AP CALCULUS AB UNIT 3 BASIC DIFFERENTIATION RULES TOTAL NAME DATE PERIOD DATE TOPIC ASSIGNMENT /18 9/19 9/24 9/25 9/26 9/27 9/28 10/1 10/2 10/3
|
|
- Brandon Clarke
- 5 years ago
- Views:
Transcription
1 NAME DATE PERIOD AP CALCULUS AB UNIT BASIC DIFFERENTIATION RULES DATE TOPIC ASSIGNMENT 0 0 9/8 9/9 9/ 9/5 9/6 9/7 9/8 0/ 0/ 0/ 0/ 0/5 TOTAL
2
3 AP Calculus AB Worksheet 9 Average Rates of Change Find the average rate of change of f ( ) = + over the interval,. Find the average rate of change of f ( ) = + over the interval Find the average rate of change of f ( ) = + cos over the interval 0,. *** Calculator *** 0,. The amount of money in a bank account, called the balance, after t years is f ( t ) = 00(.08) t dollars. a) What are the units of the rate of change of f ( t ). b) Find the average rate of change of the balance over the time interval 0,0.5. Eplain the meaning of this value in the contet of the bank account. 5 *** Calculator *** The amount of money in a bank account, called the balance, after t years is f ( t ) = 00( 0.9) t dollars. Find the average rate of change of the balance over the time interval 0,0.5. Eplain the meaning of this value in the contet of the bank account. 6 The table below shows the weight, in lbs., of a grown man over a certain time period, t, in years. Consider the first year of the data collection period as t = 0. Year Weight a) Estimate the rate of change of the man s weight over the time period 0,5. Using correct units, eplain the meaning of this value in the contet of the man s weight. b) Estimate the rate of change of the man s weight over the time period eplain the meaning of this value in the contet of the man s weight.,6. Using correct units, 7 f + h f Evalauate lim h 0 h f 5 for a) = + b) f ( ) = 7 8 Find the equation of the line that passes through, and is perpendicular to y= 7. 9 Verify that f ( ) +, = +, is either continuous or discontinuous at =.
4 AP Calculus AB Worksheet 0 Limit Definition of the Derivative and the Tangent Line Problem Find the derivative of the function using the limit process. Remember: The derivative is a new function so it requires new notation; Eample: g( ) = 5 h s = + s f = Find the slope of the tangent line to the graph of the function at the given point. f ( ) = + ; (,) 5 f ( t) = t t ;, Find the equation of the tangent line to the graph of the function at the given value of. 6 f ( ) = + +, = 7 g ( ) = ( ), = dy f ' or or y ' d 8 What is the value of k if f ( )= k, +, is everywhere continuous? 9 5+ =, 6+ 8 a) Find the discontinuities and identify them as removable or non-removable. b) Identify the vertical asymptotes and describe the behavior to the left and right of the vertical asymptote. f. Write as a piecewise For the function f ( ) c) Write a new function, g( ), that removes the removable discontinuity from function. d) Identify any horizontal asymptotes. The table below shows selected values for a function, f ( ), at various values of f ( ) 8 0) Find the average rate of change of the function over the interval,. ) Find the average rate of change of the function over the interval 5,9. ) Estimate the slope of the function when =.
5 AP Calculus AB - Worksheet Power Rule Nearly all men can stand adversity, but if you want to test a man's character, give him power. - Abraham Lincoln Learn: The Power Rule: ' f = a n ( n ) f = n a d f = f ' = 8 = d Eample: Use the Power Rule to find the derivative of each function.. y f 5 =. =. f ( ) = +. y= 5, 0 5. f ( t) = t + t 6 6. f ( ) = 7 7. y = ( + ) 8. f ( ) =, 0 9. f ( ) = ( ) 0. f =, 0. y f = a + b + c =. Find the instantaneous rate of change of each function when =.. y =, 0. y = m + b 5. f ( ) = f ( ) = y =, 0 9. y = f t = t + t Find the slope of the curves at the indicated values of. 0. = +, =. f ( ) = +, = 0, y. f ( ) = =., y = + +, = 0. Evaluate ( + ) 5( + ) + 6 ( 5 + 6) 0 lim using the power rule. 5. Find two unique functions, f ( ), such that f ' = (Answers will vary)
6 AP Calculus AB - Worksheet Tangent and Normal Lines (Power Rule) Learn: Tangent and Normal Lines to a Curve Recall: Derivative = slope of the Tangent line at that point s -coordinate Eample: f = +, = = y = ( ) f ' f ' slope of the tangent line Tangent Line: Normal Line: y = ( ) For each of the following: a) Sketch a graph - USE GRAPH PAPER!! b) Find the slope of the tangent line at the given point. c) Find the equations of the tangent line at the given point. Sketch the line. d) Find the equation of the normal to the curve at the given point. Sketch the line.. y=, (,). f ( ) = 6 (,). f ( ) =, (, ). y=, (,) Find the equations of the tangent and normal lines to the curve at the given -value. 5. y = + 8 +, = 6. y = =, 7. Find the points on the curve y = + + where the tangent is horizontal. 8. Show that the curve y = has no tangent line with slope. 9. Find an equation of the tangent line to the curve y = that is parallel to the line y= Find a parabola with equation y = a + b + c that has slope at =, slope -8 at =, and passes through the point (,5 ). Hint: You will create different equations in order to solve for a, b, and c.. Evaluate lim 0 ( + ) ( + ) ( ) using the Power Rule.
7 Worksheet Answers ) y' = ) f ' = ) = y ' = ( ) tangent: y = normal: y = ( ) ( ) ( ) f ' = ( ) tangent: y = normal: y = ( ) ) 5) 6) y' = 8 y ' = tangent: y = normal: y = y = 9 y = + + y' = + 8 y ' = ( ) tangent: y 9 = normal: y 9 = ( ) f = f '( ) = f ' = tangent: y = normal: y = y = y ( ) = 0 y ' = 6 y ' = ( ) ( ) ( ) tangent: y 0 = + normal: y 0 = + ( ) 7) y ' = = 0 8) + = 0 ( )( ) + = 0 = ; = y = ; y = 6 (, ); (, 6) y = + y 6 5 ' = = 8 = No Real Solution 9) 0) y = = y' = = = = ( ) y 8 = y = a + b + c y ' = a + b slope at = a + b = slope 8 at = a + b = 8 passes thru,5 a + b + c = 5 a+ b= a+ b= 8 b = a = + c = 5 b= c= 7 y = + 7
8 Worksheet f = a f ' = n a Learn: Package Rule n n d d The Package Rule Eample: f ( ) = ( + 7 ) f '( ) = ( 9)( )( + 7) ( ) ( 7) = 5 ( + 7) At the specified point, find the equation of the normal to the curve: f ( ) 00 at 6, f ( 6). For what value of k, is For problems -6, find f ' y k f = +, is a normal line to the graph of. Do not simplify your answer.. f ( ) = ( + ) 8. f ( ) ( ) 6 = + 7. f ( ) f 6. 5 f 9. =. = +. = 5. = ( + ) f = 8. f ( ) = + = + 0. f ( ) = +. f ( ) =. f =. f ( ) f ( ) ( )( ) = 6. 5 f =. f ( ) = + = 5 Write the equation of the normal line at the given -value. 7. f ( ) = 7 when = f ( ) = when = 5 0. f = + when = 0 f = when =
9 AP Calculus AB - Worksheet Product and Quotient Rules for Derivatives Differentiate the following functions. Do not simplify the answer... 5 g( t) = 6t. h( z) = 8z. f ( s) 5 s s 5s =. B = v Gv () = v 8 6+ t gt () = t f ( ) = = +. f( ) g( ) = 5. p = k( ) ( )( 6 5) = h( ) 5 F() t = t + = + 7. h( ) = ( 5 ) = + 8. S( ) = ( + ) 8. k( r) r ( r 7r r) t 9. H ( z) ( z 5 z )( 7z z ) = z N( z) = z f( ) = + 0. M( ) = 7 + +
10 AP Calculus AB Worksheet 5
11 AP Calculus AB Worksheet 6 Derivatives Using Tables a) Given h( ) = f ( ) + g ( ), find h' b) Given s( ) = f ( ) g ( ), find s' ( ) c) Given p( ) = f ( ) g ( ), find p' ( ) g( ) d) Given q( ) =, find q' ( 5) f ( ) e) Given c ( ) = ( f ( )) find c' ( ) f) Given m( ) = f ( g ( ) ), find h' ( 6)
12 AP Calculus AB Worksheet 7 Differentiability Describe the -values at which f is a) differentiable. b) Continuous but not differentiable. c) Neither continuous or differentiable ) ) ) ) 5) 6) Determine if the function is differentiable at the given value., f =, =, 7) +, f =, =, 8) 9) Sketch the graph of a continuous function such that: f 0 = ; ' f = for 0) Sketch the graph of a continuous function such that: f 0 = f ' 0 is undefined f ' 0 for 0 f ' 0 for 0
13 AP Calculus AB Worksheet 8 Derivatives (Power, Package, Product and Quotient Rules) Review Use the limit definition of the derivative to find f '( ) for f ( ) Find the derivative of each function below. Simplify your answer. y = + y = 5 + f ( ) = 5 f ( ) = ( )( + ) 6 y = + 5 = + Answer each question about tangents and normals. 7 Find the points on the curve y = + 0 where the tangent line is parallel to the -ais Find the slope of the normal to f = + at the point where = Find the equation of the tangent to y = + at the point where =. Find dy d for y ( )( ) = + +, then find the slope of the normal when =. If the line y = is tangent to the function f = +, what is the point of tangency? If the slope of a tangent line is 5 then what is slope of the normal line to the same curve at the same point? At what -value is y = tangent to f ( ) = +. For #-9, use the table below to find the indicated value. h( ) = f ( ) + g ( ) h 5. Given d ( ) f ( ) g ( ), find d '( ) 6. Given p( ) = f ( ) g ( ), find p' ( ). g( ) 7. Given q( ) =, find q' ( ). f ( ) 8. Given k ( ) = ( f ( ) ), find k. 9. Given c ( ) = f ( g ( ) ), find c Given, find '. =.
14 Answers: ) f ' = 5) f '( )= + + 9) y = ) dy d = 0) dy 7 ( +) ) y' = ) y' = ) 6 d = (, 7) (, 0) ) 0, ) f ' 8) 5 ) = = ( )
Grade 12 (MCV4UE) AP Calculus Page 1 of 5 Derivative of a Function & Differentiability
Grade 2 (MCV4UE) AP Calculus Page of 5 The Derivative at a Point f ( a h) f ( a) Recall, lim provides the slope of h0 h the tangent to the graph y f ( at the point, f ( a), and the instantaneous rate of
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationAP Calculus AB Worksheet - Differentiability
Name AP Calculus AB Worksheet - Differentiability MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The figure shows the graph of a function. At the
More information1 y = Recitation Worksheet 1A. 1. Simplify the following: b. ( ) a. ( x ) Solve for y : 3. Plot these points in the xy plane:
Math 13 Recitation Worksheet 1A 1 Simplify the following: a ( ) 7 b ( ) 3 4 9 3 5 3 c 15 3 d 3 15 Solve for y : 8 y y 5= 6 3 3 Plot these points in the y plane: A ( 0,0 ) B ( 5,0 ) C ( 0, 4) D ( 3,5) 4
More informationUnderstanding Part 2 of The Fundamental Theorem of Calculus
Understanding Part of The Fundamental Theorem of Calculus Worksheet 8: The Graph of F () What is an Anti-Derivative? Give an eample that is algebraic: and an eample that is graphical: eample : Below is
More informationTOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
More information5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5).
Rewrite using rational eponents. 2 1. 2. 5 5. 8 4 4. 4 5. Find the slope intercept equation of the line parallel to y = + 1 through the point (4, 5). 6. Use the limit definition to find the derivative
More informationCalculus I Sample Exam #01
Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationCALCULUS AB SECTION II, Part A
CALCULUS AB SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. pt 1. The rate at which raw sewage enters a treatment tank
More informationCalculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
More informationSection Derivatives and Rates of Change
Section. - Derivatives and Rates of Change Recall : The average rate of change can be viewed as the slope of the secant line between two points on a curve. In Section.1, we numerically estimated the slope
More informationAP Calculus AB Free-Response Scoring Guidelines
Question pt The rate at which raw sewage enters a treatment tank is given by Et 85 75cos 9 gallons per hour for t 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per
More informationDaily WeBWorK. 1. Below is the graph of the derivative f (x) of a function defined on the interval (0, 8).
Daily WeBWorK 1. Below is the graph of the derivative f (x) of a function defined on the interval (0, 8). (a) On what intervals is f (x) concave down? f (x) is concave down where f (x) is decreasing, so
More informationSummer Review Packet (Limits & Derivatives) 1. Answer the following questions using the graph of ƒ(x) given below.
Name AP Calculus BC Summer Review Packet (Limits & Derivatives) Limits 1. Answer the following questions using the graph of ƒ() given below. (a) Find ƒ(0) (b) Find ƒ() (c) Find f( ) 5 (d) Find f( ) 0 (e)
More informationPre-Calculus Mathematics Limit Process Calculus
NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Mrs. Nguyen s Initial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to find
More informationSolution to Review Problems for Midterm #1
Solution to Review Problems for Midterm # Midterm I: Wednesday, September in class Topics:.,.3 and.-.6 (ecept.3) Office hours before the eam: Monday - and 4-6 p.m., Tuesday - pm and 4-6 pm at UH 080B)
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Write the equation of the line that goes through the points ( 3, 7) and (4, 5)
More informationProperties of Derivatives
6 CHAPTER Properties of Derivatives To investigate derivatives using first principles, we will look at the slope of f ( ) = at the point P (,9 ). Let Q1, Q, Q, Q4, be a sequence of points on the curve
More informationAP Calculus Worksheet: Chapter 2 Review Part I
AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative
More informationWarmup for AP Calculus BC
Nichols School Mathematics Department Summer Work Packet Warmup for AP Calculus BC Who should complete this packet? Students who have completed Advanced Functions or and will be taking AP Calculus BC in
More informationAP Calculus I Summer Packet
AP Calculus I Summer Packet This will be your first grade of AP Calculus and due on the first day of class. Please turn in ALL of your work and the attached completed answer sheet. I. Intercepts The -intercept
More informationHelpful Website:
As we continue our journey through Calculus, there are certain skills that you learned this year which should be remembered/reviewed. Mastering these skills is crucial to your success not only in net year
More informationWork the following on notebook paper. You may use your calculator to find
CALCULUS WORKSHEET ON 3.1 Work the following on notebook paper. You may use your calculator to find f values. 1. For each of the labeled points, state whether the function whose graph is shown has an absolute
More informationAP Calculus (BC) Summer Assignment (104 points)
AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationMath 2414 Activity 1 (Due by end of class July 23) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math 44 Activity (Due by end of class July 3) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationDerivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
More informationMORE CURVE SKETCHING
Mathematics Revision Guides More Curve Sketching Page of 3 MK HOME TUITION Mathematics Revision Guides Level: AS / A Level MEI OCR MEI: C4 MORE CURVE SKETCHING Version : 5 Date: 05--007 Mathematics Revision
More informationSection 3.2 The Derivative as a Function Graphing the Derivative
Math 80 www.timetodare.com Derivatives Section 3. The Derivative as a Function Graphing the Derivative ( ) In the previous section we defined the slope of the tangent to a curve with equation y= f ( )
More informationMath 131. The Derivative and the Tangent Line Problem Larson Section 2.1
Math 131. The Derivative and the Tangent Line Problem Larson Section.1 From precalculus, the secant line through the two points (c, f(c)) and (c +, f(c + )) is given by m sec = rise f(c + ) f(c) f(c +
More informationAP Calculus AB/IB Math SL2 Unit 1: Limits and Continuity. Name:
AP Calculus AB/IB Math SL Unit : Limits and Continuity Name: Block: Date:. A bungee jumper dives from a tower at time t = 0. Her height h (in feet) at time t (in seconds) is given by the graph below. In
More informationMath Exam 1a. c) lim tan( 3x. 2) Calculate the derivatives of the following. DON'T SIMPLIFY! d) s = t t 3t
Math 111 - Eam 1a 1) Evaluate the following limits: 7 3 1 4 36 a) lim b) lim 5 1 3 6 + 4 c) lim tan( 3 ) + d) lim ( ) 100 1+ h 1 h 0 h ) Calculate the derivatives of the following. DON'T SIMPLIFY! a) y
More information1. Given the function f (x) = x 2 3bx + (c + 2), determine the values of b and c such that f (1) = 0 and f (3) = 0.
Chapter Review IB Questions 1. Given the function f () = 3b + (c + ), determine the values of b and c such that f = 0 and f = 0. (Total 4 marks). Consider the function ƒ : 3 5 + k. (a) Write down ƒ ().
More informationLimits, Continuity, and the Derivative
Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change
More informationDirections: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work
More informationRules for Differentiation Finding the Derivative of a Product of Two Functions. What does this equation of f '(
Rules for Differentiation Finding the Derivative of a Product of Two Functions Rewrite the function f( = ( )( + 1) as a cubic function. Then, find f '(. What does this equation of f '( represent, again?
More informationAP Calculus Summer Packet
AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept
More informationx 3x 1 if x 3 On problems 8 9, use the definition of continuity to find the values of k and/or m that will make the function continuous everywhere.
CALCULUS AB WORKSHEET ON CONTINUITY AND INTERMEDIATE VALUE THEOREM Work the following on notebook paper. On problems 1 4, sketch the graph of a function f that satisfies the stated conditions. 1. f has
More informationHomework 4 Solutions, 2/2/7
Homework 4 Solutions, 2/2/7 Question Given that the number a is such that the following limit L exists, determine a and L: x 3 a L x 3 x 2 7x + 2. We notice that the denominator x 2 7x + 2 factorizes as
More informationMath Worksheet 1 SHOW ALL OF YOUR WORK! f(x) = (x a) 2 + b. = x 2 + 6x + ( 6 2 )2 ( 6 2 )2 + 7 = (x 2 + 6x + 9) = (x + 3) 2 2
Names Date. Consider the function Math 0550 Worksheet SHOW ALL OF YOUR WORK! f() = + 6 + 7 (a) Complete the square and write the function in the form f() = ( a) + b. f() = + 6 + 7 = + 6 + ( 6 ) ( 6 ) +
More informationExponential Growth. b.) What will the population be in 3 years?
0 Eponential Growth y = a b a b Suppose your school has 4512 students this year. The student population is growing 2.5% each year. a.) Write an equation to model the student population. b.) What will the
More informationFinal Exam Review Sheet Algebra for Calculus Fall Find each of the following:
Final Eam Review Sheet Algebra for Calculus Fall 007 Find the distance between each pair of points A) (,7) and (,) B), and, 5 5 Find the midpoint of the segment with endpoints (,) and (,) Find each of
More informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationMr. Castle
AP CALCULUS SUMMER ASSIGNMENT Dear Prospective AP Calculus Student, Welcome to AP Calculus! This is a rigorous yet rewarding math course that will challenge you and develop your critical thinking and problem
More informationMath 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math Activity (Due by end of class Jan. 6) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationPTF #AB 07 Average Rate of Change
The average rate of change of f( ) over the interval following: 1. y dy d. f() b f() a b a PTF #AB 07 Average Rate of Change ab, can be written as any of the. Slope of the secant line through the points
More informationsin x (B) sin x 1 (C) sin x + 1
ANSWER KEY Packet # AP Calculus AB Eam Multiple Choice Questions Answers are on the last page. NO CALCULATOR MAY BE USED IN THIS PART OF THE EXAMINATION. On the AP Eam, you will have minutes to answer
More information0,0 B 5,0 C 0, 4 3,5. y x. Recitation Worksheet 1A. 1. Plot these points in the xy plane: A
Math 13 Recitation Worksheet 1A 1 Plot these points in the y plane: A 0,0 B 5,0 C 0, 4 D 3,5 Without using a calculator, sketch a graph o each o these in the y plane: A y B 3 Consider the unction a Evaluate
More information1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2.
. Find A and B so that f Ae B has a local minimum of 6 when.. The graph below is the graph of f, the derivative of f; The domain of the derivative is 5 6. Note there is a cusp when =, a horizontal tangent
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
More informationTRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)
TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents
More informationTroy High School AP Calculus Summer Packet
Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by
More information1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x
1.5 Topics: The Derivative lutions 1. Use the limit definition of derivative (the one with x in it) to find f x given f x 4x 5x 6 4 x x 5 x x 6 4x 5x 6 f x x f x f x x0 x x0 x xx x x x x x 4 5 6 4 5 6
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION Many applications of calculus depend on our ability to deduce facts about a function f from information concerning its derivatives. APPLICATIONS
More informationMath 108, Solution of Midterm Exam 3
Math 108, Solution of Midterm Exam 3 1 Find an equation of the tangent line to the curve x 3 +y 3 = xy at the point (1,1). Solution. Differentiating both sides of the given equation with respect to x,
More informationSect The Slope-Intercept Form
0 Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not
More informationWelcome to Advanced Placement Calculus!! Summer Math
Welcome to Advanced Placement Calculus!! Summer Math - 017 As Advanced placement students, your first assignment for the 017-018 school year is to come to class the very first day in top mathematical form.
More informationAB Calculus: Rates of Change and Tangent Lines
AB Calculus: Rates of Change and Tangent Lines Name: The World Record Basketball Shot A group called How Ridiculous became YouTube famous when they successfully made a basket from the top of Tasmania s
More informationPre-Calculus Midterm Practice Test (Units 1 through 3)
Name: Date: Period: Pre-Calculus Midterm Practice Test (Units 1 through 3) Learning Target 1A I can describe a set of numbers in a variety of ways. 1. Write the following inequalities in interval notation.
More informationStudent Session Topic: Average and Instantaneous Rates of Change
Student Session Topic: Average and Instantaneous Rates of Change The concepts of average rates of change and instantaneous rates of change are the building blocks of differential calculus. The AP exams
More informationCalculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ), c /, > 0 Find the limit: lim 6+
More information1 DL3. Infinite Limits and Limits at Infinity
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 78 Mark Sparks 01 Infinite Limits and Limits at Infinity In our graphical analysis of its, we have already seen both an infinite
More informationMATH 1325 Business Calculus Guided Notes
MATH 135 Business Calculus Guided Notes LSC North Harris By Isabella Fisher Section.1 Functions and Theirs Graphs A is a rule that assigns to each element in one and only one element in. Set A Set B Set
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this
More informationChapter 1/3 Rational Inequalities and Rates of Change
Chapter 1/3 Rational Inequalities and Rates of Change Lesson Package MHF4U Chapter 1/3 Outline Unit Goal: By the end of this unit, you will be able to solve rational equations and inequalities algebraically.
More informationMAC 2233 Chapter 3 Practice for the Test
Class: Date: MAC 33 Chapter 3 Practice for the Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. At which labeled point is the slope of the tangent
More informationUNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions
UNIT 3 Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions Recall From Unit Rational Functions f() is a rational function
More informationCalculus Module C09. Delta Process. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Calculus Module C09 Delta Process Copyright This publication The Northern Alberta Institute of Technology 00. All Rights Reserved. LAST REVISED April, 009 Delta Process Statement of Prerequisite Skills
More informationTangent Lines and Derivatives
The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the
More informationSet 3: Limits of functions:
Set 3: Limits of functions: A. The intuitive approach (.): 1. Watch the video at: https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-its-dc/v/itintuition-review. 3.
More informationUnit #3 Rules of Differentiation Homework Packet
Unit #3 Rules of Differentiation Homework Packet In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified
More informationAP Calculus BC. Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines
AP Calculus BC Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines Essential Questions & Why: Essential Questions: What is the difference between average and instantaneous rates of
More informationSo, t = 1 is a point of inflection of s(). Use s () t to find the velocity at t = Because 0, use 144.
AP Eam Practice Questions for Chapter AP Eam Practice Questions for Chapter f 4 + 6 7 9 f + 7 0 + 6 0 ( + )( ) 0,. The critical numbers of f( ) are and.. Evaluate each point. A: d d C: d d B: D: d d d
More informationAP Calculus AB Chapter 2 Test September 20, 2014 Show all your work.
Name AP Calculus AB Chapter Test September 0, 014 Show all your work. 1. Find 5 7 6 10 lim 5 7 A. 5/ B. -5/ C. 0 D. The limit does not eist because the function increases without bound. E. The limit does
More informationThe Fundamental Theorem of Calculus Part 3
The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative
More informationMATH The Derivative as a Function - Section 3.2. The derivative of f is the function. f x h f x. f x lim
MATH 90 - The Derivative as a Function - Section 3.2 The derivative of f is the function f x lim h 0 f x h f x h for all x for which the limit exists. The notation f x is read "f prime of x". Note that
More informationCalculus AB Topics Limits Continuity, Asymptotes
Calculus AB Topics Limits Continuity, Asymptotes Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam. Consider f x 2x 1 x 3 1 x 3
More informationChapter 1/3 Rational Inequalities and Rates of Change
Chapter 1/3 Rational Inequalities and Rates of Change Lesson Package MHF4U Chapter 1/3 Outline Unit Goal: By the end of this unit, you will be able to solve rational equations and inequalities algebraically.
More informationTopic Subtopics Essential Knowledge (EK)
Unit/ Unit 1 Limits [BEAN] 1.1 Limits Graphically Define a limit (y value a function approaches) One sided limits. Easy if it s continuous. Tricky if there s a discontinuity. EK 1.1A1: Given a function,
More informationAP Calculus Chapter 3 Testbank (Mr. Surowski)
AP Calculus Chapter 3 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.). If f(x) = 0x 4 3 + x, then f (8) = (A) (B) 4 3 (C) 83 3 (D) 2 3 (E) 2
More informationAP Calculus AB Summer Assignment
AP Calculus AB 08-09 Summer Assignment Dear Future AP Student, I hope you are ecited for the year of Calculus that we will be pursuing together! I don t know how much you know about Calculus, but it is
More information(ii) y = ln 1 ] t 3 t x x2 9
Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside
More informationARE YOU READY FOR CALCULUS?
ARE YOU READY FOR CALCULUS? Congratulations! You made it to Calculus AB! Instructions 1. Please complete the packet (see below), which will be due the day of registration. This packet will help you review
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
More informationarb where a A, b B and we say a is related to b. Howdowewritea is not related to b? 2Rw 1Ro A B = {(a, b) a A, b B}
Functions Functions play an important role in mathematics as well as computer science. A function is a special type of relation. So what s a relation? A relation, R, from set A to set B is defined as arb
More informationBE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!!
Review Guide for MAT0 Final Eam Part I. Thursday December 7 th during regular class time Part is worth 50% of your Final Eam grade. Syllabus approved calculators can be used on this part of the eam but
More informationStudents must be prepared to take a quiz on pre-calculus material by the 2 nd day of class.
AP Calculus AB Students must be prepared to take a quiz on pre-calculus material by the nd day of class. You must be able to complete each of these problems Uwith and without the use of a calculatoru (unless
More informationAP CALCULUS SUMMER REVIEW WORK
AP CALCULUS SUMMER REVIEW WORK The following problems are all ALGEBRA concepts you must know cold in order to be able to handle Calculus. Most of them are from Algebra, some are from Pre-Calc. This packet
More information(i) find the points where f(x) is discontinuous, and classify each point of discontinuity.
Math Final Eam - Practice Problems. A function f is graphed below. f() 5 4 8 7 5 4 4 5 7 8 4 5 (a) Find f(0), f( ), f(), and f(4) Find the domain and range of f (c) Find the intervals where f () is positive
More informationAP Calculus BC Summer Packet 2017
AP Calculus BC Summer Packet 7 o The attached packet is required for all FHS students who took AP Calculus AB in 6-7 and will be continuing on to AP Calculus BC in 7-8. o It is to be turned in to your
More informationTopic 6: Calculus Integration Markscheme 6.10 Area Under Curve Paper 2
Topic 6: Calculus Integration Markscheme 6. Area Under Curve Paper. (a). N Standard Level (b) (i). N (ii).59 N (c) q p f ( ) = 9.96 N split into two regions, make the area below the -ais positive RR N
More informationA.P. Calculus Summer Assignment
A.P. Calculus Summer Assignment This assignment is due the first day of class at the beginning of the class. It will be graded and counts as your first test grade. This packet contains eight sections and
More information(a) During what time intervals on [0, 4] is the particle traveling to the left?
Chapter 5. (AB/BC, calculator) A particle travels along the -ais for times 0 t 4. The velocity of the particle is given by 5 () sin. At time t = 0, the particle is units to the right of the origin. t /
More informationLimits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
More informationAlgebra 2 Unit 9 (Chapter 9)
Algebra Unit 9 (Chapter 9) 0. Spiral Review Worksheet 0. Find verte, line of symmetry, focus and directri of a parabola. (Section 9.) Worksheet 5. Find the center and radius of a circle. (Section 9.3)
More informationUnit 4: Rules of Differentiation
Unit : Rules of Differentiation DAY TOPIC ASSIGNMENT Power Rule p. Power Rule Again p. Even More Power Rule p. 5 QUIZ 5 Rates of Change p. 6-7 6 Rates of Change p. 8-9 7 QUIZ 8 Product Rule p. 0-9 Quotient
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. The only way to guarantee the eistence of a it is to algebraically prove it.
More informationMath 75B Practice Problems for Midterm II Solutions Ch. 16, 17, 12 (E), , 2.8 (S)
Math 75B Practice Problems for Midterm II Solutions Ch. 6, 7, 2 (E),.-.5, 2.8 (S) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual
More informationFundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.
.5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus.5 COMPLEX NUMBERS 1. Understand that - 1 is an imaginary number denoted by the letter i.. Evaluate the square root
More information