WHAT OUR COLLEAGUES SAY:
..The Barron's book is good but I think the book PREPARING FOR THE AP CALCULUS (BC) EXAMINATION by George Best and Richard Lux is even better. It contains six full length examinations with lots of questions where the functions are defined by graphs or numerical data values..
- Helen Schneider, Renton, WA
Are you looking for a calculus text that supports the use of technology and the ideas of calculus reform? [One of the approved texts for the AP Audit, this best selling book focuses
on conceptual understanding and thinking skills.]
Our AP exam prep materials are updated annually to prepare your students for the current exam
Nov 16, 2020 Past Common and Final Exams To submit an update to the information below, please email Alison Boldero at aboldero@njit.edu. Past Common and Final Exams. Exam Start Times: Mountain Time: 12:00 pm. Mountain time 2:00 pm. United States Government and Politics. Tues, May 12: Calculus AB. Wednesday, May 13. Updated 4 September 2020. The 2021 AP Exams test dates are from May May 3-7 & May 11-14. The chart below shows test times for the CENTRAL DAYLIGHT TIME zone. Visit the AP websites 2021 AP Exam Schedule to see test times in other time zones. COURSE SPECIFIC INFO: Visit the AP website for course specific information.
NOW AVAILABLE - Teacher's Guide with CD to accompany the first and second editions of Calculus: Concepts and Calculators
- Teacher notes for each section
- Chapter review and test questions
- Multiple choice and free response questions for each chapter
- MAC or Windows CD that includes all chapter review/test questions and a sample AP (AB) and (BC) examination
The Trigonometry Workbook is a full text for a one semester course in trigonometry. The book is specifically designed to utilize technology including the TI-83 calculator, and The Geometer's Sketchpad.
A Brief Introduction to Logic, Digital Circuits and Sets
Calculus Concepts & Calculators
(Approved text for AP Audit)
This best selling book focuses on conceptual understanding and thinking skills. The graphing calculator is an integral part of the text and allows students to manipulate and compare graphical, numerical and algebraic representations of the ideas. Instructor's Manual and Solution Manual are available too.
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Preparing for the 2021 AP Calculus (AB) and (BC) Examinations
Exam Updated Information Ap Calculus Syllabus
The AB and BC preparation books have been updated to reflect the recent changes prescribed by the College Board. New topics have been tested; multiple-choice questions have four answer choices.
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Try sample questions B/C
1001 tracks.
PreCalculus Transformed Extending Transformations to Explore PreCalculus
Pre-Calculus Transformed highlights the under-explored role of transformations in visualizing, interpreting, and understanding pre-calculus concepts.
Click here for more information explorations in Calculus
Explorations in Calculus has been written as an enrichment to an AP Calculus course. Graphs are a very important component of calculus and are crucial in helping students visualize and gain insight into traditional calculus topics.
AP Calculus with the TI-89 Graphing Calculator
One of the most powerful and sophisticated calculators available, the TI-89 offers students and teachers the opportunity to bring graphical, numerical, symbolic processing into the teaching and learning of calculus. This book is designed to support the effective and appropriate use of the TI-89 in an AP calculus course.
Introduction to Fractals and Chaos 2nd Edition
Fractals and Chaos is an exploration of discrete dynamical systems, using primarily the concepts of precalculus mathematics and the technology of the TI-83//84 graphing calculator.
OUR NEWEST ADDITIONS
Chapter Tests andReview Exercises
by George Best
This new book is intended to be used as a resource in an introductory calculus course. A rich collection of review exercises and chapter tests, the book is designed to supplement any calculus text that is aligned with the College Board’s AP Course Description for AB and BC calculus .
Explorations in Calculus
Exporations in Calculus has been written as an enrichment to an AP Calculus course. Graphs are a very important componet of calculus and are crucial in helping students visualize and gain insight in traditonal calculus topics.
Exploring Algebra, Precalculus and Statistics with the TI-Nspire CAS
The TI-Nspire CAS is a document oriented machine that packs a lot of power. It has a learning curve, given the sophisticated keyboard and the number of features it supports. This book has been written to support the learning of mathematical concepts.
Calculus I Examples For Exam II
| If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page. | QuickTime free download. |
Here is a Geometer's Sketchpad video with audio (Larger Version) of me describing approximating the slope of a tangent line with a secant line.
Here is a Quicktime Animation showing a secant line approximating a tangent line.
Here is a Winplot demonstration of a secant line that can be used to approach a tangent line. You may need to download the file to your desktop and then use the freeware Winplot to open the file (by opening Winplot, clicking on Window, clicking on 2-dim, clicking on File, clicking on Open, and then opening SecantSlope from your desktop. You can use the sliders to vary the values of A and H. The A-value gives you the x-value at a point on the graph of the function you are looking at. The H-value gives you the x-value of another point on the graph of the function you are looking at in terms of A + H (so H is your delta x). The secant line through the points on the graph of the given function with x-values of A and A + H is graphed and its slope displayed at the top of the graph. The default function is f(x) = 3sin(x). This function is named FN and you can edit the definition of FN by clicking on Equa, choosing User functions .. , highlighting the function and editing its definition. Changing the definition to x^2 for example will produce a different graph.
Computing the derivative using the limit definition of derivative
| Derivative Approximation The following points are the points pictured on the graph of the function at the right: (-0.5,10.875), (-0.4,10.816), (-0.3,10.693), (-0.2,10.512), (-0.1,10.279), (0,10), (0.1,9.681),(0.2,9.328), (0.3,8.947), (0.4,8.544), (0.5,8.125). Approximate the derivative of the function at the points (-0.3,10.693), (0,10), and (0.4,8.544). |
Finding the Equation of a Tangent Line
Find an equation of the line tangent to the graph of the function f(x) = x3 - 2x2 - 3x + 10 at the point (-1,10). Click on the picture at the right to see a Quicktime animation of tangent lines as x goes from -2 to 2.4. |
| Differentiation Example--Product Rule (and Chain Rule) | What would the graph look like for x greater than 4? |
| Differentiation Example--Quotient Rule | What do you think the graph looks like outside the viewing window shown above? |
| Differentiation Example--Chain Rule (Twice) | Do you think a graphing calculator could draw an accurate graph of this function over an x-interval of [0,10]? |
Use implicit differentiation to find the equations of the tangent and normal lines to the graph of the relation whose equation is given below at the point (4,2).
| In the graph below the tangent line is drawn in blue and the normal line is drawn in red. |
Use implicit differentiation to find an equation of the line tangent to the graph of the given relation at the indicated point.
Click on the picture to enlarge. |
Three more implicit differentiation examples
DPGraph picture of z = xcos(y) + 2y - y2 and z = c. When c = 0 the intersection of the two graphs will be the graph of this relation. |
DPGraph picture of z = x2cos(y) + 2xy - xy2 and z = c. When c = 0 the intersection of the two graphs will be the graph of this relation. Compare the graph on the right to the graph in example 1 and note the difficulty atx = 0. This graph was constructed using Maple. Can you tell what the problem is around the y-axis? Click on the graph for a larger Maple image. |
DPGraph picture of z = 2xy - y2 + 3 + 3xy3 - cos(xy2) and z = c. When c = 0 the intersection of the two graphs will be the graph of this relation. |
| Quicktime AnimationAverage and Instantaneous SpeedAnimation--No Scales |
Click on the picture to see an animation. |
| Just as she is entering a 20 mph speed zone, Emily notices a police car up ahead parked a little way off the road. Emily applies the brakes gently to begin slowing down (she does not want to be too obvious). She first hits the brakes when the front of her car is at position zero in the picture above and she slows down for three seconds with a constant negative acceleration. She travels a total of 111 feet during these three seconds and at the end of the three seconds the front of her car is level with the radar gun being used to clock her speed. It turns out that the radar began clocking her motion at the instant she first started to slow down. The tables below give distance traveled in feet and time in seconds from the time when she first applied the brakes. |
Time 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7
1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 | Distance 0.00 3.99 7.96 11.91 15.84 19.75 23.64 27.51 31.36 35.19 39.00 42.79 46.56 50.31 54.04 57.75 61.44 65.11 68.76 72.39 76.00 79.59 83.16 86.71 90.24 93.75 97.24 100.71 104.16 107.59 111.00 | Time 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 . . . . 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 | Distance 0.0000 0.3999 0.7996 1.1991 1.5984 1.9975 2.3964 2.7951 3.1936 3.5919 3.9900 . . . . . . 107.5900 107.9319 108.2736 108.6151 108.9564 109.2975 109.6384 109.9791 110.3196 110.6599 111.0000 |
| What was Emily's average speed in feet per second and in miles per hour from time t = 0 seconds to time t = 3 seconds? What would you estimate to have been Emily's speed in feet per second and in miles per hour at time t = 0 seconds? What would you estimate to have been Emily's speed in feet per second and in miles per hour at time t = 3 seconds? Estimate Emily's speed in feet per second and in miles per hour at time t = 2 seconds. I will call the position function s(t). Answers Average speed from t = 0 to t = 3 seconds: Speed at t = 0 seconds: That guess turns out to be pretty good since the radar, measuring to an accuracy of 7 digits to the right of the decimal, indicated that s(0.0000001) = 0.000004. Speed at t = 3 seconds: This guess also seems to be pretty good since the radar, measuring to an accuracy of 7 digits to the right of the decimal, indicated that s(2.9999999) = 110.9999966. Can you see why this makes the 34 ft/sec estimate of the speed at t = 3 seconds seem pretty accurate? Speed at t = 2 seconds: 3 tips to maximize your weebly websites. Since we have less precise data around t = 2 seconds it would probably be more accurate to look at s(2.1) and s(1.9). Do you think Emily got pulled over and if so do you think she got a ticket or a warning? Do we have enough information to make an educated guess at the answer to that question? The idea for this example was inspired by a terrific presentation on speed from Calculus Quest. Notice that in this example the average speed over the three second time interval is equal to the average of the estimated instantaneous speeds at the beginning and end of the time interval. Would this always be the case? Would it be possible for someone to be traveling at 40 ft/sec at time t = 0 seconds, 34 ft/sec at time t = 3 seconds, and yet the average speed over this 3 second interval be 38 ft/sec or even be 42 ft/sec? EC Find an equation representation of the position function, s(t), that would nicely fit the data given above. |
| Average and Instantaneous Speed The position function shown below gives the position of an object in feet as a function of time in seconds. Determine when the object is traveling to the right (positive direction) and when the object is traveling to the left (negative direction). Determine the average speed from t = 1 to t = 5. Determine the instantaneous speed at t = 1 and at t = 5. | The graph below shows position (not distance traveled) as a function of time. Click here or on the picture to see a linear motion animation. In the linear motion animation the animated point on the left vertical axis represents the elapsed time. Quicktime animation |
A Vertically Launched Projectile
Quicktime Animation (In the animation the point traveling down the vertical axis indicates the speed of the projectile.)
The function above (called a position function) gives the height above the ground (in feet) of a projectile launched vertically (straight up) with an initial velocity of 192 ft/sec and an initial height of 100 feet above the ground (air resistance is being ignored here). The variable 't' represents time in seconds with t = 0 corresponding to the moment the projectile was launched. Here are the types of questions you need to be able to answer.
(A) What is the average speed of the projectile during the first 3 seconds after launch?
(B) What is the average speed of the projectile from t = 1 second to t = 4 seconds?
(C) What is the instantaneous speed of the projectile at t = 3 seconds?
(D) What will be the maximum height attained by the projectile?
(E) How long until the projectile comes back down and hits the ground?
Solutions
(A) We need to find the total distance traveled during the first three seconds and divide this total distance traveled by the time interval (3 seconds). This means we need to compute
(B)
Exam Updated Informationap Calculus 1
(C) The derivative of the position function will give us the instantaneous speed. Thus we need to evaluate the derivative at t = 3 seconds (i.e., substitute 3 for t).
(D) At some point in time the projectile is going to stop going up and start coming back down. At the instant it stops going up its speed will be zero ft/sec. When it starts coming back down its speed (velocity in the sense that direction is now being considered) will be negative. At the instant when the speed is zero the projectile will be at its highest point (maximum height). We need to find the value of time 't' when the derivative (which gives velocity) is zero. We can then evaluate the position function (which gives height above the ground) at that value of t to find the maximum height.
(E) The projectile will hit the ground when the value of the position function is zero.
The t-value of -0.5 would correspond to before the projectile was launched and is not in the domain of our position function describing this motion. The answer must come from the positive solution to the equation above. Thus the projectile comes back down and hits the ground in 12.5 seconds.
| Section 2.6 Number 2a |
| In the picture above the blue point is the point where x = 3. Click on the picture (Quicktime version) to see an animation of a point moving along the graph of the parabola as a function of time (t) where |