10.4 Usubstitution Trig Functionsap Calculus PORTABLE

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3 Determining Intervals on Which a Function is Increasing or Decreasing5 4 Using the First Derivative Test to Determine Relative Local Extrema5.

8 Determining Limits Using the Squeeze Theorem1 9 Connecting Multiple Representations of LimitsMid-Unit Review - Unit 11.. 6 includes horizontal tangent lines, equation of the normal line, and differentiability of piecewise)2.. )$$ = displaystyle{2 cdot sqrt{ cos^2 theta } } $$ $$ = displaystyle{2 cdot Big|cos theta Big| } $$ (Assume that $ displaystyle - frac{pi}{2} le theta le displaystyle frac{pi}{2} $ so that $ cos theta ge 0 $.

2 Defining Limits and Using Limit Notation1 3 Estimating Limit Values from Graphs1.. 8 Ratio Test for Convergence 10 9 Determining Absolute or Conditional Convergence 10.. 1 Using the Mean Value Theorem5 2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points5.. 5 Using the Candidates Test to Determine Absolute (Global) Extrema5 6 Determining Concavity of Functions over Their Domains 5.. 12 Integrating Using Linear Partial Fractions (BC topic) 6 13 Evaluating Improper Integrals (BC topic)6.

1 Exploring Accumulation of Change6 2 Approximating Areas with Riemann Sums6 3 Riemann Sums, Summation Notation, and Definite Integral Notation6.. 8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation6 9 Integrating Using Substitution6.. Lessons will begin to appear starting summer 2020 BC Topics are listed, but there will be no lessons available for SY 2020-2021Unit 0 - Calc Prerequisites (Summer Work)0.. 7 Defining Polar Coordinates and Differentiating in Polar Form 9 8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve 9.

3 Arc Lengths of Curves (Parametric Equations) 9 4 Defining and Differentiating Vector-Valued Functions 9.. 2 includes equation of the tangent line)2 3 Estimating Derivatives of a Function at a Point2.. 13 Removing Discontinuities1 14 Infinite Limits and Vertical Asymptotes1 15 Limits at Infinity and Horizontal Asymptotes1.. 12 Lagrange Error Bound 10 13 Radius and Interval of Convergence of Power Series 10.. 1 Modeling Situations with Differential Equations 7 2 Verifying Solutions for Differential Equations 7.. 4 Separation of VariablesReview - Unit 10Unit 11 - Area and Volume11 1 Area Between Two Curves11.. 1 Interpreting the Meaning of the Derivative in Context4 2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration4.. 1 Implicit Differentiation6 2 Related Rates6 3 Optimization​ Review - Unit 6Unit 7 - Approximation Methods7.. Recall that if$$ x = f(theta) , $$$$ dx = f'(theta) dtheta $$For example, if$$ x = sec theta , $$then$$ dx = sec theta tan theta dtheta $$The goal of trig substitution will be to replace square roots of quadratic expressions or rational powers of the form $ displaystyle frac{n}{2} $ (where $ n $ is an integer) of quadratic expressions, which may be impossible to integrate using other methods of integration, with integer powers of trig functions, which are more easily integrated.. 10 Exploring Types of Discontinuities1 11 Defining Continuity at a Point1 12 Confirming Continuity Over an Interval1.. 14 Finding Taylor Maclaurin Series for a Function 10 15 Representing Functions as a Power Series Review - Unit 8Version #2​The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day.. 6 Comparison Tests for Convergence 10 7 Alternating Series Test for Convergence 10.. It is a method for finding antiderivatives of functions which contain square roots of quadratic expressions or rational powers of the form $ displaystyle frac{n}{2}$ (where $n$ is an integer) of quadratic expressions.. Choose from 500 different sets of trig functions calculus ab trigonometric flashcards on Quizlet.. 6 Calculating Higher-Order DerivativesReview - Unit 3Unit 4 - Contextual Applications of Differentiation4.. 1 Extrema on an Interval5 2 First Derivative Test5 3 Second Derivative TestReview - Unit 5Unit 6 - Implicit Differentiation6.. 3 Using Accumulation Functions and Definite Integrals in Applied Contexts 8 4 Area Between Curves (with respect to x) 8.. Since $$ displaystyle (adjacent)^2 + (opposite)^2 = (hypotenuse)^2 longrightarrow $$ $$ (adjacent)^2 + (x)^2 = (2)^2 longrightarrow adjacent = sqrt{4-x^2} longrightarrow $$ $$ cos theta = displaystyle{ adjacent over hypotenuse }= displaystyle{ sqrt{4-x^2} over 2 } $$ Then $$ displaystyle{ 2 theta + 2 sin theta cos theta } + C = 2 arcsin Big( frac{x}{2} Big) + 2 cdot displaystyle{ x over 2} cdot displaystyle{ sqrt{4-x^2} over 2} $$$$ = displaystyle 2 arcsin Big( frac{x}{2} Big) + frac{1}{2} x cdot sqrt{4-x^2} + C $$ When using the method of trig substitution, we will always use one of the following three well-known trig identities : (I) $ 1 - sin^2 theta = cos^2 theta $ (II) $ 1 + tan^2 theta = sec^2 theta $ and (III) $ sec^2 theta - 1 = tan^2 theta $.. 9 includes a revisit of particle motion and determining if a particle is speeding up/down.. 4 Chain Rule3 5 Trig DerivativesReview - Unit 3Unit 4 - More Deriviatvies4 1 Derivatives of Exp.. For example, if we start with the expression$$ displaystyle{ sqrt{4-x^2} } $$ and let$$ x = 2 sin theta , $$ then$$ displaystyle{ sqrt{4-x^2} } = displaystyle{ sqrt{4-(2 sin theta)^2 } } $$$$ = displaystyle{ sqrt{4-4 sin^2 theta } } $$$$ = displaystyle{ sqrt{4 (1- sin^2 theta ) } } $$$$ = displaystyle{ sqrt{4} cdot sqrt{1- sin^2 theta } } $$(Recall that $ cos^2 theta + sin^2 theta = 1 $ so that $ 1- sin^2 theta = cos^2 theta $.. 3 The nth Term Test for Divergence 10 4 Integral Test for Convergence 10 5 Harmonic Series and p-Series 10.. Examples of such expressions are $$ displaystyle{ sqrt{ 4-x^2 }} and displaystyle{(x^2+1)^{3/2}} $$The method of trig substitution may be called upon when other more common and easier-to-use methods of integration have failed.. 1 Rectangular Approximation Method7 2 Trapezoidal Approximation MethodReview - Unit 7Unit 8 - Integration8.. ) $$ = displaystyle{ 4 int frac{1}{2}(1+ cos 2 theta) , d theta } $$$$ = displaystyle{ 2 int (1+ cos 2 theta) , d theta } $$$$ = displaystyle{ 2 ( theta + frac{1}{2} sin 2 theta) } + C $$ $$ = displaystyle{ 2 theta + sin 2 theta } + C $$(Recall that $ sin 2 theta = 2 sin theta cos theta $.. 1 The 2nd FTC9 2 Trig Integrals9 3 Average Value (of a function)9 4 Net ChangeReview - Unit 9Unit 10 - More Integrals10.. 3 Rates of Change in Applied Contexts Other Than Motion4 4 Introduction to Related Rates4.. 10 Derivatives of tan(x), cot(x), sec(x), and csc(x)Review - Unit 2Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions3.. )$$ = displaystyle{ 2 theta + 2 sin theta cos theta } + C $$We need to write our final answer in terms of $ x$.. 3 Sketching Slope Fields7 4 Reasoning Using Slope Fields 7 5 Euler's Method (BC topic) 7.. Trig substitution assumes that you are familiar with standard trigonometric identies, the use of differential notation, integration using u-substitution, and the integration of trigonometric functions.. 7 Using L'Hopital's Rule for Determining Limits of Indeterminate FormsReview - Unit 4Unit 5 - Analytical Applications of Differentiation5.. Lessons and packets are longer because they cover more material Unit 0 - Calc Prerequisites (Summer Work)0.. 13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC topic) Review - Unit 8Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics) 9.. 7 Derivatives of cos(x), sin(x), e^x, and ln(x)2 8 The Product Rule2 9 The Quotient Rule2.. 1 Defining Convergent and Divergent Infinite Series 10 2 Working with Geometric Series 10.. 9 Finding the Area of the Region Bounded by Two Polar Curves Review - Unit 9 Unit 10 - Infinite Sequences and Series (BC topics) 10.. 1 Things to Know for Calc0 2 Summer Packet0 3 Calculator SkillzUnit 1 - Limits1 1 Limits Graphically1.. 11 Washer Method: Revolving Around the x- or y- Axis8 12 Washer Method: Revolving Around Other Axes 8.. 1 Defining and Differentiating Parametric Equations 9 2 Second Derivatives of Parametric Equations 9.. 2 Volume - Disc Method11 3 Volume - Washer Method​ 11 4 Perpendicular Cross SectionsReview - Unit 11 FINDING INTEGRALS USING THE METHOD OF TRIGONOMETRIC SUBSTITUTION The following integration problems use the method of trigonometric (trig) substitution.. and Logs4 2 Inverse Trig Derivatives4 3 L'Hopital's RuleReview - Unit 4Unit 5 - Curve Sketching5.. 9 Disc Method: Revolving Around the x- or y- Axis 8 10 Disc Method: Revolving Around Other Axes 8.. 7 Selecting Procedures for Determining Limits (1 7 includes rationalization, complex fractions, and absolute value)1.. 6 General Solutions Using Separation of Variables7 7 Particular Solutions using Initial Conditions andSeparation of Variables7.. 1 Summer PacketUnit 1 - Limits and Continuity1 1 Can Change Occur at an Instant?1.. 1 Definite Integral8 2 Fundamental Theorem of Calculus (part 1)8 3 Antiderivatives (and specific solutions)Review - Unit 8Unit 9 - The 2nd Fundamental Theorem of Calculus9.. 4 Connecting Differentiability and Continuity2 5 Applying the Power Rule2 6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple (2.. 1 Average Value of a Function on an Interval 8 2 Position, Velocity, and Acceleration Using Integrals 8.. 10 Alternating Series Error Bound 10 11 Finding Taylor Polynomial Approximations of Functions 10.. ) $$ = displaystyle{2 cos theta } $$ and$$ dx = 2 cos theta d theta $$Thus,$$ displaystyle{ int sqrt{4-x^2} , dx } $$could be rewritten as$$ displaystyle{ int sqrt{4-x^2} , dx } = displaystyle{ int 2 cos theta cdot 2 cos theta , d theta } = displaystyle{ 4 int cos^2 theta , d theta } $$(Recall that $ cos 2 theta = 2 cos^2 theta -1 $ so that $ cos^2 theta = displaystyle frac{1}{2}(1+ cos 2 theta) $.. 6 Applying Properties of Definite Integrals6 7 The Fundamental Theorem of Calculus and Definite Integrals 6.. 5 Area Between Curves (with respect to y) 8 6 Area Between Curves - More than Two Intersections Mid-Unit Review - Unit 8 8.. Version #1​The course below follows CollegeBoard's Course and Exam Description.. 7 Cross Sections: Squares and Rectangles 8 8 Cross Sections: Triangles and Semicircles 8.. 1 Defining Average and Instantaneous Rate of Change at a Point2 2 Defining the Derivative of a Function and Using Derivative Notation (2.. Since $ x = 2 sin theta $, it follows that$$ sin theta = displaystyle{ x over 2} = displaystyle{ opposite over hypotenuse } $$ and $$ theta = arcsin Big(displaystyle frac{x}{2} Big) $$Using the given right triangle and the Pythagorean Theorem, we can determine any trig value of $ theta $.. 1 Power Rule3 2 Product and Quotient Rules3 3 Velocity and other Rates of Change3.. We also discuss the use of graphing Mathematics 104—Calculus, Part I (4h, 1 CU) Course Description: Brief review of High School Calculus, methods and applications of integration, infinite series, Taylor's theorem, first order ordinary differential equations.. 4 Estimating Limit Values from Tables1 5 Determining Limits Using Algebraic Properties (1.. 10 Integrating Functions Using Long Division​ and Completing the Square 6 11 Integrating Using Integration by Parts (BC topic) 6.. 5 Integrating Vector-Valued Functions 9 6 Solving Motion Problems Using Parametric and Vector-Valued Functions 9.. 16 Intermediate Value Theorem (IVT)Review - Unit 1Unit 2 - Differentiation: Definition and Fundamental Properties2.. 1 Average Rate of Change2 2 Definition of the Derivative2 3 Differentiability [Calculator Required]​ Review - Unit 2 Unit 3 - Basic Differentiation3.. 8 Exponential Models with Differential Equations 7 9 Logistic Models with Differential Equations (BC topic)Review - Unit 7Unit 8 - Applications of Integration 8.. )5 10 Introduction to Optimization Problems5 11 Solving Optimization Problems5 12 Exploring Behaviors of Implicit RelationsReview - Unit 5Unit 6 - Integration and Accumulation of Change6.. Use of symbolic manipulation and graphics software in Calculus Note: This course uses Maple®.. 1 Slope Fields10 2 u-Substitution (indefinite integrals)10 3 u-Substitution (definite integrals)10.. 5 Solving Related Rates Problems 4 6 Approximating Values of a Function Using Local Linearity and Linearization4.. This chapter reviews the basic ideas you need to start calculus The topics include the real number system, Cartesian coordinates in the plane, straight lines, parabolas, circles, functions, and trigonometry.. 1 The Chain Rule3 2 Implicit Differentiation3 3 Differentiating Inverse Functions3.. 4 The Fundamental Theorem of Calculus and Accumulation Functions6 5 Interpreting the Behavior of Accumulation Functions​ Involving AreaMid-Unit Review - Unit 66.. 8 Sketching Graphs of Functions and Their Derivatives5 9 Connecting a Function, Its First Derivative, and Its Second Derivative (5.. 2 Limits Analytically1 3 Asymptotes1 4 ContinuityReview - Unit 1 Unit 2 - The Derivative2.. Learn trig functions calculus ab trigonometric with free interactive flashcards.. 14 Selecting Techniques for AntidifferentiationReview - Unit 6Unit 7 - Differential Equations 7.. 7 Using the Second Derivative Test to Determine ExtremaMid-Unit Review - Unit 55.. 5 includes piecewise functions involving limits)1 6 Determining Limits Using Algebraic Manipulation1.. 4 Differentiating Inverse Trigonometric Functions3 5 Selecting Procedures for Calculating Derivatives3.

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