Quick Review Sheet for A.P. Calculus Exam

Transcription

1 Quick Review Sheet for A.P. Calculus Exam Name AP Calculus AB/BC Limits Date Period 1. Definition: 2. Steps in Evaluating Limits: - Substitute, Factor, and Simplify 3. Limits as x approaches infinity If taking the limit of a Rational expression, Divide by the highest power of x. 4. L'Hopital's Rule 5. Trig Limits Continuity 1

2 1. Definition Derivatives 1. Definition: 2

3 Applications of Derivatives Analysis of Functions 1. Increasing / Decreasing If f'(x) > 0 Then f(x) is increasing. If f'(x) < 0 Then f(x) is decreasing. If f'(x) = 0 Then f(x) is constant. 2. Concavity If f''(x) > 0, then f(x) is concave up. (Mr. Happy Face) If f''(x) < 0, then f(x) is concave down. (Mr. Frowny) Points of inflection occur when the concavity changes. Test: If there is a point of inflection, the second derivative is zero. BUT just because the second derivative is zero doesn't guarantee a point of inflection. 3. Relative Extrema 1st derivative test: Test points on each side of the critical points found by substituting in the first derivative. If the value of the derivative of the point to the left of the critical point is positive and the value of the derivative for the point to the right is negative, then the critical point is a relative maximum. If the value of the derivative of the point to the left of the critical point is negative and the value of the derivative for the point to the right is positive, then the critical point is a relative minimum. 3

4 If the values of the derivative of the points to the left and the right of the critical point are the same (i.e., both positive or both negative), then the critical point is a point of inflection. 2nd derivative test: Take the first derivative and set it equal to zero to solve for critical points. Take the second derivative of the function. Substitute the critical point in the second derivative. If this value is negative, the critical point is a relative maximum. If this value is positive, the critical point is a relative minimum. If this value is zero, the critical point is a possible point of inflection. Test points on either side of the critical point by substituting them into the second derivative to verify that the concavity changed. 4. Mean Value Theorem 5. Exponential Growth and Decay 4

5 Newton's Method Implicit Differentiation Related Rates Applied Maxima / Minima Slope Fields Integration Fundamental Theorem of Calculus: Some Basic Indefinite Integrals sin x dx = -cos x + C cos x dx = sin x + C sec² x dx = tan x + C csc x cot x dx = - csc x + C sec x tan x dx = sec x + C csc² x dx = - cot x + C 5

6 Methods of Integration If you cannot simply integrate using the basic formulas above, try one of the following methods: 1.Integration by substitution: a method by which some variable is substituted for part of the function f(x) (generally some complicated function within the larger function) in order to make integration simpler. 2. Integration using long division: If the integral involves a quotient in which the degree of the numerator is greater than or equal to the degree of the denominator, divide the numerator by the denominator. 3. Integration by parts: a method of integrating two functions multiplied together (the opposite of the product rule for derivatives), following the formula: 4. Integration by Trig Substitution: If the integral contains trig expressions, try substituting some of the basic trig identities: If the integral contains the sum or difference of two squares, set up right triangles and make appropriate trig substitutions. 5. Numerical Integration: When symbolic methods fail, use of some numerical approximation method will give useful answers along a specified interval. Most calculators enact these methods to give extremely exact answers by using very tiny subdivisions. a). Riemann Sums with Rectangles: Rectangular Approximation Method. 6

7 b). Trapezoidal Rule: Trapezoidal approximation. Applications of Integration 1.Rectilinear Motion Recall the difference between Displacement and Total Distance. 2. Average Value of a Function 3. Area Between Two Curves 7

8 4. Volumes of Solids of Revolution 5. Volumes of Cross-Sections 8

9 6. Length of an Arc Miscellaneous Geometry Formulas 1. Volumes 2. Surface Area 3. Area and Circumference Formulas Miscellaneous Algebra Formulas 1. Quadratic Formula 9

10 Trigonometric Formulas SOHCAHTOA Identities listed above Some Geometry Figures: Graphs 10