Section I Multiple Choice 45 questions. Section II Free Response 6 questions

Transcription

1

2 Section I Multiple Choice 45 questions Each question = 1.2 points, 54 points total Part A: No calculator allowed 30 questions in 60 minutes = 2 minutes per question Part B: Calculator allowed 15 questions in 45 minutes = 3 minutes per question Section II Free Response 6 questions Each question = 9 points, 54 points total Part A: Graphing calculator is REQUIRED 2 questions in 30 minutes = 15 minutes per question Part B: No calculator allowed 4 questions in 60 minutes = 15 minutes per question You are allowed to return to Part A but without a calculator

3 It depends on you goal passing is a 3 3 = Qualified to receive college credit 4 = Well qualified to receive college credit 5 = Extremely well qualified to receive credit For AP Calculus AB (it varies) points = 3 (sometimes it starts at 40) points = points = 5

4 Differentiation Integration AP Calculus AB

5 Derivatives and tangent lines Implicit Differentiation Optimization Rates (including Physics)

6 Antiderivatives Area under the curve Integration Area / Volume Differential equations

7 Limits Theorems

8

9 Derivative = slope of the tangent line at any point Tangent lines are used to approximate a function value You must know the differentiation rules!!!

10

11

12

13 Implicit Differentiation Differentiate with respect of x Remember to include the dy/dx whenever a y is being differentiated. Move the terms with a dy/dx to the left side of the equation, all others to the right. Factor out the dy/dx (GCF) Solve for dy/dx When finding the second derivative, substitute dy/dx with the expression in x and y.

14 When differentiating in terms of time (t), the derivative becomes the rate of change of the independent variable. Physics One dimensional motion only 3 quantities: position, velocity and acceleration Velocity = rate of change of the position (derivative) Acceleration = rate of change of the velocity (derivative) To go backwards, integrate! Related Rates Rates of change of 2 or more related variables that are changing with respect of time. The chain rule and implicit differentiation are used. Formulas for volume are given. Only 2 variables should be in the formula before differentiating.

15

16 To find the maximum or minimum value First derivative: f (x) Critical points: f (x) = 0 or undefined f is increasing: f (x) > 0 f is decreasing: f (x) < 0 Relative maximum at c: f (x) changes from + to Relative minimum at c: f (x) changes from to + Second derivative: f (x) Points of inflection: f (x) = 0 or undefined AND changes signs. f is concave UP: f (x) > 0 f is concave DOWN: f (x) < 0 Second derivative test: For a critical point c Relative maximum: f (c) < 0 Relative minimum: f (c) > 0 For absolute extrema, check the endpoints!!!

17

18 Question #11 Question #12

19

20

21

22 You must know all the rules!! You must know how to use u-substitution Remember differentiation and integration are inverses (they undo each other)

23

24

25 The definite integral represents the net change Geometry can be used to evaluate a definite integral. Riemann sums and Trapezoids can be used to estimate a definite integral.

26

27

28 Area between two curves = top bottomfunction (where a and b are the x values of the intersection points) Volume = integrate the area of the cross-section Most likely a Free Response Question b a

29 An equation consisting of dy/dx, x and/or y Slope field: the graph of a differential equation consisting of small line segments. To solve: use separation of variables Often a Free Response Separate the variables using Solving a differential equation: Separate the variables using multiplication and/or division only Integrate both sides Add +C to the x side only Use the given condition to find the value of C Substitute the value of C Solve for y

30

31

32 Extreme Value Theorem If f is continuous on [a, b], then f must attain a maximum and a minimum. Intermediate Value Theorem (often in FRQ s) If f is continuous on [a, b], then there is a c on (a,b) such that f(c) is between f(a) and f(b) Mean Value Theorem (often in FRQ s) If f is continuous and differentiable on [a, b], then there is a c on (a, b) such that f '( c) f ( b) f b a ( a) The slope of the derivative is equal to the slope of the secant.

33 Fundamental Theorem of Calculus Given that F(x) is an anti-derivative of f(x), then: b a f ( x) dx F( a) F( b) 2 nd Fundamental Theorem of Calculus Differentiation and Integration undo each other b f '( x ) dx f ( b ) f ( a ) a d dx g ( x) a f ( t) dt f ( g( x)) g ( x) Average Value of a function The average value of f on [a, b] is 1 b f ( x) dx b a a

34

35 Limits Determining limits algebraically & graphically Continuity & differentiability Using L Hospital s rule

36

37

38 Accumulation Rates Table A function for the rate in and another for the rate out are given Amount in or out, total amount, maximize or minimize the amount. A table of values is given. Approximations for the derivative and integrals. Area / Volume Graph of the derivative An area R is defined or given in a graph The value of the area, volume by revolution or cross-sections. A graph of the derivative of a function is given Function values, critical and inflection points, relative and absolute extrema. Physics The velocity function is given, along with a position point. Find the acceleration, position, direction of movement, distance. Differential Equation The equation of the derivative is given (dy/dx) Slope field, equation of tangent line, solve the equation, 2 nd derivative.

39

40

41

42

43

44

45

46

47

48

49

50

51

52 At home: Find released AP exams: 2008, 2003, 1998, 1997, 2008 Practice Exam Use these exams to practice multiple choice questions. Go to and use the released FRQ s to practice. Use the scoring guidelines to score yourself Figure out the topics you need help, and research them online At school: Ask your teacher to use the secured exams to practice multiple choice questions in class Ask your teacher to use the released FRQ s to practice in class. Discuss with your teacher any topics that you think needs refreshing.

53 Sources Calculus 9 th edition by R.Larson 2008 AP Calculus Released Exam