  2

b) 10  e0.1t  14

c) ln x  2

d) 3  2 ln x  7

120

Example 5: Find the derivative of each function: a) f ( x )  ln 5 x



2 b) f ( x )  ln 5 x  4



c) f ( x )  x ln x

d) f ( x ) 

ln x x

d) f ( x )  ln 3 x  1

2 2 e) f ( x )  ln  x x  1   

Example 6: Determine the relative extrema of the function f ( x)  x  2 ln x .

121

Change-of-base formula: log a x 

ln x ln a

Example 7: Find the derivative of each function: a) f ( x )  log 2 5 x b) f ( x )  5x

c) f ( x )  5x

2

Example 8: The temperatures T (o F ) at which water boils at selected pressures p (pounds per square inch) can be modeled by T  87.97  34.96 ln p  7.91 p Find the rate of change of the temperature when the pressure is 60 pounds per square inch.

122

Example 9: A sample contains 1 gram of radium. How much radium will remain after 1000 years? (The half-life of Radium is 1599 years)

123

Example 10: Find the half-life of a radioactive material if after 1 year 99.57% of the initial amount remains.

124

Example 11: The effective yield is the annual rate I that will produce the same interest per year as the nominal rate r compounded n times per year. a) For a rate r that is compounded continuously, show that the effective yield is i  er  1 .

b) Find the effective yield for a nominal rate of 6%, compounded continuously.

125

Example 12: On the Richter scale, the magnitude R of an earthquake of intensity I is ln I  ln I 0 given by R  , where I 0 is the minimum intensity used for comparison. ln10 Assume I 0  1 . a) Find the intensity of the 1906 San Francisco earthquake in which R  8.3 .

b) Find the factor by which the intensity is increased when the value of R is doubled.

c) Find

dR . dt

126

Lecture Section 24

5.1

Objectives

Assignment 5.1: 1-19 odd, 23, 25, 33-41 odd, Antiderivatives and Indefinite Integrals 49-61 odd, 71, 75, 77

Understanding Goals: 1. 2. 3. 4.

Understand the definition of antiderivative. Understand how to use integral sign for antiderivatives. Understand how to use basic integration rules to find antiderivatives of functions. Understand how to fine particular solutions of indefinite integrals using initial conditions. 5. Understand how to use antiderivatives to solve real-life problems. Question to think about: We know f ( x )  x 2 has derivative f ' ( x )  2 x . Is x 2 the only function whose derivative is 2x ?

Integration is the “inverse” of differentiation Differentiation is the “inverse” of integration

 F '( x)dx  F ( x)  C d  f ( x)dx   f ( x)  dx   127

Example 1: Find a)  5 dx

b)  2 x dx

c)



x dx

d)  2 dt

e)  dx

128

1

f)

x

g)



7

dx



x  x dx

h)  ( x  1) dx

3 2 i)  (2 x  5 x  x  1) dx

j)



x 1 dx x

129

Differential equation: an equation involves x, y and derivatives of y.

Particular solution of differential equation:

130

1 , x  0 and find the particular x2 solution that satisfies the initial condition f (1)  0 . Example 2: Find the general solution of f '( x ) 

Example 3: Solve the differential equation a) f '( x )   2 x  3  2 x  3 , f (3)  0

b) f "( x)  x 2 , f '(0)  6, f (0)  3

131

Example 4: A ball is thrown upward with an initial velocity of 64 ft/s from an initial height of 80 feet. a. Find the position function giving the height s as a function of the time t. b. When does the ball hit the ground?

132

Lecture Section 25 5.2

Objectives The General Power Rules

Assignment 5.2: 1-27 odd, 35-47 odd

Understand Goals: 1. Understand how to use the General Power Rule to find indefinite integrals. 2. Understand how to use substitution to find indefinite integrals. 3. Understand how to use the General Power Rule to solve real-life problems.

Example 1: Find each indefinite integral. 5 a)  2(2 x  1) dx

2 3 b)  (3 x  1)( x  x ) dx

c)  2 x x 2  9 dx

133

6 x

d)

 (1  3x

e)

 4 x

dx

2 2

)

2

1  dx  2 x  

Multiply and divide integrand by a necessary constant multiple based on the Constant Multiple Rule:  kf ( x) dx  k  f ( x) dx Example 2: Find 2

a)

2  x( x  1) dx

b)

 x (3x 3

4

 1) 2 dx

2 2 c)  8(3  4 x ) dx

134

d)  5 x x 2  1 dx

e)





3 2



2

x  4  x  dx  

Change of variables: Example 3: Find a)  2 x -1 dx

b)

x

2 x -1 dx

135

Example 4: Find the indefinite integral difference in the forms of the answers.

 x(2 x

2

 1) 2 dx in two ways. Explain any

1) Simple Power Rule

2) General Power Rule

136

Lecture Section Objectives 26 5.3 Exponential and Logarithmic Integrals

Assignment 5.3: 1-27 odd, 39-57 odd

Understanding Goals: 1. Understand how to use the Exponential and Log Rule to find indefinite integrals.

Example 1: Find each indefinite integral: x a)  3e dx

3x b)  3e dx

x c)  (e  x ) dx

2 x 1 dx d)  e

3 x 1 dx e)  e

137

x f)  3xe dx 2

x g)  4 xe dx 2

Example 2: Find each indefinite integral: 2

a)

x

b)

x

c)

3x 2  x3 dx

dx

2x dx 5

2

138

2

d)

 2 x  1 dx

e)

 4 x  1 dx

f)

x

h)

4 x 2  3x  2 dx  x2

i)

e

1

3x dx 4

2

2 dx 1

x

x2  2x  5 j)  dx x 1

139

Example 3: Because of an insufficient oxygen supply, the trout population in a lake is dying. The population’s rate of change can be modeled by t  dP  125e 20 dt where t is the time in days. When t  0 , the population is 2500. a) Write an equation that models the population P in terms of the time t.

b) What is the population after 15 days?

c) According to this model, how long will it take for the entire trout population to die?

140

Lecture Section 27

5.4

Objectives

Assignment 5.4: 1-45 eoo, 53, 57, Area and the Fundamental Theorem of Calculus 61-73 odd, 75, 81, 91

Learning Objectives: 1. Understand the definition of definite integral and the difference between indefinite integral and definite integral. 2. Use the Fundamental Theorem of Calculus to evaluate definite integrals. 3. Find the average values of functions over closed intervals. 4. Use properties of even and odd functions to help evaluate definite integrals. 5. Use definite integrals to solve marginal analysis and annuities problems. We know how to use geometric formulas to find area of simple regions such as rectangles, triangles and circles. How can we find the area of nonstandard regions such as the region R shown in the Figure shown below?

141

3

Example 1: Evaluate  4 x dx . 0

\

Proof of the Fundamental Theorem of Calculus Let  be a partition of [a, b] with a = x0 < x1 < x2 < ... < xn-1 < xn = b Using this partition, F (b)  F (a ) can be rewritten as

n

  F (x )  F (x ) i 1

i

i 1

By the Mean Value Theorem, there exists a number in each subinterval (call it F ( xi )  F ( xi 1 ) ci) such that F '(ci )  xi  xi 1 Because F' is f, F '(ci )  f (ci ) . We let xi = xi - xi-1, which means we can rewrite n

the sum, above, as

F (b)  F (a )   f (ci )xi i 1

142

b

Taking the limit as x  0 , F (b)  F (a )   f ( x )dx a

Note: The definite integral do not necessarily represent areas and can be negative, zero, or positive.

143

Example 2: Find the area of the region bounded by the x-axis and the graph of f ( x)  x 2  1 , 2  x  3 .

Example 3: Evaluate each definite integral 2

3x a)  e dx 0

144

5

b)   2

1 dx x

4

c)  3 x dx 1

2

d)

 2 x  1 dx

1

145

Example 4: The velocity v of blood at a distance r from the center of an artery of radius R can be modeled by v  k ( R 2  r 2 ) where k is a constant. Find the average velocity along a radius of the artery. (Use 0 and R as the limits of integration.)

146

Example 5: Evaluate each definite integral 1

a)

x

4

dx

1

2

b)

x

5

dx

2

Example 6: You deposit $2000 each year for 15 years in an individual retirement account (IRA) paying 10% interest. How much will you have in your IRA after 15 years?

147

Lecture Section 28

5.5

Objectives The Area of a region Bounded by Two Graphs

Assignment 5.5: 1-7 odd, 15-29 odd, 35, 37 , 51

Learning Objectives: 1. Use integral to find the areas of regions bounded by two graphs. 2. Solve real-life problems using the areas of regions bounded by two graphs.

148

The Area between Curves Case I : The area between two continuous functions y = f ( x ) and y = g ( x ) on the interval [a, b] with f ( x ) ≥ g ( x )

b

b

a

a

A    f ( x )  g ( x)  dx or A   [ upper function    lower function  ] dx,

a xb

Case II: The area between two continuous functions x  f  y  and x  g  y  on the interval [c, d] with f  y   g  y 

d

d

c

c

A   [ f  y   g  y  ] dy or A   [ right function    left function  ] dy ,

c yd

b

* Note that if f  g the area between f and g is

  f  x   g  x  

dx , regardless of the

a

signs of f and g.

149

Guidelines: 1. Sketch the curves to determine which function is the upper function. 2. Set the functions equal to find their intersections if necessary. 3. Set up the definite integral and evaluate. 2

Example 1: Find the area bounded by y  xe  x , y  2 x  1, x  2, and the y -axis.

150

Example 2: Determine the area of the region enclosed by f  x   x 2 , g  x  

x

151

Example 3: Determine the area of the region bounded by y  2 x 2  10, y  4 x  16, x  2 and x  5

152

Sometimes it is easier to integrate with respect to y than with respect to x to find the area between two curves. Example 4: Determine the area of the region bounded by x   y 2  10 and x   y  2  2

153

Example 5: An epidemic was spreading such that t weeks after its outbreak it had 2 infected N1 (t )  0.1t  0.5t  150 , 0  t  50 people. Twenty-five weeks after the outbreak, a vaccine was developed and administered to the public. At that point, the 2 number of people infected was governed by the model N 2 (t )  0.2t  6t  200 . Approximate the number of people that the vaccine prevented from becoming ill during the epidemic.

154

Lecture Section 29

5.6

Objectives The Definite Integral as the Limit of a Sum – the Midpoint Rule

Assignment 5.6: 1-12 odd, 17-25 odd

Learning Objective: 1. Use the Midpoint Rule to approximate definite integrals. Sometimes you cannot use the Fundamental Theorem of Calculus to evaluate a definite integral unless you can find an antiderivative of the integrand. For example, 3  x 1  x dx ,  1  x3 dx . In those cases, you can approximate the value of the

integral using some approximation techniques, like Midpoint Rule, Trapezoidal Rule and Simpson’s Rule. Example 1: Use five rectangles to approximate the area of the region bounded by the graph of f ( x)   x 2  5 , the x-axis, and the lines x  0 and x  2 .

155

Note: as n increases, the approximation tends to improve. Example 2: Use the Midpoint Rule with n  4 to approximate the area of the region bounded by the graph of f and the x-axis over the interval. f ( x)  2 x  x 3 ,  0, 1

156

Example 3: Use the Midpoint Rule with n  4 to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. f ( y )  4 y  y 2 ,  0, 4

157

The midpoint approximation becomes better and better as n increases. In fact, the limit of this sum as n approaches infinity is exactly equal to the definite integral on the closed interval  a, b  . b

 f ( x)dx  lim  f ( x )  f ( x )   f ( x ) x a

n 

1

2

n

158

Lecture Section 30 6.5

Objectives The Trapezoidal Rule and Simpson’s Rule

Assignment 6.5: 1-12 odd, 17-25 odd

Learning Objectives: 1. Use the Trapezoidal Rule and Simpson’s Rule to approximate definite integrals. 2. Understand how to analyze the sizes of the errors when approximating definite integrals with the Trapezoidal Rule and Simpson’s Rule. b

In this section, we will develop two methods for estimating

 f  x  dx by thinking of the a

integral as an area problem and using known shapes to estimate the area under the curve.

ba . n Then on each subinterval we will approximate the function with a straight line that is equal to the function values at either endpoint of the interval. Here is a sketch of this case for n  4 . Each of these objects is a trapezoid and the area of the trapezoid in the interval Vx [ xi 1, xi ] is given by Ai   f  xi 1   f  xi   2  We will break up the interval [a, b] into n subintervals of width Vx , where Vx 

So, if we use n subintervals the integral is approximately,

159

b

Vx Vx Vx  f ( x0 )  f ( x1 )   f ( x1 )  f ( x2 )    f ( xn 1 )  f ( xn ) 2 2 2 a Upon doing a little simplification we arrive at the general Trapezoid Rule. b Vx a f ( x)dx  2 [ f ( x0 )  2 f ( x1 )  2 f ( x2 )   2 f ( xn1 )  f ( xn )] ba where Vx  and xi  a  i Vx n Note that all the function evaluations, with the exception of the first and last, are multiplied by 2.

 f ( x)dx 

160

In the Trapezoid Rule we approximated the curve with a straight line. For Simpson’s Rule we are going to approximate the function with a quadratic and we’re going to require that the quadratic agree with three of the points from our subintervals. Below is a sketch of this using n  6 . Each of the approximations is colored differently so we can see how they actually work.

161

Notice that each approximation actually covers two of the subintervals. This is the reason for requiring n to be even. It can be shown that the area under the approximation on the intervals [ xi 1, xi ] and [ xi , xi 1 ] is, x Ai   f  xi 1   4 f  xi   f  xi 1   3 If we use n subintervals the integral is then approximately, b Vx Vx a f ( x)dx  3  f ( x0 )  4 f ( x1 )  f ( x2 )  3  f ( x2 )  4 f ( x3 )  f ( x4 ) L 

Vx  f ( xn2 )  4 f ( xn 1 )  f ( xn ) 3

Upon simplifying we arrive at the general Simpson’s Rule. b

Vx [ f ( x0 )  4 f ( x1 )  2 f ( x2 )  4 f ( x3 )   2 f ( xn 2 )  4 f ( xn 1 )  f ( xn )] 3 a ba where n is even and Vx  n Note that all the function evaluations at points with odd subscripts are multiplied by 4 and all the function evaluations at points with even subscripts (except for the first and last) are multiplied by 2.

 f ( x)dx 

162

Example 1: Using n  4 and all three rules to approximate the value of the following integral. 2

e

x2

dx

0

a. Maple gives the following value for this integral 2

e

x2

dx =16.45262776

0

2

b. Midpoint Rule:

e

x2

dx 

0

2

x c. Trapezoid Rule:  e dx  2

0

2

x d. Simpson’s Rule:  e dx  0

2



2 2 2 1 0.252 e  e0.75  e1.25  e1.75 2











2 2 2 2 1/ 2 02 e  2e0.5  2e1  2e1.5  e 2 2

2 2 2 2 1/ 2 02 e  4e0.5  2e1  4e1.5  e 2 3

163

None of the estimations in the previous example are all that good. The best approximation in this case is from the Simpson’s Rule and yet it’s still had an error of almost 1. To get a better estimation we would need to use a larger n. So, for completeness sake here are the estimates for some larger value of n. n 4 8 16 32 64 12 8

Midpoint Approx. 14.485612 5 15.905676 7 16.3118539 16.417170 9 16.443746 9 16.450406 5

Trapezoid Error Approx. 1.967015 20.644559 2 1 0.5469511 17.565085 8 0.140773 16.735381 9 2 0.035456 16.523617 8 6 0.008880 16.470394 9 2 0.002221 16.457070 2 6

Simpson’s Error Approx. 4.191931 17.353626 3 5 1.1124580 16.538594 7 0.282753 16.458813 5 1 0.070989 16.453029 8 7 0.017766 16.452653 5 1 0.004442 16.452629 8 4

Error 0.9009987 0.0859669 0.0061853 0.0004019 0.0000254 0.0000016

Example 2: Cardiac output, the volume of blood pumped by a person’s heart over an interval of time can be measured by injecting dye into a vein near the heart. Cardiac Rt F t output can be measured by the quantity . F(L/min) represents the cardiac  C (t )dt 0

output(flow rate), R(mg/s) is the constant rate at which the dye is injected into the vein, and C(t) (mg/dL) is the concentration of the dye in the bloodstream at time t(sec). a. Use the following data and the trapezoidal rule to estimate area under the concentration curve of the 30 second measurement period. Time (sec) C(mg/dl)

0 0

5 10

10 36

15 35

20 15

25 13

30 8

164

b. Use Simpson’s rule to estimate area under the concentration curve of the 30 second measurement period.

c. If the flow at 30 seconds is 5L/min, find R. Use the estimate for area under the concentration curve determined by Simpson’s Rule.

Example 3: Clearance measures the volume of blood that flows through an organ of elimination per unit time from which all drug is extracted. Clearance can be determined by dividing the dosage of a drug by the area under the concentration curve (IV). dose Cl   C (t )dt

a. Find the clearance for a 50 mg dosage of a drug using the following times and respective concentrations. Time(hr)

1

C(mg/dl)

1. 8

1. 5 1. 5

2 1. 2

2. 5 1. 2

3 1.1

165

b. Find the clearance for a 35 mg dosage of a drug using the following times and respective concentrations. Time(hr)

0.5

C(  g/L)

12 0

1. 0 76

1. 5 42

2. 0

2.5

25

15

166

Read Example 3 on page 432.

167

Lecture Section 31

6.1

Objectives Integration by Substitution

Assignment 6.1: 1-45 eoo, 55, 57, 64

Learning Objectives: 1. Use the basic integration formulas and substitution to find indefinite integrals and evaluate definite integrals. 2. Understand how to use integration to solve real-life problems.

168

Example 1: Use the substitution u  x  1 to find the indefinite integral

Example 2: Find

x

 ( x  1)

2

dx .

e5 x  1  e5 x dx .

169

Example 3: Find the indefinite integral

5

Example 4: Evaluate

 1

x

x  5 dx .

x dx . 2x 1

170

Example 5: The probability of recall in an experiment is modeled by b 15 P (a  x  b)   x 1  x dx , where x is the percent of recall. What is the probability of 4 a recalling between 40% and 80%?

171

Lecture Section 32

6.3

Objectives Partial Fractions and Logistic Growth

Assignment 6.3: 1-43 eoo, 55, 56, 57, 59

Learning Objectives: 1. Use partial fractions to find indefinite integrals. 2. Use logistic growth functions to model real-life situations. Compare the following two integrals: Case 1: The numerator is the derivative of the denominator or a constant multiple of the derivative of the denominator,

x

2x 1 1 dx u  x 2  x  6  du  ln | x 2  x  6 |  c x6 u

2

Case 2: The numerator is not the derivative of the denominator and it is not a constant multiple of the derivative of the denominator. How do we evaluate it? 3 x  11 dx 2  x6

x

If we notice that 3x  11 4 1   2 x  x 6 x 3 x  2 Then the integral can be evaluated.

172

Example 1: Write the partial fraction decomposition for

3 x  11 dx and evaluate it. 2  x6

x

173

Example 2: Find

5 x 2  20 x  6  x3  2 x 2  x dx .

174

Example 3: Find

x 4  5 x 3  6 x 2  18 dx .  x3  3x 2

175

176

Deriving the Logistic Growth Function Example 4: Solve the equation

dy  ky (1  y ) . Assume y  0 and 1  y  0 . dt

177

Example 5: On a college campus, 50 students return from semester break with a dN 100e 0.1t  2 contagious flu virus. The virus has a history of spreading at a rate of dt  1  4e0.1t  where N is the number of students infected after t days. (a) Find the model giving the number of students infected with the virus in terms of the number of days since returning from semester break. (b) If nothing is done to stop the virus from spreading, will the virus spread to infect half the student population of 1000 students? Explain your answer.

178

Lecture Section

35-36

7.3-7.4

Objectives Functions of Several Variables Partial Derivatives

Assignment 7.3: 1-13 eoo, 15 7.4: 1-27 eoo, 29, 35, 39, 43, 45, 55, 57, 63

Understanding Goals: 3. Understand how to evaluate functions of several variables. 4. Understand how to find the domains and ranges of functions of several variables. 5. Understand how to find the first partial derivatives and higher-order derivatives of functions of two or more variables. 6. Understand how to find the slopes of surfaces in the x- and y-directions and use partial derivatives to answer questions about real-life situations. Many quantities in science, business, and technology are functions of several variables.

Example 1: Evaluate Functions of Several Variables Find the function values of f ( x, y ) . a) For f ( x, y )  x 2  2 xy , find f (2, 1) .

179

b) For f ( x, y, z ) 

2x2 z , find f ( 3, 2,1) . y3

Example 2: Find the domain and range of the function f ( x, y )  64  x 2  y 2 .

Example 3: Find

z z and for z  2 x 2  4 x 2 y 3  y 4 . y x

180

2

Example 4: Find the first partial derivative of f ( x, y )  xe x y and evaluate each at the point (1, ln 2) .

181

Graphical Interpretation of Partial Derivatives:

Example 5: Find the slope of the surface given by z  x 2  4 y 2 at the point  2, 1, 8  in (a) the x-direction and (b) the y-direction.

182

Example 6: Find the three partial derivatives of the function w  xe xy  2 z .

183

Example 7: Find the second partial derivatives

2 2 2 z 2 z  z  z , , , and of 2 xy x 2 y y x

z  xe y  ye x .

184

Lecture Section 34 None

Objective Zero Order Differential Equations

Assignment Attached

Zero-order Process  

Modeled by a linear function A  A0  kt , where k is the rate constant. dA  k  kA0 Rate of change is constant:  dt

Example 1. Solve the decreasing rate equation for A.

185

Example 2. Sketch the linear function A and the rate equation.

Example 3. Determine the half-life formula of a zero-order process.

186

Example 4. The eliminating rate of alcohol by humans is constant at 0.02/hr. a. If the initial percent blood alcohol is 0.162, write the equation that can be used to determine the amount of alcohol at any hour after ingestion (solve the rate equation for A).

b. The legal blood alcohol limit in many states is 0.08. At what time is the percent alcohol 0.08?

187

ASSIGNMENT: 1. Suppose the absorption rate by some organism of a chemical is constant at 0.12/h. a. If the initial concentration in the blood is 0.02 mg/dL, find the equation that can be used to determine the amount of chemical present at any time. b. If the amount of chemical exceeds 0.10 mg/dL, the animal will die. When will this happen? c. When is the concentration double the initial concentration?

188

Lecture Section 34 None

Objective First Order Differential Equations

Assignment Finish the problems on this worksheet

Quick-check: How do you know if a process is zero-order?

First-order Process •  

 kt Modeled by a exponential function A  A0 e , where k is the rate constant. dA  kA1 Rate of change for a decreasing 1st order process is :  dt dA  kA1 Rate of change for a increasing 1st order process is : dt

1. Solve the decreasing rate equation for A.

189

2. Sketch the exponential function and the rate equation.

3. Determine the half-life formula for a decreasing 1st order process.

190

4. The rate of a chemical reaction at any time (s) during the reaction is given by dA  1.3  103 A dt a. If the initial concentration of the reactant was 1.2 M, find the function A.

b. Find the half life of the chemical.

191

5. The rate of change of the concentration of cyclosporine in the blood plasma t minutes after an intravenous dose is given by dC  9.2 104 C dt a. Determine the concentration function if the initial concentration is 0.1 mg/dL. What is the concentration after 1 hour?

b. What is the half-life of cyclosporine?

c. Sketch the concentration function on the appropriate domain.

192

6. A certain industrial machine depreciates so that its value after t years becomes Q(t )  20, 000e 0.4t dollars. (Courtesy of Applied Calculus, Hoffman, L.) a. What is the rate equation for the value of the machine?

b. When will the machine be worth half of its initial value?

193

Lecture Section 34 None

Objective Second Order Differential Equations

Assignment

Quick-check: How do you know if a process is first-order?

Second-order Process 

Modeled by a rectangular hyperbolic function

A

1 kt 

1 , where k is the rate A0

constant.  

dA  kA2 dt dA  kA2 Rate of change for a increasing 2nd order process is : dt Rate of change for a decreasing 2nd order process is : 

1. Solve the rate equation of a decreasing 2nd order process for A.

1/[A] = 1/[A]0 + k t

2. Graph the rectangular hyperbolic function and the rate equation.

194

3. Determine the half-life formula of A.

1. The decomposition of HI is a second-order process with rate constant k  30 L / mol min at 443oC . a. Determine the equation that models the chemical decomposition.

195

b. Determine the half-life of HI.

2. The rate of change of the number of bacteria remaining following treatment with an antibiotic is proportional to the number of bacteria squared. a. If the initial number of bacteria is 1200, find the equation that can be used to determine the number of bacteria at any time (hours).

196

b. When will the number of bacteria be half of the initial amount?

197