AP Calculus ABAP Student Curriculum Review, Spring 2018

Lisa Hausercmshauser4math@gmail.com

https://sites.google.com/site/mslhauser/

Exam Structure

Exam ScoringMultiple Choice: If you get all the questions correct, the maximum points from the multiple choice section are 45 x 1.20 = 54.

1. Number correct x 1.20 = score for multiple choice.

2. There is no deduction for incorrect answers!!!

Free Response: Six questions, each worth 9 points for a total of 54 points.

3. Part I: 2 questions with graphing calculator. 15 minutes per question.

4. Part II: 4 questions without graphing calculator. 15 minutes per question. Access to all the questions, but without a calculator

Big Ideas

● Limits

● Derivatives

● Integrals

1. Enduring Understandings about Limits1. The concept of a limit can be used to understand the behavior of functions.

2. Continuity is a key property of functions that is defined using limits.

2. Enduring Understanding about Derivatives

1. The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

2. A function’s derivative, which is itself a function, can be used to understand the behavior of the function.

3. The derivative has multiple interpretations and applications including those that involve instantaneous rates of change.

4. The Mean Value Theorem connects the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval.

3. Enduring Understandings about Integrals1. Antidifferentiation is the inverse process of differentiation.

2. The definite integral of a function over an interval is the limit of the Reimann sum over that interval and can be calculated using a variety of strategies.

3. The Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration.

4. The definite integral of a function over an interval is a mathematical tool with many interpretations and applications involving accumulation.

5. Antidifferentiation is an underlying concept involved in solving separable differential equations. Solving separable differential equations involves determining a function or relation given its rate of change.

3.1 Antidifferentiation is the Inverse of DifferentiationAn antiderivative of a function f is a function g whose derivative is f.

3.1 Antidifferentiation is the Inverse of DifferentiationThe rules of differentiation provide the foundation for finding antiderivatives.

Example 11.

Example 22.

3.2 Definite Integral as the Limit of a Reimann SumFinding area under a curve using rectangles.

Visualize the rectangles

As the number of rectangles increases, the approximate area under the curve gets closer to the actual area under the curve.

Finding Reimann Sums Area of a Rectangle: f(x)*�x Area of a Trapezoid: ½*�x*(f(x1) + f(x2))

Left: �x*(f(x1) + f(x2)+ f(x3) + …+ f(xn))

Right: �x*(f(x0) + f(x1)+ f(x2) + …+ f(xn-1))

Trapezoidal Approximation: ½*�x*(f(x0) + 2f(x1) + 2f(x2) + …+ 2 f(xn-1) + f(xn))

IMPORTANT: These formulas only apply when the subintervals are of equal length. When subintervals are of different length, each individual area must be calculated first, and then all the areas are added up.

Example 11.

Example 22.

3.2 Definite Integral as the Limit of a Reimann SumHence,

We now define the most general situation as follows:If f is continuous on [a,b], and:

1. The interval [a,b] is divided into n sub-intervals of equal width △x, with △x=(b−a)/n, and

2. The endpoints of these sub-intervals are x0=a,x1,x2,...,xn=b, and3. x1*,x2*,…,xn* are any sample points in these sub-intervals, then the definite

integral of from x=a to x=b is

provided the limit exists.

Example 11

Example 22.

3.2 Definite Integral Evaluated Using Geometry***Area is always POSITIVE.***

BUT, when evaluating a definite integral using geometry, area above the x-axis is considered POSITIVE while area below the x-axis is considered NEGATIVE.

Break the integral into smaller pieces that can be calculated using geometry and then add the “areas.”

Use properties of integrals

Example 11.

3.2 Properties of Definite Integrals

Know these

lower limit < upper limit

Example1.

Example 22.

3.3 Functions Defined by Integrals(14_04)

3.3 The Fundamental Theorem of Calculus (FTC) Part I. This is a BIG deal!!!

We’ve connected INTEGRALS and DERIVATIVES!

***Remember to use the CHAIN RULE and to USE A NEGATIVE when the lower bound has x.***

Examples1.

Examples 2. Find F’(x)

3.3 Fundamental Theorem of Calculus - Part 2This is how we are going to evaluate definite integrals!!

Example 2.

Example3.

3.3 Techniques of IntegrationU-Substitution.

1. Let u = g(x). 2. Find du/dx. 3. Solve for du. 4. If evaluating a definite integral, find upper and lower limits of integration

in terms of u. 5. Substitute. 6. Solve integral. I7. f finding an indefinite integral, substitute g(x) for u and remember “+C.”

Examples - Definite Integral 1.

2.

Examples - Indefinite Integral1.

“+C” for ALL INDEFINITE INTEGRALS!

Example 1Algebraic Manipulation - Division

3.4 Accumulation● A function defined as an integral represents an accumulation of a rate of

change.● The definite integral of the rate of a change of a quantity over an interval gives

the net change of that quantity over that interval.

Example 1 1.

Example 213_01

Scoring Guideline13_01

Example 310B_03

Scoring Guidelines 10B_03

3.4 Average ValueDemonstration

Average Value Average height of the graph of a function.

For y = f(x) over the domain [a,b], the formula for average value is given by

Mean Value Theorem of Integrals

Example (Average Rate of Change ≠ Average Value) 2.

Example 22.

Rectilinear MotionDerivatives: Position -> Velocity -> Acceleration

Integrals: Acceleration -> Velocity -> Position

Example 11.

Example 213_02

Scoring Guidelines13_02

Displacement vs. Total DistanceDisplacement: Distance from original position.

Total Distance: All areas are considered positive.

ExampleMultiple Choice

Example 112_06

Scoring Guidelines12_06

Example 22.

3.4 Area and Volume: Area Between Two Curves● Sketch the graphs● Find the points of intersection (where the graphs meet) to determine the

endpoints of integration.● Divide the area into vertical or horizontal strips● Integrate.

Example1. 05_01

Scoring Guidelines05_01

Volume by RevolutionDisc Method - NO SPACE between CURVE and AXIS of REVOLUTION

Wolfram Demonstration Project…

REMEMBER π !!!

***r = (top - bottom) or (right - left)***

ExampleMultiple Choice

Volumes by RevolutionWasher Method: GAP between CURVE and AXIS of REVOLUTION

R = Draw a line from the axis to the outer edge of the solid. Use (top-bottom) or (right-left) to find the length.

r = Draw a line from the axis to the inner edge of the solid. Use Use (top-bottom) or (right-left) to find the length.

Example06_01

Scoring Guidelines06_01

Volumes of Solids with Known Cross-SectionsVolume = Accumulation of the Volume of Little Prisms = Accumulation of Area•∆x

Example07_01

Scoring Guidelines07_01

ExampleArea and Volume

all together…

Scoring Guidelines

Separable Differential Equations● To solve a separable differential equation:

● Separate variables through multiplication and/or division ONLY.

● Integrate both side.

● Remember the constant C which only needs to be placed on one side.

● Finding a particular solution involves finding “C” with the given conditions.

● Solve for y.

ExampleMultiple Choice

Example10_06

Scoring Guidelines06_10

Newton’s Law of CoolingFormula:

Newton’s Law of Cooling - Example1. Something like it….

Slope Fields - “Slopitos”● Slope fields present the general solution of a differential solution (with all the

C’s).

● A particular solution can be graphed by tracing the values on the slope field.

● Evaluate dy/dx at different points. Pay close attention to when dy/dx = 0 (horizontal tangent lines) and when dy/dx is undefined (vertical tangent lines).

● Consider the sign of x and y in each quadrant.

● Consider whether dy/dx is only in terms of x or only in terms of y.

ExampleMultiple

Choice

Example08_05

Scoring Guidelines08_05

Differential Equations - ProportionalCan you solve the differential equation in this question?

Test Taking Do’s and Don’ts

Do’s and Don’ts● Constructing exponential models according to rate of change

Never round intermediate steps

Set calculator toRADIANS!!!

Do’s and Don’ts

Testing TipsYou’ve taken tests before, but remember:

● Attempt as many questions as possible in the given time. Do not spend too much time on a single question. Hunt for the questions you know how to answer.

● Make sure that you record multiple choice questions properly on the answer sheet. Leave no questions blank!

● Bring your graphing calculator with fresh batteries or a fresh charge. ● Bring a sweater, extra pencils and a good eraser.