Name:
[Pages:1]
Calculus AB
Wasatch High School
2010-2011
Student Name _______________________
Summer Algebra Review
Name________________
|Monday |Wednesday |
|7/26/10 9:00-11:00 AM |7/28/10 9:00-11:00 AM |
|Numeracy, Laws of Arithmetic, Area Formulas |Functions and Graphs, Domain and Range, Piece-wise and Composite |
| |functions, Transformations, Symmetry or Even and Odd functions |
|8/2/10 9:00-11:00 AM |8/4/10 9:00-11:00 AM |
|Exponential Functions, Exponential Growth and Decay, Applications|Functions and Logarithms, Inverses, Properties of Logarithms, |
| |Applications |
|8/9/10 9:00-11:00 AM |8/11/10 9:00-11:00 AM |
|Polynomials and Rational Functions, Linear and Nonlinear |Trigonometric Functions, Radian Measure, Graphs, Trigonometric |
|Inequalities—Sign charts, Introduction to Limits and Continuity |Identities, Inverse Trigonometric Functions |
|8/16/10 9:00-11:00 AM |8/18/10 9:00-11:00 AM |
|Linear Functions, Slope as a Rate of Change, Parallel and |Introduction to Calculus, Average and Instantaneous Rate of |
|Perpendicular Lines, Applications |Change. Definition of a Derivative. |
Algebra and Arithmetic
Simplify WITHOUT a calculator. Show all of your work or briefly explain your thoughts if you use mental arithmetic.
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Area Formulas
Convert every shape into a rectangle (or half of a rectangle) to find the formula for the area of each figure in terms of the given lengths.
1. Parallelogram Formula
[pic]
2. Triangle Formula
[pic]
Can you use the triangle in the rectangular box shown below to find the formula for the area of the triangle in terms of the given dimensions of the rectangle? Explain.
[pic]
3. Trapezoid Formula
[pic]
4. Circle (Circumference = [pic]) Formula
[pic]
The volume of any right solid is the area of its base times its height.
5. Draw and label the dimensions of a right cylinder, then write a formula for its volume.
Lines
K
Practice
1. Find the slope of the line that passes through the points ( 3, 2 ) and ( -5, 8 ).
2. Find the general form of the equation whose graph is parallel to the graph of [pic]and is satisfied by the ordered pair [pic]
3. Are the points [pic] solutions to a linear equation? Explain.
□ Read pages 3-8 Practice Problems: sec 1.1 pp 9-10
Find the coordinate increments from A to B.
1. [pic] 3. [pic]
Let L be the line determined by the points A and B.
a) Plot A and B. b) Find the slope of L. c) Draw the graph of L.
5. [pic] slope=__________
graph
7. [pic] slope=__________
graph
9. Write an equation for (a) the vertical line and (b) the horizontal line through point [pic].
(a) (b)
Write the point-slope equation for the line through the point P with slope m.
13. [pic] 15. [pic]
19. Write the slope-intercept equation for the line with slope [pic]and [pic]
21. Write a general linear equation for the line through the points [pic].
25. The line contains the origin and the point in the upper right corner of the grapher screen. Write an equation for the line.
[pic]
[-10, 10] by [-25, 25]
Find the (a) slope and (b) y-intercept, and (d) graph the line.
27. [pic] a) slope =________ b) y-intercept c) graph below
29. [pic] a) slope =________ b) y-intercept c) graph below
32. Write an equation for the line through [pic] that is (a) parallel to [pic], and (b) perpendicular to [pic]
a) parallel equation:______________________ b) perpendicular equation:___________________________
35. A table of values is given for the linear function [pic] Determine [pic]and[pic]
|x |[pic] |
|1 |2 |
|3 |9 |
|5 |16 |
[pic] [pic]
37. Find the value of y for which the line through [pic]and [pic]has slope [pic]
Functions and Graphs
Objective: At the end of the lesson you should be use functions to describe how one variable depends on another.
Even Functions
Odd Functions
Absolute Value Function
Composite Functions
Without a calculator graph each function then: a) Label all intercepts and asymptotes. State whether it is even or odd. Find its domain and range. b) Do the same for the given transformation.
1a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
1b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
2a. [pic]
Domain:
Range:
Intercepts:
Even or Odd, Justification:
Asymptote(s):
2b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
3a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
3b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
4a. [pic]
Domain:
Range:
Intercepts:
Even or Odd, Justification:
Asymptote(s):
4b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
5a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
5b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
6a. [pic]
Domain:
Range:
Intercepts:
Even or Odd, Justification:
Asymptote(s):
6b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
7a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
7b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Period:
Root(s):
8a. [pic]
Domain:
Range:
Intercepts:
Even or Odd, Justification:
Asymptote(s):
Period:
8b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
Period:
9a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
Period:
9b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
Period:
10a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
10b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
11a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
11b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
12a. [pic]
Domain:
Range:
Intercepts:
Even or Odd Justification:
Asymptote(s)
Root(s):
12b. [pic]
Domain:
Range:
Intercepts:
Asymptote(s):
Root(s):
13. Graph[pic]
Even or Odd, Justification:
14. Graph[pic]
Even or Odd, Justification:
15. Graph [pic]
Domain:
Range:
Even or Odd Justification:
16. Graph [pic]
Domain:
Range:
□ Read pages 12-18 Practice section 1.2 pp 19-21 2,5,9,16,21-30,34,37-40,44,67,69,70
2. Write a formula for the height [pic]of an equilateral triangle as a function of its side length [pic], then use the formula to find the height of an equilateral triangle of side length 3 m.
Draw and label below [pic] [pic]
(Always draw a described figure!)
(a) Identify the domain and range, then (b) sketch the graph of the function.
5. [pic] (a) Domain:__________ (b)
Range:____________
9. [pic] (a) Domain:__________ (b)
Range:____________
Use your calculator to (a) identify the domain and range and (b) draw the graph of the function. Compare your answers with another class member. Have her/him initial below.
16. [pic] (a) Domain:__________ (b)
Range:____________
Initials ____________
16(continued). Describe how you could find the domain and range of [pic]without a calculator.
Use the definitions to determine whether the function is even, odd, or neither. Write a short justification for each answer.
21. Example: [pic]is even since [pic]
22. [pic]
23. [pic]
24. [pic]
25. [pic]
26. [pic]
27. [pic]
28. [pic]
29. [pic]
30. [pic]
34. Graph the piecewise-defined function [pic]
Compare with a classmate. Initials____
Sketch the figures 37-40 from page 19 of your text then use the vertical line test to determine whether the curve is the graph of a function.
37. 38.
39. 40.
44. Write a piecewise formula for the function.
Group Activity
Work in a group of 3 or more of your classmates to complete each graph assuming that the graph is (a) even (b) odd. Only a portion of the graph is defined on [pic]is shown on page 21 of your text. Two initials of students from your group are required for each graph.
67. (a) even 67. (b) odd
Initials ____, _____ Initials ____, _____
69. (a) even 69. (b) odd
Initials ____, _____ Initials ____, _____
70. (a) even 70. (b) odd
Initials ____, _____ Initials ____, _____
Sec. 1.3
Objective: At the end of the lesson you should be able to graph, simplify, and solve exponential equations.
□ Read pages 22-25 Practice section 1.3 pp 26-29 2-10 even, 13-18,20,24,25,27,34,35,36
For exercises 2-18 compare and discuss your work for each problem with a classmate. Obtain initials after you feel you understand the problem.
3. Example: Graph the function [pic]. Since [pic], the graph of [pic]will look almost identical. The domain is all real numbers so try some x values close to the origin to create a T chart. The function has a horizontal asymptote at [pic]since y gets closer and closer to [pic]as x approaches [pic]
|x |y |
|-2 |25 |
|-1 |7 |
|0 |1 |
|1 |[pic] |
|2 |[pic] |
Draw a smooth curve through the points and beyond.
2. Graph [pic] State its domain and range.
Domain:_________________
Range____________________
Classmate initials _________
4. Graph [pic] State its domain and range.
Domain:_________________
Range____________________
Classmate initials_________
6. Rewrite [pic]in base 2. ____________________ classmate initials ____
8. Rewrite [pic]in base 3. _______________________Classmate initials____
10. Use a graph on your calculator to find the zero of [pic] Describe how you did it.
Match the function with its graph. Do not to use a graphing calculator.
13. [pic] 14. [pic] 15. [pic] 16. [pic] 17. [pic] 18. [pic]
___ ___ ____ ____ _____ ______
[pic] [pic] [pic]
(a) (b) (c)
[pic] [pic] [pic]
(d) (e) (f)
20. Population in Virginia The table gives the population of Virginia for several years. Classmate initials ____
|Year |Population (thousands)d |
|1998 |6901 |
|1999 |7000 |
|2000 |7078 |
|2001 |7193 |
|2002 |7288 |
|2003 |7386 |
a. Compute the ratios of the population in one year by the population in the previous year.
b. Based on part (a), create an exponential model for the population of Virginia as a function of time (year).
c. Use your model in part (b) to predict the population of Virginia in 2008.
24. If John invests $2400 in a savings account at 6% APR, how long will it take until John’s account has a balance of $4150?
Initials ______
25. Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.25% compounded annually.
27. Determine how much time is required for an investment to triple in value if interest is earned at the rate of 6.25% compounded continuously.
Complete each table for the function.
34. [pic] 35. [pic] 36. [pic]
| | |Change |
|x |y |[pic] |
| 1 | | |
|2 | | |
| 3 | | |
|4 | | |
| | |Change [pic]|
|x |y | |
|1 | | |
|2 | | |
|3 | | |
|4 | | |
|x |y |Ratio |
| | |[pic] |
|1 | | |
|2 | | |
|3 | | |
|4 | | |
Initials____ Initials____
Additional Practice
1. Solve [pic]without a calculator.
Initials______
Act Review
Sec. 1.5
Objective: At the end of the lesson you should be able to graph, simplify, and solve logarithmic equations.
K
□ Read pages 37-43 Practice section 1.5 pp 44-45 1-6,8,10,13,15,17,23,34,37,46,47
1. [pic] 2. [pic]
[pic]
One to one? Yes No One to one? Yes No
3. [pic] 4. [pic]
[pic] [pic]
One to one? Yes No One to one? Yes No
5. [pic] 6. [pic]
[pic] [pic]
One to one? Yes No One to one? Yes No
8. Determine if [pic] has an inverse. Explain. Initials________
10. Determine if [pic] has an inverse. Explain. Initials________
Find [pic]and verify that [pic]
13. [pic]
15. [pic]
17. [pic]
24. [pic] Initials________________
34. Solve [pic]. Check your solution graphically by finding the root with your calculator.
Initials____________
37. Solve [pic] for y.
46. Radioactive Decay The half-life of a certain radioactive substance is 12 hours. There are 8 grams present initially.
(a) Express the amount of substance remaining as a function of time t.
(b) When will there be 1 gram remaining?
Initials________
More Practice with Logarithms
1. Write [pic]as a single logarithm.
Initials____
2. Show why the expression [pic] is equivalent to[pic].
Initials_____
3. Solve [pic]
Initials______
4. Solve [pic] without a calculator
Initials______
Act Review
Sec. 1.6
Objective: At the end of the lesson you should be able to graph, simplify, and solve trigonometric functions
K
Act Review
□ Read pages 46-51 Practice section 1.6 pp52-55 1,5-8,9,13,14,17,18,24,27,28
1. Find the indicated arc length. _________________________
[pic]
Determine if the function is even or odd. Use the definition of an even or odd function to explain.
5. secant ___________ 6. tangent __________ 7. cosecant ____________ 8. cotangent___________
6. Initials______ 8. Initials______
9. Find all of the trigonometric values of [pic] given [pic].
[pic]
Sketch the position of [pic]
Determine (a) the period, (b) the domain, (c) the range, and (d) draw the graph of the functions without a calculator.
13. [pic] 14. [pic] initials_________
(a) period ________ (a) period________
(b) domain __________ (b) domain __________
(c) range ___________ (c) range __________
(d) (d)
[pic] [pic]
Specify (a) the period, (b) the amplitude, and (c) identify the viewing window that is shown.
17. [pic] 18. [pic] initials___________
[pic] [pic]
(a) period _______ (a) period _______
(b) amplitude__________ (b) amplitude__________
(c) viewing window_________________ (c) viewing window_________________
24. Group activity (2-3 students): The table gives the average monthly temperatures for St. Louis for a 12-month period starting with January. Model the monthly temperature with an equation of the form
[pic]
y in degrees Fahrenheit, t in months, as follows
|Time (months) |Temperature ([pic]F ) |
|1 |34 |
|2 |30 |
|3 |39 |
|4 |44 |
|5 |58 |
|6 |67 |
|7 |78 |
|8 |80 |
|9 |72 |
|10 |63 |
|11 |51 |
|12 |40 |
(a) Find the value of b assuming that the period is 12 months period___________
(b) How is the amplitude [pic]related to the difference [pic]
(c) Use the information in (b) to find [pic] [pic]
(d) Find [pic]and write an equation for [pic] _______________________________________
(e) Check your work by graphing [pic]on a scatter plot of the data with your calculator.
(a) – (e) initials _________
_________
Find the radian measure for 27. [pic]____________ and 28. [pic]________________initials_____
Ch. Sec. 2.1
Objective: At the end of the lesson you should be able to find limits of various functions
□ Read pages 59-65 practice: 2.1 page 66 1,3,7-37odd
An object is dropped from rest from the top of a tall building falls [pic] feet in the first t seconds.
1. Find the average speed during the first 3 seconds of fall.
3. Find the speed at t = 3 seconds by using the formula [pic], then substitute smaller and smaller values for h to get a better approximation for the speed of the object at t = 3 seconds.
[pic]
|Length of the |Average Speed for Interval, |
|Time Interval, |[pic] |
|[pic] | |
|1 | |
|0.1 | |
|0.01 | |
|0.001 | |
|0.0001 | |
|0.00001 | |
What number does [pic]approach as the time interval goes to zero?
Determine the limit by substitution. Support graphically.
7. [pic]= 9. [pic]=
11. [pic] 13. [pic]
Explain why you cannot use substitution to determine the limit. Find the limit if it exists.
15. [pic]
17. [pic]
Determine the limit graphically. Confirm algebraically.
19. [pic][pic] (hint: Factor and simplify)
21. [pic]
23. [pic] (hint: Expand [pic])
25. [pic]
27. [pic]
29. Use a graph to show that [pic]does not exist. Explain.
31. [pic]= 33. [pic] 35. [pic]
37. Which of the statements are true about the function [pic]graphed below, and which are false?
[pic]
[pic] [pic] [pic]
[pic] [pic]
[pic] [pic] [pic]
[pic] [pic]
ACT review:
In the diagram below, a car starts at point A and travels 13 miles east, then 12 miles north, and finally 4 miles west to point D. If the car traveled a straight path from D to A, how may miles would this trip be?
[pic]
Ch. Sec. 2.2
Objective: At the end of the lesson you should be able to find limits that involve infinity.
ACT practice problem
The symbol ^ is defined by the equation A ^ B = AB – A. For example, 5 ^ 2 = (5)(2) – 5 = 5. What does 2 ^ (3 ^ 1) equal?
A. 0
B. -2
C. 2
D. 4
E. -4
□ Read pages 70-75 Practice: pp76-77 1,3,7,9,11,15,19,23,24,29,30,35-38,41-43,45,53
Use graphs and tables to find (a) [pic]and (b) [pic](c) Identify all horizontal asymptotes.
1. [pic] (a) [pic]=_____________ (b) [pic]=___________
(c) horizontal asymptotes________________________________
3. [pic] (a) [pic]=_____________ (b) [pic]=___________
(c) horizontal asymptotes________________________________
7. [pic] (a) [pic]=_____________ (b) [pic]=___________
(c) horizontal asymptotes________________________________
Find the limit and confirm your answer using the Sandwich Theorem.
9. [pic]
11. [pic]
Use graphs and tables to find the limits.
15. [pic] Graph Table
| |[pic] |
|x | |
| | |
| | |
| | |
| | |
| | |
19. [pic] Graph Table
| |[pic] |
|x | |
| | |
| | |
| | |
| | |
| | |
Find [pic] and [pic]
23. [pic] [pic]=__________________________ [pic]_____________________________
24. [pic] [pic]=__________________________ [pic]_____________________________
Initials_______
(a) Find the vertical asymptotes of the graph of [pic] (b) Describe the behavior of [pic]to the left and right of each vertical asymptote.
29. [pic]
(a)
(b)
(c)
30. [pic] Initials__________
(a)
(b)
(c)
Match the function with the graph of its end behavior model.
35. [pic] 36. [pic] 37. [pic] 38.[pic]
[pic] [pic] [pic] [pic]
(a) _______ (b)_________ (c) ___________ (d)___________
(a) Find a power function end behavior model [pic] (b) Identify any horizontal asymptotes
41. [pic] (a) Power function___________________ (b) horizontal asymptotes_____________
42. [pic] (a) Power function___________________ (b) horizontal asymptotes_____________
Initials___________
43. [pic] (a) Power function___________________ (b) horizontal asymptotes_____________
45. For [pic]find:
(a) a simple basic function as a right end behavior model_______________________________________________
(b) a simple basic function as a left end behavior model______________________________ _________________
53. For [pic]find:
(a) [pic] (b) [pic]
(c) [pic] (d) [pic]
Ch. Sec. 2.3
Objective: At the end of the lesson you should be able understand continuity
ACT practice problem
The ratio of the measures of the 3 angles of a triangle is 1:2:3. What is the measure of the largest angle in degrees?
A. 45
B. 77
C. 90
D. 108
E. 135
□ Read pages 78-83. Practice: page 84 1-12,19-31odd
Find the points of continuity and points of discontinuity of the function. Identify each type of discontinuity.
Function continuity discontinuity type
1. [pic] _____________________ _____________________ ___________________________
2. [pic] _____________________ _____________________ ___________________________
Initials_________
3. [pic] _____________________ _____________________ ___________________________
4. [pic] _____________________ _____________________ ___________________________
Initials_________
5. [pic] _____________________ _____________________ ___________________________
6. [pic] _____________________ _____________________ ___________________________
Initials__________
7. [pic] _____________________ _____________________ ___________________________
8. [pic] _____________________ _____________________ ___________________________
Initials__________
9. [pic] _____________________ _____________________ ___________________________
10. [pic] _____________________ _____________________ ___________________________
Initials__________
11. Use the function [pic]defined and graphed below to answer the questions.
[pic]
(a) Does [pic]exist? (b) Does [pic]exist?
(c) Does [pic] (d) Is [pic]continuous at [pic]
(a) Find each point of discontinuity (b) Which of the discontinuities are removable? Not removable? Give reasons for your answers.
Function discontinuity points removable not removable explanation
19. [pic] ________________, _____________, ______________, ______________________
21. [pic] ________________, _____________, ______________, ______________________
23. [pic]
________________, _____________, ______________, ______________________
Give a formula for the extended function that is continuous at the indicated point.
25. [pic]
27. [pic]
29. [pic]
31. Explain why [pic]is continuous.
Use Theorem 7 to show that the given function is continuous on its domain.
33. [pic]
35. [pic]
Ch. Sec. 2.4
Objective: At the end of the lesson you should be able to fine the average rate of change of a function of a function. You will also be able to find the equation of a tangent line at a given point for a function.
K
ACT practice problem
If all of the positive integers that are factors of 18 are added together, what is the sum?
A. 12
B. 18
C. 21
D. 36
E. 39
□ Read pages 87-91. Practice: 2.4 page p92 1, 9-12,15,16,23,29,38
1. Find the average rate of change of [pic]over each interval.
(a) [pic] (b) [pic]
In exercises 9-12, at the indicated point find (a) the slope of the curve, (b) an equation of the tangent, and (c) an equation of the normal. (d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window.
9. [pic]at[pic]
Graphs
Window [ , ]
(a) slope=_______________ (b) tangent line_______________________ (c) normal line______________________
10. [pic]at[pic]
Graphs
Window [ , ]
Initials_______
(a) slope=_______________ (b) tangent line_______________________ (c) normal line______________________
11. [pic]at[pic]
Graphs
Window [ , ]
(a) slope=_______________ (b) tangent line_______________________ (c) normal line______________________
12. [pic]at[pic]
Graphs
Window [ , ]
Initials_______
(a) slope=_______________ (b) tangent line_______________________ (c) normal line______________________
Determine whether the curve has a tangent at the indicated point. If it does, give its slope. If not, explain why not.
15. [pic] 16. [pic]
Initials________
23. Free Fall An object is dropped from the top of a 100-m tower. Its height above ground after t sec is [pic]m. How fast is it falling 2 sec after it is dropped?
29. Horizontal Tangent At what point is the tangent to [pic]horizontal?
38. Multiple Choice Find the average rate of change of [pic]over the interval [pic]
(A) -5 (B) 1/5 (C) 1/4 (D) 4 (E) 5 Initials_____________
Sec. 3.1
Objective: At the end of the lesson you should be able to find the instantaneous rate of change of a function by differentiation.
Use the definition
[pic]to find f’(2), given [pic]
ACT practice problem
For all x > 1, [pic]
A. [pic]
B. 1
C. [pic]
D. [pic]
E. 1 + x
□ Read Pages 99-104, practice: 3.1 page p105 3,5,7,11,13-16,17,21
3. Use the definition[pic] to find the derivative of [pic]at [pic]
5. Use the definition[pic] to find the derivative of [pic]at [pic]
7. Use the definition[pic] to find the derivative of [pic]at [pic]
11. Use the definition[pic] to find [pic]
Match the graph of the function with the graph of the derivative shown below.
[pic] [pic] [pic] [pic]
(a)_____ (b)_____ (c) _____ (d)_____
13. [pic] 14. [pic] 15. [pic] 16. [pic]
17. If [pic] and [pic] find and equation of (a) the tangent line, and (b) the normal line to the graph of [pic]at the point where [pic]
(a) tangent line equation
(b) normal line equation
21. Daylight in Fairbanks The viewing window below shows the number of hours of daylight in Fairbanks,
Alaska, on each day for a typical 365-day period from January 1 to December 31. Answer the following questions by estimating slopes on the graph in hours per day. For the purposes of estimation, assume that each month has 30 days.
[pic]
[pic]
(a) On about what date is the amount of daylight increasing at the fastest rate? What is that rate?
(b) Do there appear to be days on which the rate of change in the amount of daylight is zero? If so, which ones?
(c) On what dates is the rate of change in the number of daylight hours positive? Negative?
Sec. 3.2
Objective: At the end of the lesson you should be able to determine when a derivative fails to exist.
K
Prove: Differentiability implies continuity.
ACT practice problem
If i = p + prt, then p = ?
A. [pic]
B. [pic]
C. [pic]
D. [pic]
E. [pic]
□ Read pages 109-113 practice: 3.2 page p114 1-4,9,10,19,25,35
Compare the right-hand and left-hand derivatives to show that the function is not differentiable at the point P. Find all points where [pic]is not differentiable.
1. 2.
[pic]
[pic] [pic]
[pic]
Right-hand derivative____________ Right-hand derivative____________
Left-hand derivative____________ Left-hand derivative____________
Point(s) where not diff. ____________________ Point(s) where not diff. ____________________
3. 4.
[pic] [pic]
[pic]
[pic] [pic]
[pic]
[pic]
[pic]
Right-hand derivative____________ Right-hand derivative____________
Left-hand derivative____________ Left-hand derivative____________
Point(s) where not diff. ____________________ Point(s) where not diff. ____________________
9. The graph of a function over a closed interval D is given. At what domain points does the function appear to be
[pic]
(a) differentiable?_______________________
(b) continuous but not differentiable?__________________
(c) neither continuous nor differentiable?_______________
10. The graph of a function over a closed interval D is given. At what domain points does the function appear to be
[pic]
(a) differentiable?_______________________
(b) continuous but not differentiable?__________________
(c) neither continuous nor differentiable?_______________
Find the numerical derivative of the given function at the indicated point. Use [pic] Is the function differentiable at the indicated point? Explain.
[pic]
19. [pic] 25. [pic]
35. Find all values of x for which [pic]is differentiable.
Sec. 3.3
Objective: At the end of the lesson you should be able to use the rules for differentiation to take derivates of a variety of functions.
ACT practice problems
If x + 2y = 10 and x – 2y = 2, then x + y = ?
A. -2
B. 2
C. 4
D. 5
E. 8
What is the slope of the line 2x – 3y = 18 ?
A. 2
B. -2
C. -6
D. [pic]
E. [pic]
□ Read pages 116-123 practice: 3.3 p124 1-6,7,9,11,13,15,17,19,21,23,27,33
Find [pic]
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic]
Find the x value locations of the horizontal tangents of the curve.
9. [pic] 11. [pic]
13. Let [pic] Find [pic]
(a) by applying the product rule (b) by multiplying the factors and then differentiating.
Find [pic] Support your answer graphically.
15. [pic] 17. [pic]
19. [pic] 21. [pic]
23. Suppose [pic]and [pic] are functions of x that are differentiable at [pic]and that [pic] Find the values of the following derivatives at [pic]
(a) [pic] (b) [pic]
(c) [pic] (d) [pic]
27. Find an equation for the line tangent to the curve of [pic]at [pic]
33. Find the first four derivatives of [pic]
[pic]
[pic]
[pic]
[pic]
Sec. 3.4
Objective: At the end of the lesson you should be able to use the rules for differentiation to calculate rates of change.
ACT practice problems
If the lenths of BC = AC = AD = DE in the diagram below, then x = ?
[pic]
A. 10
B. 20
C. 30
D. 40
E. 45
□ Read pages 127-134 Home practice: sec 3.4 p135 1-5,9,16
1. (a) Write the volume V of a cube as a function of the side length [pic]
(b) Find the (instantaneous) rate of change of the volume with respect to a side [pic]
(c) Evaluate the rate of change of [pic] at [pic]and [pic]
(d) If [pic]is measured in inches and [pic]is measured on cubic inches, what units would be appropriate for [pic]
2. (a) Write the area [pic]of a circle as a function of the circumference [pic] Initials____________
(b) Find the (instantaneous) rate of change of the Area [pic]with respect to the circumference [pic]
(c) Evaluate the rate of change of [pic] at [pic]and [pic]
(d) If [pic]is measured in inches and [pic]is measured in square inches, what units would be appropriate for [pic]
3. (a) Write the area [pic]of an equilateral triangle as a function of the side length [pic]
(b) Find the (instantaneous) rate of change of the Area [pic]with respect to a side [pic]
(c) Evaluate the rate of change of [pic]at [pic]and [pic]
(d) If [pic]is measured in inches and [pic]is measured in square inches, what units would be appropriate for [pic]
4. A square of side length [pic]is inscribed in a circle of radius r. Initials______________
(a) Write the area [pic]of the square as a function of the radius r.
(b) Find the (instantaneous) rate of change of the area [pic]with respect to the radius r of the circle.
(c) Evaluate the rate of change of [pic]at [pic]and [pic]
(d) If r is measured in inches and [pic]is measured in square inches, what units would be appropriate for [pic]
5. Group Activity The coordinates [pic]of a moving body for various values of [pic]are given.
[pic](sec) |0 |0.5 |1.0 |1.5 |2.0 |2.5 |3.0 |3.5 |4.0 | |[pic](ft) |12.5 |26 |36.5 |44 |48.5 |50 |48.5 |44 |36.5 | |
(a) Plot [pic]versus [pic], and sketch a smooth curve through the given points. Initials ______ ________
[pic]
(b) Assuming that this smooth curve represents the motion of the body, estimate the velocity at
[pic]
9. Particle Motion The accompanying figure shows the velocity [pic]of a particle moving on a coordinate line. Write a brief explanation below each answer.
[pic]
(a) When does the particle
move forward?_____________________________ move backward?______________________________________
speed up?__________________________________ slow down?__________________________________________
(b) When is the particle’s acceleration
positive?___________________________________negative?____________________________________________
zero?______________________________________
(c) When does the particle move at the greatest speed?______________________________________________
(d) When does the particle stand still for more than an instant?________________________________________
Sec. 3.5
Objective: At the end of the lesson you should be able to find and use the derivatives of the six basic trig functions.
K
ACT practice problems
1. If [pic] for all x > 0, then t = ?
A. 0
B. 4
C. 5
D. 7
E. 16
2. What is the solution of
[pic]?
A. x >12
B..x > -4
C. x < -4
D. x > -12
E. x < 6
□ Read pages 141-145 Assignment 3.5 pg 146 1-11,14,18,21,27,47
Find [pic] Use your graphing calculator to support your analysis if you are unsure of your answer.
1. [pic] 2. [pic] Initials___ 3. [pic]
4. [pic] Initials____ 5. [pic] 6. [pic] Initials___
7. [pic] 8. [pic] Initials____ 9. [pic]
10. [pic] Initials____
11. A weight hanging from a spring bobs up and down with position function [pic] ([pic]in meters, [pic]in seconds). What are its velocity and acceleration at time [pic] Describe its motion.
Velocity _________________ Acceleration_________________________
Description of its motion__________________________________________________________________________
14. A body is moving in simple harmonic motion with position function [pic]([pic]in meters, [pic]in seconds).
(a) Find the body’s velocity, speed, and acceleration at time [pic]
[pic]_________________________
Speed at [pic]_______________________
[pic]_________________________
(b) Find the body’s velocity, speed, and acceleration at time [pic]
[pic]_________________________
Speed at [pic]____________________
[pic]_________________________
Initials____
18. A body is moving in simple harmonic motion with position function [pic]([pic]in meters, [pic]in seconds). Find the jerk at time t. Initials____
21. Find the equations for the lines that are tangent and normal to the graph of [pic] at [pic]
27. Show that the graphs of [pic]and[pic] have horizontal tangents at [pic]
47. Multiple Choice Which of the following is an equation of the normal line to [pic] at [pic]
(A) [pic] (B) [pic] (C) [pic]
(D) [pic] (E) [pic]
Ch. Sec. 3.6
Objective: At the end of the lesson you should be able to find the derivative of a composite function by using the chain rule.
Find [pic] if [pic]
ACT practice problems
1. If [pic], then x = ?
A. [pic]
B. [pic]
C. [pic]
D. [pic]
E. [pic]
2. Steve saves 10% on a regularly priced $40 radio but still must pay 5% sales tax on the reduced price. What is the total amount that he must pay?
A. $41.80 B..$38.20 C. $38.00 D. $37.50 E. $35.00
□ Read Pages 148-152 Assignment 3.6 p153 1-8,11,13,14,21,24,35,36
Use the given substitution and the Chain Rule to find [pic]
1. [pic] 2. [pic] Initials____
3. [pic] 4. [pic] Initials____
5. [pic] 6. [pic] Initials____
7. [pic] 8. [pic] Initials____
11. An object moves along the x-axis so that its position at [pic]is given by [pic] Find the velocity of the object as a function of t.
Find [pic] If you are unsure of your answer, use NDER to support your computation.
13. [pic] 14. [pic] Initials____
21. [pic] 24. [pic] Initials____
Find the value of [pic] at the given value of x.
35. [pic] 36. [pic]
Initials_____
Sec. 3.7
Objective: At the end of the lesson you should be able to use implicit differentiation to find derivatives of relations that are not solved explicitly for y.
□ Read Pages 157-162 HW Assignment: 3.7 pp162-168 1-23 odd,29
Find [pic]
1. [pic] 3. [pic]
5. [pic] 7. [pic]
Find [pic]and find the slope of the curve at the indicated point.
9. [pic] 11. [pic]
Find where the slope of the curve is defined.
13. [pic] 15. [pic]
Find the lines that are (a) tangent and (b) normal to the curve at the given point.
17. [pic] 19. [pic]
21. [pic]
23. [pic]
29. Use implicit differentiation to find [pic] and then [pic] for [pic]
3.8
Objective: At the end of the lesson you should be able to find the derivatives of inverse trigonometric functions
Graph: [pic] [pic][pic] [pic]
State the domain for each function graphed above.
Find the derivative of [pic] using implicit differentiation.
[pic]
← Read pp 165-169 p170 1-21odd
Find the derivative of y with respect to the appropriate variable.
1. [pic] 3. [pic]
5. [pic] 7. [pic]
A particle moves along the x-axis so that its position at any time [pic] is given by [pic] Find the velocity at the indicated value of t.
9. [pic] 11. [pic]
Find the derivatives of y with respect to the appropriate variable.
13. [pic] 15. [pic]
17. [pic] 19. [pic]
21. [pic]
3.9
Objective: At the end of the lesson you should be able to find the derivatives of inverse trigonometric functions
Graph: [pic] [pic][pic] [pic] [pic]
4d pp178-179 1-28
Ch. Sec. 4.1
Objective: At the end of the lesson you should be able to use the derivative to find extreme values
HW Assignment: 5a pp193-195 1-10,11-29odd
Find the extreme values and where they occur.
1. 2.
[pic] [pic]
3. 4.
[pic] [pic]
Identify each x-value at which any absolute extreme value occurs. Explain how you answer is consistent with the Extreme Value Theorem. Sketch the graphs shown on page 194.
5. 6. 7.
8. 9. 10.
Use analytic methods to find the extreme values of the function on the interval and where they occur.
11. [pic] 13. [pic]
15. [pic] 17. [pic]
Find the extreme values of the function and where they occur.
19. [pic] 21. [pic]
23. [pic] 25. [pic]
27. [pic] 29. [pic]
Ch. Sec. 4.2
Objective: At the end of the lesson you should be able to understand and apply the mean value theorem
Lesson problems:
HW Assignment: p202 1-11
a) State whether or not the functions satisfies the hypotheses of the Mean Value Theorem on the given interval, and
b) if it does, find each value of c in the interval (a , b) that satisfies the equation
[pic]
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
The interval [pic]is given. Let [pic] and [pic] Write an equation for
(a) the secant line [pic].
(b) the tangent line to [pic] in the interval [pic]that is parallel to [pic]
9. [pic] 10. [pic]
11. Speeding A trucker handed in a ticket at a toll booth showing that in 2 h she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?
Ch. Sec. 4.3
Objective: At the end of the lesson you should be able to analyze functions using the first and second derivatives.
D
HW Assignment: p214 1-19odd, 23,29,33,37,41,47,48,51
Ch. Sec. 4.4
Objective: At the end of the lesson you should be able to solve optimization problems using the derivative.
Sample problem:
HW Assignment: 4.4I p226 1-10
Solve the problems analytically. Support your answer graphically.
1. Finding Numbers The sum of two nonnegative numbers is 20. Find the numbers if
(a) the sum of their squares is:
as large as possible as small as possible.
(b) one number plus the square root of the other is:
as large as possible as small as possible.
2. Maximizing Area What is the largest possible are for a right triangle whose hypotenuse is 5 cm long, what are its dimensions?
3. Maximizing Perimeter What is the smallest perimeter possible for a rectangle whose area is 16 [pic], and what are its dimensions?
4. Finding area Show that among all rectangles with an 8-m perimeter, the one with largest area is a square.
Inscribing Rectangles The figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.
(a) Express the y-coordinate of P in terms of x. [Hint: Write an equation for line AB]
(b) Express the area of the rectangle in terms of x.
(c) What is the largest area the rectangle can have, and what are its dimensions?
6. Largest Rectangle A rectangle has its base on the x-axis and its upper two vertices on the parabola [pic]. What is the largest area the rectangle can have, and what are its dimensions?
7. Optimal Dimensions You are planning to make an open rectangular box from an 8- by 15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume?
8. Closing off the first Quadrant You are planning to close off a corner of the first quadrant with a line segment 20 units long running from [pic]to [pic] Show that the area of the triangle enclosed by the segment is largest when [pic]
9. The Best Fencing Plan A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?
10. The Shortest Fence A 216-[pic]rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?
Ch. Sec. 4.4
Objective: At the end of the lesson you should be able to solve optimization problems using the derivative.
Another sample problem:
HW Assignment: 4.4II: p226 11,13,14,15,19,21,22,23,36,37
11. Designing a Tank Your iron works has contracted to design and build a 500-ft3, square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible.
a) What dimensions do you tell the shop to use?
b) Writing to Learn Briefly describe how you took weight into account.
13. Designing a Poster You are designing a rectangular poster to contain 50 in2 of printing with a 4-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used?
14. Vertical Motion The height of an object moving vertically is given by [pic] with s in ft and t in sec. Find:
a) the objects velocity when [pic] and
b) its maximum height and when it occurs, and
c) its velocity when [pic]
15. Finding an Angle Two sides of a triangle have lengths a and b, and the angle between them is [pic] What value of [pic] will maximize the triangle’s area? [Hint: [pic]]
19. Designing a Suitcase A 24-by 36-in. sheet of cardboard is folded in half to form a 24-by 18-in. rectangle as shown in the figure. Then for congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.
[pic]
The sheet is then unfolded.
[pic]
a) Write a formula [pic]for the volume of the box.
b) Find the domain of [pic]for the problem situation and graph [pic]over its domain.
c) Use a graphical method to find the maximum volume and value of x that gives it.
d) Confirm your result in part c) analytically.
e) Find a value of x that yields a volume of 1120 in3.
f) Writing to Learn Write a paragraph describing the issues that arise in part b)
21. Inscribing Rectangles A rectangle is to be inscribed under the arch of the curve [pic]from [pic]to [pic] What are the dimensions of the rectangle with largest area, and what is the largest area?
23. Maximizing Profit Suppose [pic]represents revenue and [pic]represents cost, with x measured in thousands of units. Is there a production level that maximizes profit? If so, what is it?
36. Inscribing a Cone Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.
37. Strength of a Beam The strength S of a rectangular wooden beam is proportional to its width times the square of tis depth.
a) Find the dimensions of the strongest beam tha can be cut from a 12-in. diameter cylindrical log.
b) Writing to Learn Graph S as a function of the beam’s width w, assuming the proportionality constant to be k = 1. Reconcile what you see with your answer in part a)
c) Writing to Learn On the same screen, graph S as a function of the beam’s depth d, again taking k = 1. Compare the graphs with one another and with your answer in part a). What would be the effect of changing to some other value of k? Try it.
Ch. Sec. 4.5
Objective: At the end of the lesson you should be able to use the idea of local linearity to find approximations.
Practice problems:
HW Assignment: 4.5 p242 1-27odd
(a) Find the linearization [pic]of [pic]at [pic].
(b) How accurate is the approximation [pic]
1. [pic]
3. [pic]
5. [pic]
7. Show that the linearization of [pic]at x = 0 is [pic].
9. Use the linear approximation [pic]to find an approximation for the function [pic]for values of x near zero.
a) [pic] b) [pic] c) [pic]
11. Approximate [pic]by using a linearization centered at an approximate nearby number.
13. Approximate [pic]by using a linearization centered at an approximate nearby number.
15. Use Newton’s method to estimate all real solutions of [pic]accurate to 6 decimal places.
17. Use Newton’s method to estimate all real solutions of [pic]accurate to 6 decimal places.
19. For [pic]
a) find dy b) evaluate dy for x = 2 and dx = 0.05
21. . For [pic]
a) find dy b) evaluate dy for x = 1 and dx = 0.01
23. . For [pic]
a) find dy b) evaluate dy for [pic]and dx = -0.1
25. For [pic]
a) find dy b) evaluate dy for x = 0and dx = 0.01
27. Find the differential [pic]
Ch. Sec. 4.6
Objective: At the end of the lesson you should be able to set up and solve related rate problems.
Sample problem:
ACT:
1. In the circle below centered at O, if [pic]is 6 centimeters long and the perimeter of [pic] is 16 centimeters, what is the area of the circle in square centimeters?
[pic]
A) 25π
B) 36π
C) 50π
D) 64π
E) 100π
2. If the cosine of [pic] is [pic], what is the area of [pic] below?
[pic]
A) 6
B) 12
C) 20
D) 24
E) 40
HW Assignment: 4.6 pp251-254 9,11,13,14,16,19,21,22,29,33
Ch. Sec. 5.1
Objective: At the end of the lesson you should be able to use approximation methods for estimating areas under curves
Key ideas:
Investigation:
HW Assignment: 6a pp271 problems 17&18
Ch. Sec. 5.2
Objective: At the end of the lesson you should be able to use the geometric interpretation of the definite integral to evaluate certain integrals.
Key vocabulary and ideas:
Practice problem:
HW Assignment: 5.2 pp282-284 problems 1-27odd
Ch. Sec. 5.3
Objective: At the end of the lesson you should be able to use the properties of integrals to evaluate definite integrals.
Key vocabulary and ideas:
Numerical Integration:
HW Assignment: 5.3 pp290-292 problems 1,3,11-35odd,47
Ch. Sec. 5.4
Objective: At the end of the lesson you should be able to use the fundamental theorem of calculus to evaluate definite integrals.
Key vocabulary and ideas:
Practice problem:
HW Assignment: 5.4 pp302-305 problems 1,3,7,9,11,13,17,21,23,31,35,41,43,45,47,52
Ch. Sec. 5.5
Objective: At the end of the lesson you should be able to apply the trapezoidal rule to approximate definite integrals.
Key vocabulary and ideas:
Practice problem:
HW Assignment: 6e page 312 1-9
Ch. Sec. 6.1
Objective: At the end of the lesson you should be able to solve simple differential equations, and to graph the slope field which indicates the family of curves of the general solution.
Key vocabulary and ideas:
Practice problem:
1. Find the general solution to the exact differential equation: [pic]
13. Solve the initial value problem explicitly: [pic] and u = 1 when x = 1
HW Assignment: 6.1 pp327-330 1-19odd,25-28,33,35-40
Ch. Sec. 6.2
Objective: At the end of the lesson you should be able to evaluate indefinite and definite integrals by making the appropriate substitution.
Useful trigonometric identities:
Use a change of variables to turn [pic]into a familiar integral then evaluate it.
HW Assignment: 6.2 17-41odd,(53-65 odd, check your answers with fnInt)
Ch. Sec. 6.2
Objective: At the end of the lesson you should be able to evaluate indefinite and definite integrals by making the appropriate substitution.
Useful trigonometric identities:
For practice, try these problems again on your own:
48. Use substitution to evaluate: [pic]
50. Evaluate: [pic]
52. Evaluate: [pic]
HW Assignment: 6.2 18-42even,47-52,(54-66even, check your answers with fnInt)
Ch. Sec. 6.4
Objective: At the end of the lesson you should be able to solve differential equations by separation of variables, and to solve exponential growth and decay problems.
Key ideas:
For practice, try these problems again on your own:
3. Use separation of variables to solve the initial value problem: [pic] and y = 2 when x = 2.
11. Find the solution of the differential equation [pic] where k = 1.5, y(0)=100.
21. Half-life: The radioactive decay of Sm-151 (an isotope samarium) can be modeled by the differential equation [pic], where t is measured in years. Find the half-life of Sm-151.
HW Assignment: 6.4 pp357-361 1-27odd,31,35,45
Ch. Sec. 7.1
Objective: At the end of the lesson you should be able to solve application problems with the use of the definite integral.
Key ideas:
For practice, try these problems again on your own:
3. The function [pic] is the velocity in m/sec of a particle moving along the x-axis. Use analytical methods to find:
a) When the particle is moving to the right, to the left, and stopped.
b) The particle’s displacement (net change in position) for the given interval.
c) The total distance traveled by the particle.
17. The graph of the velocity of a particle moving on the x-axis is given. The particle starts at x=2 when t=0.
a) Find where the particle is at the end of the trip.
b) Find the total distance traveled by the particle.
v (m/sec)
[pic]t (sec)
HW Assignment: 8a: pp386-389 1-15odd, 17-20,23,24,27
Ch. Sec. 7.2
Objective: At the end of the lesson you should be able find the area between two curves
Key ideas:
5. Find the area between the two curves. [pic]
HW Assignment: 7.2 pp395-398 1-19odd, 35,37
Ch. Sec. 7.3
Objective: At the end of the lesson you should be able find the volumes of solids of known cross sections.
Key ideas:
Practice:
Let A be the area in the 1st quadrant bounded by the graph of [pic], the x-axis, and the graph of x=4. Find the volume of the solid generated when A is revolved:
1. About the x-axis 2. About the y-axis
3. Find the volume of the solid that lies between planes perpendicular to the x-axis at x = -1 and x = 1. The cross sections perpendicular to the x-axis between these planes are squares whose diagonals run from the semicircle [pic] to the semicircle[pic].
HW Assignment: 7.3 pp395-398 1-25odd
Holiday Travel Investigation
[pic]
Graphs of LRAM and RRAM
-----------------------
3
3
ACT Practice
1. [pic]
a. 328
b. 468
c. 1,000
d. 1,240
e. 23,496
2. A regular hexagon is inscribed in a circle with a radius of 5 centimeters. Which of the following statements is true about the perimeter of the hexagon?
A. It is less than 25 centimeters
B. It equals 25 centimeters
C. It equals 30 centimeters
D. It equals 50 centimeters
E. It is greater than 50 centimeters
3. The greatest common factor of 6 and 10 is multiplied by the least common multiple of 6 and 8. What is the product?
A. 24
B. 36
C. 48
D. 60
E. 120
[pic]
Given [pic] [pic] and [pic] find the following composite functions, then state the domain and range.
1. [pic]
2. [pic]
3. [pic]
Graph the absolute value function[pic], then state its domain and range.
Numerical Derivative of [pic]
NDER [pic]
Differentiability
1. On a 30-question test, Toni earns 5 points for every correct answer and loses 3 points for every incorrect answer. If she answered all the questions, and her score was 6, how many did she answer correctly?
A. 48
B. 25
C. 14
D. 12
2. [pic]
A. -11
B. -7
C. -5
D. 5
E. 11
3. In a rectangle of area 12 square meters, the length of the base is represented by (x + 2) meters and the height by ( x + 3 ) meters. What is the length of the base of the rectangle in meters?
A. 2
B. 3
C. 4
D. 5
E. 6
Exponential Functions
Exponential function Example
Base
Rules for Exponentials:
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]=
Half-life: [pic]
Exponential Growth and Decay:
Compounded Interest:
The number e:
Continuously Compounded Interest:
1. On a 30-question test, Toni earns 5 points for every correct answer and loses 3 points for every incorrect answer. If she answered all the questions, and her score was 6, how many did she answer correctly?
A. 48
B. 25
C. 14
D. 12
2. [pic]
A. -11
B. -7
C. -5
D. 5
E. 11
3. In a rectangle of area 12 square meters, the length of the base is represented by (x + 2) meters and the height by (x + 3) meters. What is the length of the base of the rectangle in meters?
A. 2
B. 3
C. 4
D. 5
E. 6
Functions and Logarithms
One-to-one function
Horizontal line test
Inverse of f
Identity function: (A way to test whether two functions are inverses of each other.)
Writing [pic] as a function of x example: [pic]
1. 1.
2. 2.
3. 3.
4. 4.
Exponential form
Logarithmic form
Natural logarithm
Common logarithm
Inverse properties
Base a: _________________________, ___________________________
Base e: _________________________, ___________________________
Properties of logarithms
1. 2. 3.
Proof Proof Proof
Change of base formula
THEOREM 7 If [pic]is continuous at c and g is continuous at [pic] then the composite [pic]is continuous at c.
Trigonometric Functions
Special triangles
[pic] [pic]
Unit circle
[pic]
Exact trigonometric ratios
Graphs (12)
Transformations: [pic]
Even and Odd trigonometric functions
Period
Frequency
Modeling trigonometric functions from data
1. If A = 40[pic], then [pic]
A. -3
B. 0
C. 2
D. 3
E. 4
2. What does [pic]equal?
A. [pic]
B. [pic]
C. [pic]
D. 6
E. 8
3. Jan received grades of 85, 92, and 100 on her first 3 tests. What must she get on her fourth test to have an average of exactly 90?
A. 82
B. 83
C. 87
D. 90
E. 95
Introduction to limits
Average speed
Instantaneous speed
Formal definition of a limit.
Assume f is defined in a neighborhood of c and let c and L be real numbers. The function f has limit L as x approaches c if, given any positive number (, there is a positive number ( such that for all x,
[pic]
Properties of limits:
1. Sum Rule:
2. Difference Rule:
3. Product Rule:
4. Constant Multiple Rule:
5. Quotient Rule:
6. Power Rule:
Polynomial Functions
Rational Functions
Right-hand limit
Left-hand limit
Two-sided limit
Sandwich theorem
A. 34
B. 29
C. 24
D. 15
E. 14
Limits involving infinity
Horizontal asymptote
Vertical asymptote
End behavior
End behavior model
Properties of limits:
1. Sum Rule:
2. Difference Rule:
3. Product Rule:
4. Constant Multiple Rule:
5. Quotient Rule:
6. Power Rule:
Sandwich theorem
Continuity
Intuitive notion of continuity
Continuous at an interior point c
Continuous at a left endpoint a
Continuous at a right endpoint b
Discontinuous
Point of discontinuity
removable
Jump Discontinuity
Infinite discontinuity
Oscillating discontinuity
Removing a discontinuity (extended function)
Intermediate value for continuous functions
Rates of Change and Tangent Lines
Average rate of change
Slope
Tangent line to a curve
Difference quotient
Normal line
Instantaneous rate of change
Speed
Derivative of a Function
Definition of Derivative
Definition (Alternative) x = a
Notation
Relationships of f and f’
Where f’ does not exist
Locally linear
Symmetric difference quotient
Numerical derivative
Graphing a derivative using NDER
Intermediate Value Theorem for Derivatives
Rules for Differentiation
Derivative of a constant
Power rule
The constant multiple rule
The sum and difference rule
The product rule
The quotient rule
Higher order derivatives
Velocity and Other Rates of Change
Instantaneous Rate of Change (definition of the derivative)
Displacement
Average velocity
Instantaneous velocity
Speed
Acceleration
Marginal Cost and Marginal Revenue
Derivatives of Trigonometric Functions
Making a conjecture with NDER about the derivative of the sine function.
Confirm the derivative of the sine with the definition of a derivative.
[pic]
Jerk
Chain Rule
Relating Derivatives
Chain Rule
Outside-Inside Rule
Implicit Differentiation
1. x = tan y
2. [pic]
3. Fine the slope of [pic] at (-3,-4).
1. [pic]
21. Find the equation of the a) tangent line b) normal line of [pic] at (-1,0)
1. Find dy/dx, [pic]
2. [pic]
Derivatives of Inverse Trigonometric
Functions
1. [pic] 4. [pic]
2. [pic] 5. [pic]
3. [pic] 6. [pic]
Derivatives of Exponential and Logarithmic
Functions
1. [pic]
2. [pic]
3. [pic]
3. [pic]
4. [pic]
16. [pic]
30. At what point on the graph of [pic] is the tangent line perpendicular to the line[pic]?
Extreme Values of Functions
1. Absolute extreme values (global extreme):
2. Local extreme values (relative extreme):
3. Critical values:
1. [pic]
21. Find the equation of the a) tangent line b) normal line of [pic] at (-1,0)
Find the values of a for:
1. f”(a) = 0
2. f(a) = 0
3. f”(a) < 0
4. f”( ) > 0
Find the extreme values of the function and where they occur.
15. [pic]
25. [pic]
Mean Value Theorem
1. Mean Value Theorem:
2. Secant line:
3. Tangent line:
Write the conclusion below:[pic]
State whether or not the function satisfies the hypotheses of the Mean Value Theorem on the given interval, and if it does, find each value of c in the interval (a,b) that satisfies the equation:
[pic][pic]
2. [pic]on [0,1]
8. [pic]on [-1,3]
Derivative Tests
First derivative test:
1. At a critical point c:
a) f ’ changes sign from + to –
b) f ’ changes sign from – to +
c) f ’ does not change sign
2. At a left endpoint a:
a) f ’ < 0
b) f ’ > 0
3. At right endpoint b:
a) f ’ < 0
b) f ’ > 0
Concavity test:
a) y” > 0
b) y” < 0
Point of inflection:
Second derivative test for local Extrema:
a) f ’(c) = 0 and f ”(c) < 0:
a) f ’(c) = 0 and f ”(c) < 0:
Applications of Derivatives
1. Read the problem carefully until you understand what is asked.
2. Sketch a diagram (graph) to represent the problem.
3. Find a formula or equation that describes the situation.
4. Take the derivative then find where it is zero or undefined (critical values). Be sure to consider the endpoints.
5. Solve the model.
6. Interpret the solution using correct units and vocabulary.
Find the area of the largest rectangle that can be inscribed between the graph of [pic]and the x-axis.
Applications of Derivatives
1. Read the problem carefully until you understand what is asked.
2. Sketch a diagram (graph) to represent the problem.
3. Find a formula or equation that describes the situation.
4. Take the derivative then find where it is zero or undefined (critical values). Be sure to consider the endpoints.
5. Solve the model.
6. Interpret the solution using correct units and vocabulary.
What are the dimensions of the lightest open-top right cylindrical can that will hold a volume of 1000 cm3?
Newton’s Method and Linearization
Linearization
If [pic]is differentiable at [pic], then the equation of the tangent line,
[pic],
defines the linearization of [pic]at [pic].
The point [pic] is at the center of the approximation.
Newton’s Method:
Newton’s Method with a calculator:
Estimating change with differentials:
Use Newton’s method to estimate all of the zeros of [pic]
The radius of a circle is increased from 3.00 to 3.05m. Estimate the resulting change in area.
Related Rates
1. Read the problem carefully until you understand what is asked.
2. Sketch a diagram (graph) to represent the problem.
3. Find a formula or equation that describes the situation. Don’t substitute values for variables that represent changing quantities in the model yet.
4. Differentiate both sides of the equation with respect to time t. Every term should be divided by dt.
5. Now substitute values for any quantities that depend on time.
6. Interpret the solution using correct units and vocabulary.
19. Sliding Ladder A 13ft. ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5ft/sec.
See the diagram on page 252.
a) How fast is the top of the ladder sliding down the wall at that moment?
b) At what rate is the area of the triangle formed by the ladder, wall, and ground changing at that moment?
c) At what rate is the angle [pic] between the ladder and the ground changing at that moment?
Finite Sums
RAM
LRAM
RRAM
MRAM
Try this on your next road trip (while someone else drives!):
1. Record the miles on the odometer
2. Take periodic readings of the speed as indicated by the speedometer. Be sure to record the lapsed time for each reading. You decide at what precise time during the interval to write down the speed, but be consistent. The intervals do not have to be the same duration.
3. After you collect an adequate amount of data, record the miles on the odometer again.
4. Multiply each time interval (converted to hours) by the speed that you recorded for that interval.
5. Add these products.
6. Compare this number with the total distance traveled, which will be the difference of the odometer readings.
7. Summarize your findings.
8. Create a chart of your data, a graph of the results, and a brief description.
Definite Integrals
Partition:
Subintervals:
Riemann Sum:
[pic]
[pic]=___________________
[pic]
Area under the curve interpretation of [pic]:
The integral of a constant [pic]
Evaluate [pic]
Definite Integrals and Antiderivatives
1. Order of integration:
2. Zero:
3. Constant Multiple:
4. Sum and Difference:
5. Additivity:
6. Max-Min Inequality:
7. Domination:
Average (Mean) Value for Definite Integrals
Connecting Differential and Integral Calculus
Find the value of [pic] using the (NINT) function on your calculator
Definite Integrals
Fundamental Theorem of Calculus:
proof:
Fundamental Theorem, Part 2
Finding total area:
Find the derivative of [pic]
Trapezoidal Rule
Area of a rectangle: Area of a parallelogram: Area of a trapezoid:
Find the exact area of the indicated trapezoid:
[pic]
Trapezoidal rule:
Use the Trapezoidal rule with n=4 to:
a) Approximate the value of [pic]
b) Use concavity to determine if the approximation is an over estimation or an under estimation.
c) Find the integral’s exact value to check your answers.
Slope Fields
Differential equation:
General solution:
Initial condition (value):
Particular solution:
Slope Field: Construct a slope field for [pic] at the indicated points. Graph the solution through the origin (0,0).
[pic]
Antidifferentiation by Substitution
Liebniz Notation: Explain the significance of dx in terms of the limit of Riemann sum.
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
Pythagorean identities: Double angle identities
[pic] [pic]
Antidifferentiation by Substitution
(Continued)
2/1/07
Pythagorean identities:
[pic]
Double angle identities:
[pic]
Exponential Growth and Decay
Separable Differential Equations:
The Law of exponential Change:
Compounded interest:
Half-life:
Integral as Net Change
Net change vs. total change
Strategy for Modeling with integrals:
Areas in the Plane
Typical rectangle:
Integrating with respect to y or x:
Making the choice:
Volumes
2/1/07
Typical slice:
Washer method:
Shell method:
Increments
Slope
Point-Slope Equation
Slope-Intercept Equation
General Linear Equation
Determine if the following functions are even, odd, or neither. Use the definitions to explain. Sketch the graph of each to check your answer.
1. [pic]
2. [pic]
3. [pic]
Find the natural domain for each function.
1. [pic]
2. [pic]
3. [pic]
If [pic], what are the possible values for [pic]?
Range: [pic]
Domain: [pic]
Dependent variables
Independent variables ___________ _____________
(output) (input)
[pic]
Range
Domain
Natural domain
Graph of a function
Boundary Points
Open and Closed Intervals
Interval Notation
[pic]
[pic]
[pic]
[pic]
[pic]
y
x
................
................
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