**Remember, if it’s pink it’s a link!**

## #1: No Put Downs!

## #2: No Sleeping!

## #3: Technology is a TOOL, not a TOY!

**If you have not done so already, please sign up for the Remind alert system.**

**Text number: 81010**

**Text message: @apcalcuu**

**Grade Weighting: **

**Quizzes: 20%**

**Tests: 50%**

**Activities & Projects: 10%**

**Final Exam: 20%**

**Supplies Needed:**

### Paper and pencil – whichever type works best for you.

**Graphing calculators (TI Nspire CAS) will be provided in class; however, will not be sent home.**

## **Cheat Sheets**

# Click on the link below for detailed standards for AP Calculus AB & BC:

# AP Calculus Big Ideas Frameworks

**The following materials are provided to assist you in being successful in AP Calculus AB/BC:**

**old tests, keys to old tests, and videos**

**Prerequisite Skills**

**Pacing**: 1 instructional day, 1 assessment day

Study Guide for Prerequisite Skills Test

Solutions to Study Guide for Prerequisite Skills Test

**See the review of Pre-Calculus video below.**

Properties of Logarithms Video

Simplifying Trigonometric Expressions Video

Solving Logarithmic and Exponential Inequalities Video

**Limits**

**Pacing: **10 instructional days, 1 assessment day

**Many calculus concepts are developed by first considering a discrete model and then the consequences of a limiting case. Therefore, the idea of limits is essential for discovering and developing important ideas, definitions, formulas, and theorems in calculus. Students must have a solid, intuitive understanding of limits and be able to compute various limits, including one-sided limits, limits at infinity, the limit of a sequence, and infinite limits. They should be able to work with tables and graphs in order to estimate the limit of a function at a point. Students should know the algebraic properties of limits and techniques for finding limits of indeterminate forms, and they should be able to apply limits to understand the behavior of a function near a point. Students must also understand how limits are used to determine continuity, a fundamental property of functions.**

** **

**Content Vocabulary:**

**Limit** – The value that a function or expression approaches as the domain variable(s) approach a specific value. Limits are written in the form

**Continuous Function** – A function with a connected graph.

**Piecewise Continuous Function** – A function made up of a finite number of continuous pieces. Piecewise continuous functions may not have vertical asymptotes. In fact, the only possible types of discontinuities for a piecewise continuous function are removable and step discontinuities.

**Intermediate Value Theorem (IVT)** – A theorem verifying that the graph of a continuous function is connected.

**Discontinuous Function** – A function with a graph that is not connected.

**Removable Discontinuity** – A hole in a graph. That is, a discontinuity that can be “repaired” by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can be made connected by filling in a single point. Formally, a removable discontinuity is one at which the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point.

**Essential Discontinuity** – Any discontinuity that is not removable. That is, a place where a graph is not connected and cannot be made connected simply by filling in a single point. Step discontinuities and vertical asymptotes are two types of essential discontinuities. Formally, an essential discontinuity is a discontinuity at which the limit of the function does not exist.

**One-Sided Limit** – Either a limit from the left or a limit from the right.

**Asymptote** – A line or curve that the graph of a relation approaches more and more closely the further the graph is followed. Note: Sometimes a graph will cross a horizontal asymptote or an oblique asymptote. The graph of a function, however, will never cross a vertical asymptote.

**Infinite Limit** – A limit that has an infinite result (either ∞ or –∞ ), or a limit taken as the variable approaches ∞ (infinity) or –∞ (minus infinity). The limit can be one-sided.

**Textbook Suggested Problems & Solutions**

**I recommend that you look at the problems that are suggested and determine which ones YOU need to work. It is not expected that every student will need to work every problem or that every student will need to work the same problems. When you have chosen the problems you need to work, attempt the problems, check your solutions using the links provided below, and bring your questions with you to class.**

Section | Page(s) | Suggested Problem Numbers |

2.2 | 74 – 76 | 1 – 27 , 38, & 47 – 53 |

2.3 | 80 | 1 – 22 & 27 – 30 |

2.4 | 88 – 89 | 1 – 6 , 17 – 19 , 57 – 60 , & 67 – 78 |

2.5 | 94 – 95 | 1 – 28 , 30 , 34, 37 – 41 |

2.6 | 99 | 17 – 48 |

2.7 | 105 | 7 – 26 |

2.8 | 109 | 1 – 4 |

Ch Review | 117 – 119 | 5 – 26 , 28 , 29 , 31 – 34 , 51 – 54 , 57 , 58 – 60 , 64, 68 , & 72 |

**Video Reources**

Finding Limits With Graphs Video

Finding Limits by Factoring Video

Limits Review Video (Mixed Methods)

## Old Tests

Limits 2009-2010 Test with Solutions

2010-2011 Limits Test #1 Video

2010-2011 Limits Test #2 Video

2010-2011 Limits Test #’s 3-4 Video

2010-2011 Limits Test #’s 5-9 Video

2010-2011 Limits Test #’s 10-13 Video

2010-2011 Limits Test #’s 14 & 15 Video

Review of Limits Techniques Video

2013-2014 Limits Test w Solutions

2014-2015 Limits Test w: solutions

2015-2016 Limits Test Solutions and Scoring

**The Derivative**

**Pacing: **25 instructional days, 2 assessment days

**Using derivatives to describe the rate of change of one variable with respect to another variable allows students to understand change in a variety of contexts. In AP Calculus, students build the derivative using the concept of limits and use the derivative primarily to compute the instantaneous rate of change of a function. Applications of the derivative include finding the slope of a tangent line to a graph.**

## Content Vocabulary

**Derivative** – A function which gives the slope of a curve; that is, the slope of the line tangent to a function. The derivative of a function *f* at a point *x* is commonly written *f *‘(*x*).

**Slope of a curve** – A number which is used to indicate the steepness of a curve at a particular point. The slope of a curve at a point is defined to be the slope of the tangent line. Thus the slope of a curve at a point is found using the derivative.

**Tangent line** – A line that touches a curve at a point without crossing over. Formally, it is a line which intersects a differentiable curve at a point where the slope of the curve equals the slope of the line.

**Difference quotient** – For a function *f*, the formula . This formula computes the slope of the secant line through two points on the graph of *f*. These are the points with *x*-coordinates *x* and *x* + *h*. The difference quotient is used in the definition the derivative.

**Differentiable** – A curve that is smooth and contains no discontinuities or cusps. Formally, a curve is differentiable at all values of the domain variable(s) for which the derivative exists.

**Instantaneous Rate of Change** – The rate of change at a particular moment. Same as the value of the derivative at a particular point. For a function, the instantaneous rate of change at a point is the same as the slope of the tangent line. That is, it’s the slope of a curve. Note: Over short intervals of time, the average rate of change is approximately equal to the instantaneous rate of change.

**Implicit Differentiation** – A method of finding the derivative of a function or relation in which the dependent variable is not isolated on one side of the equation. For example, the equation *x*² + *xy* – *y*² = 1 represents an implicit relation.

**Logarithmic Differentiation** – A method for finding the derivative of functions such as *y* = *x*sin *x* and , where the derivative is cumbersome or impossible with direct differentiation techniques.

**Textbook Suggested Problems & Solutions**

**I recommend that you look at the problems that are suggested and determine which ones YOU need to work. It is not expected that every student will need to work every problem or that every student will need to work the same problems. When you have chosen the problems you need to work, attempt the problems, check your solutions using the links provided below, and bring your questions with you to class.**

Section | Page(s) | Suggested Problem NumbersDisregard any directions to find the equation of the tangent line! |

3.1 | 126 | 27 – 38 |

3.2 | 139 | 1 – 19 , & 21 – 36 |

3.3 | 147 – 148 | 1 – 45 |

3.5 | 163 | 1 – 26 |

3.6 | 167 | 5 – 24 |

3.7 | 175 | 11 – 22 , & 29 – 49 |

3.8 | 181 | 3 – 10 , & 23 – 28 |

3.9 | 187 – 188 | 1 – 24 , & 37 – 50 |

3.10 | 192 | 9 – 20 |

Ch Review | 203 – 205 | 5 – 12 , 25 , 28 – 42 , 45 – 52 , 55 – 68 , 73 – 75 , 82, 85 – 90 , 95 , 101 – 106 , & 109 – 111 |

Ch 3 Review Exercises Solutions

## Video Resources

Definition of the Derivative Video

Definition of the Derivative Video #2

Definition of the Derivative Homework Solutions Video

Product and Quotient Rule Video

Derivatives of Trig Functions Video

Higher Order Derivatives Video

Implicit Differentiation Video

**Old Tests**

**Combined Derivative Tests**

2013-2014 Derivative Test w Solutions (On #3, I left off my 15 exponent on the last line… it’s on the line above)

2013-2014 Derivative Test #’s 1 – 3 ShowMe Video

2013-2014 Derivative Test #’s 4 – 8 ShowMe Video

2013 – 2014 Derivative Test #’s 9 – 14 ShowMe Video

2013 – 2014 Derivative Test #’s 15 – Bonus ShowMe Video

**Derivative Test Part 1**

2006-2007 Der Part 1 With Solutions

2007-2008 Der Part 1 With Solutions

2008-2009 Der Part 1 With Solutions

Der Test 1 2009-2010 TEST ONLY

2010-2011 Der Part 1 With Solutions

2010-2011B Der Part 1 With Solutions

2014-2015 Derivative Test 1 Solutions The clean copy is not available, it died in computer crash!

2016-2017-derivative-test-1-solutions-rubric

**Derivative Test Part 2**

Der 2 Test 2006-2007 – Test Only

2007-2008 Derivative 2 Test with Key

2008-2009 Derivatives Test 2 with Key

2009-2010 Der Part 2 With Solutions

2010-2011 Der Part 2 With Solutions

2010-2011B Der Part 2 With Solutions

2012-2013 Der Part 2 With Solutions

2014-2015 Derivative Test 2 Solutions The clean copy is not available, it died in computer crash!

2015-2016 Derivative Test Part 2

2015-2016 Derivative Test 2 Scoring Rubric

2016 – 2017 Derivatives Test Part 2 with Rubric

**Applications of the Derivative**

**Pacing: **25 instructional days, 2 assessment days

**Applications of the derivative include finding the slope of a tangent line to a graph at a point, analyzing the graph of a function (for example, determining whether**** ****a function is increasing or decreasing and finding concavity and extreme values), and solving problems involving rectilinear motion. Students should be able to use different definitions of the derivative, estimate derivatives from tables and graphs, and apply various derivative rules and properties. In addition, students should be able to solve separable differential equations, understand and be able to apply the Mean Value Theorem, and be familiar with a variety of real-world applications, including related rates, optimization, and growth and decay models.**

## Content Vocabulary

**Average rate of change** – The change in the value of a quantity divided by the elapsed time. For a function, this is the change in the *y*-value divided by the change in the *x*-value for two distinct points on the graph. Note: This is the same thing as the slope of the secant line that passes through the two points.

**Instantaneous rate of change** – The rate of change at a particular moment. Same as the value of the derivative at a particular point. For a function, the instantaneous rate of change at a point is the same as the slope of the tangent line. That is, it’s the slope of a curve. Note: Over short intervals of time, the average rate of change is approximately equal to the instantaneous rate of change.

**Related rates** – A class of problems in which rates of change are related by means of differentiation. Standard examples include water dripping from a cone-shaped tank and a man’s shadow lengthening as he walks away from a street lamp.

**Projectile Motion** – A formula used to model the vertical motion of an object that is dropped, thrown straight up, or thrown straight down.

**Optimization** – the process of finding the greatest or least value of a function for some constraint, which must be true regardless of the solution. In other words, optimization finds the most suitable value for a function within a given domain.

**Increasing Function** – A function with a graph that goes up as it is followed from left to right. For example, any line with a positive slope is increasing. Note: If a function is differentiable, then it is increasing at all points where its derivative is positive.

**Decreasing Function** – A function with a graph that moves downward as it is followed from left to right. For example, any line with a negative slope is decreasing. Note: If a function is differentiable, then it is decreasing at all points where its derivative is negative.

**Extrema** – An extreme value of a function. In other words, the minima and maxima of a function. Extrema may be either relative (local) or absolute (global). Note: The first derivative test and the second derivative test are common methods used to find extrema.

**Absolute Maximum** – The highest point over the entire domain of a function or relation.

**Relative Maximum** – The highest point in a particular section of a graph.

**Absolute Minimum** – The lowest point over the entire domain of a function or relation.

**Relative Minimum** – The lowest point in a particular section of a graph.

**Concave** – A shape or solid which has an indentation or “cave”. Formally, a geometric figure is concave if there is at least one line segment connecting interior points which passes outside of the figure.

**Concave up** – A graph or part of a graph which looks like a right-side up bowl or part of an right-side up bowl.

**Concave down** – A graph or part of a graph which looks like an upside-down bowl or part of an upside-down bowl.

**Point of inflection** – A point at which a curve changes from concave up to concave down, or vice-versa. Note: If a function has a second derivative, the value of the second derivative is either 0 or undefined at each of that function’s inflection points.

**Mean Value Theorem (MVT) –** A major theorem of calculus that relates values of a function to a value of its derivative. Essentially the theorem states that for a “nice” function, there is a tangent line which is parallel to any secant line.

**L’Hopital’s Rule** – A technique used to evaluate limits of fractions that evaluate to the indeterminate expressions and . This is done by finding the limit of the derivatives of the numerator and denominator. Note: Most limits involving other indeterminate expressions can be manipulated into fraction form so that L’Hôpital’s rule can be used.

**Rolle’s Theorem** – A theorem of calculus that ensures the existence of a critical point between any two points on a “nice” function that have the same *y*-value.

**Acceleration** – The rate of change of velocity over time. For motion along the number line, acceleration is a scalar. For motion on a plane or through space, acceleration is a vector.

**Critical Point** – A point (*x*, *y*) on the graph of a function at which the derivative is either 0 or undefined. A critical point will often be a minimum or maximum, but it may be neither. Note: Finding critical points is an important step in the process of curve sketching.

**Cusp** – A sharp point on a curve. Note: Cusps are points at which functions and relations are not differentiable.

**Extreme Value Theorem** – A theorem which guarantees the existence of an absolute max and an absolute min for any continuous function over a closed interval.

**Speed** – Distance covered per unit of time. Speed is a nonnegative scalar. For motion in one dimension, such as on a number line, speed is the absolute value of velocity. For motion in two or three dimensions, speed is the magnitude of the velocity vector.

**Below is a motion simulation. It requires Java.**

**Below is a derivative graph simulation. It requires Java.**

**Textbook Suggested Problems & Solutions**

**I recommend that you look at the problems that are suggested and determine which ones YOU need to work. It is not expected that every student will need to work every problem or that every student will need to work the same problems. When you have chosen the problems you need to work, attempt the problems, check your solutions using the links provided below, and bring your questions with you to class.**

**Old Tests**

2005 app der test w: solutions

2005-2006 app der test w: solutions

2007-2008 app der test w solutions

2008-2009 app der test w solutions

2012-2013Applications of the Derivative Test Solutions

2013-2014 App Deriv Test w: Solutions

**2015-2016 Applications of the Derivative Test**

**2015-2016 App of Deriv Scoring Rubric**

**2016-2017 Applications of the Derivative Test with Scoring Rubric**

**The Integral**

**Textbook Suggested Problems & Solutions**

**Old Tests**

2003-2004 Int Test 1 Rev with Solutions – Skip #14 (this question has been moved to applications of integration)

2003-2004 int test 1 with solutions

2005-2006 int test 1 with solutions

2005-2006 int test 2 with solutions

2006-2007 int test 1 with solutions

2007-2008IntTest1WithSolutions – Skip #1 and #6 (these questions have been moved to applications of integration)

2007-2008 int test 2 with solutions

2009-2010IntTest1WithSolutions

2010-2011 int test 1 with solutions – Skip #11 part a (this question has been moved to applications of integration)

Int2Test2010-2011 (without solutions)

2011-2012IntegrationTest2WithSolutions

2012-2013 int test 1 with solutions

Int Test 1 Extra Practice with Solutions

2013-2014IntTest1WithSolutions

2014-2015 Integration Test 1 with Answers

2015-2016 Integration Test 1 Scoring Rubric

2016-2017 Integration Test 1 Rubric

**Applications of Integration**

**Old Tests**

2010-2011AppIntTestWithRubrics

2016-2017 Applications of Integration Test

2016-2017 Applications of Integration Test Solutions & Scoring Rubric

# AP Prep / Final Exam Review (For Calc AB)

McGrawll Hill 5 Steps to a 5 Online Practice

Answers to APEX 1, APEX 2, & PR 1

**Integration Techniques**

** Below is a vector addition simulation. It requires Java.**