Limits & Continuity 1

The limit of f(x) as x approaches c from the left.

Limits & Continuity 2

The limit of f(x) as x approaches c from the right.

Limits & Continuity 3

The y-value that the graph is approaching on both sides of the x-value c.

Limits & Continuity 4 The Limit D.oes N.ot E.xist

1.Oscillating 2.Approaching + or - 3.The function does not approach the same value from the left as the right.

Limits & Continuity 5 Limits of Rational Functions at Vertical Asymptotes 4 Cases: (c is located at the vertical asymptote x = c) ABCDABCD

ABCDD.N.E. onlyor D.N.E.

Limits & Continuity 6 Evaluating Limits Using Direct Substitution

Plug it in!

Limits & Continuity 7 Evaluating Limits Using Factor & Simplify

Limits & Continuity 8 Evaluating Limits Using Conjugates

Limits & Continuity 9 Evaluating Limits Using Common Denominators

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Limits & Continuity 11

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Limits & Continuity 16 Limits of Rational Functions as x approaches infinity 3 cases

Limits & Continuity 17

Definition of “e”

Limits & Continuity 18

Definition of “e”

Limits & Continuity 19

1

Limits & Continuity 20

0

Limits & Continuity 21 Continuity A function is continuous at the point x = a if and only if: 1) 2) 3)

Limits & Continuity 22 Open Interval

(a, b) *endpoints not included

Limits & Continuity 23 Closed Interval

[a, b] *endpoints are included

Limits & Continuity 24 Continuity on a Closed Interval

A function is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and

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Limits & Continuity 26

Limits & Continuity 27

Limits & Continuity 28

Basic Derivative Mechanics 29 Average Rate of Change

Basic Derivative Mechanics 29 (same as average velocity, different from average value)

Derivatives 30 Definition of Derivative

Derivatives 30 Slope of the tangent line

Derivatives 31 Definition of Derivative at a specific x value

Derivatives 31

Derivatives 32 Point-Slope form of a Line

Derivatives 32

Derivatives 33 Normal Line

Derivatives 33 Perpendicular to the tangent line at the point of tangency (slope is opposite sign and reciprocal)

Derivatives 34 Situations in which the Derivative fails to exist 1) 2) 3) 4)

Derivatives 34 1)The function does not exist. (ex: holes, asymptotes, gaps, discontinuities) 2) 3) Cusps (Sharp Corners – the function isn’t “smooth,” ex: absolute value) 4) Points of Vertical Tangency

Derivatives 35

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Derivatives 37 Power Rule

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Derivatives 47 Product Rule Product Rule with 3 factors

Derivatives 47 Product Rule with 3 factors

Derivatives 48 Quotient Rule

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Derivatives 50

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Derivatives 51

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Derivatives 52

Derivatives 53 Chain Rule Derivative of a composition of functions where f is the outside function and g is the inside function

Derivatives 53 1)Take the derivative of the outside function. 2)Copy the inside function. 3)Multiply by the derivative of the inside function.

Advanced Derivative Mechanics Implicit Differentiation 54

Advanced Derivatives 1.Differentiate both sides with respect to x. (When you take the derivative of something with a y in it you do the derivative as normal, but then multiply it by the derivative of y with respect to x (dy/dx). 2.Collect all dy/dx terms on one side. 3.Factor dy/dx out. 4.Solve for (isolate) dy/dx.

Advanced Derivatives Logarithmic Differentiation 55

Advanced Derivatives

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Advanced Derivatives

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Advanced Derivatives

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Advanced Derivatives

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Advanced Derivatives

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Advanced Derivatives

If f and g are inverse functions, 66

1. Find the derivative of f(x). 2. The derivative of its inverse is

Advanced Derivatives Indeterminate Forms 67

Advanced Derivatives

L’Hôpital’s Rule 68

Advanced Derivatives If the limit yields an indeterminate form, *Can be repeated on same problem if you get indeterminate forms again!

Graphical Analysis Critical Number And Critical Point 69

Graphical Analysis Critical Numbers: x values where Critical Point:

Graphical Analysis Increasing 70

Graphical Analysis

Decreasing 71

Graphical Analysis

Relative or Local Minimum 72

Graphical Analysis

Relative or Local Maximum 73

Graphical Analysis

Extrema Extremum 74

Graphical Analysis Extrema: (plural) maximums and minimums Extremum: (singular) a maximum or minimum

Graphical Analysis First Derivative Test 75

Graphical Analysis

Concave Up (Positive Curvature) 76

Graphical Analysis Like a Bowl

Graphical Analysis Concave Down (Negative Curvature) 77

Graphical Analysis Like a Rainbow

Graphical Analysis P.oints O.f I.Nflection (P.O.I.) 78

Graphical Analysis The curvature changes

Graphical Analysis Second Derivative Test 79

Graphical Analysis

Absolute or Global Maximum 80

Graphical Analysis The BIGGEST y-coordinate on a graph (cannot be infinity, so sometimes there are none) *Check all critical points and endpoints to find the absolute maximum!

Graphical Analysis Absolute or Global Minimum 81

Graphical Analysis The smallest y-coordinate on a graph (cannot be negative infinity, so sometimes there are none) *Check all critical points and endpoints to find the absolute minimum!

Graphical Analysis Intermediate Value Theorem 82

Graphical Analysis If a function is continuous and y is between f(a) and f(b), there exists at least one number x=c in (a,b) such that f(c) = y.

Graphical Analysis Mean Value Theorem 83

Graphical Analysis If a function is continuous and differentiable on (a,b) there is at least one number x=c in (a,b) such that

Graphical Analysis Rolle’s Theorem 84

Graphical Analysis If a function is continuous and differentiable on (a,b) and f(a) = f(b) there is at least one number x=c in (a,b) such that

Graphical Analysis Extreme Value Theorem 85

Graphical Analysis If a function is continuous on a closed interval [a, b], Then it has both an absolute minimum and an absolute maximum.

Derivative Applications Related Rates 86

Derivative Applications *Plugging in non-constant quantities before differentiating is a NO-NO! Draw a picture. Make a list of all known and unknown rates and quantities. Label each quantity that changes with time. Relate the variables in an equation. Differentiate with respect to time. *EVERY VARIABLE WILL GENERATE A RATE!* Substitute the known quantities and rates in and solve.

Derivative Applications Local Linearity (Tangent to a Curve) 87

Derivative Applications Given:

Derivative Applications Differential dy 88

Derivative Applications

Maximum and Minimum Application Problems 89

Derivative Applications 1.Find a formula for the quantity to be maximized/ minimized (only 2 variables). 2.Find an interval of possible values based on restrictions. 3.Set f ’(x) = 0 and solve. 4.The max/min will be at the answer to step 3 (or at one of the endpoints of the interval, if applicable).

Derivative Applications: Motion Position Function 90

Derivative Applications: Motion Tells where a particle is along a straight line

Derivative Applications: Motion Velocity Function 91

Derivative Applications: Motion Describes the change in position Positive Velocity is moving to the right or up Negative Velocity is moving to the left or down

Derivative Applications: Motion Acceleration Function 92

Derivative Applications: Motion Describes the change in velocity

Derivative Applications: Motion Speed

Derivative Applications: Motion Absolute Value of Velocity Always Positive!

Derivative Applications: Motion Displacement ***We will learn another method for this in the Spring Semester!*** 94

Derivative Applications: Motion Final Position – Initial Position

Derivative Applications: Motion Total Distance Traveled ***We will learn another method for this in the Spring Semester!*** 95

Derivative Applications: Motion Add the absolute values of the differences in position between all resting points.

Derivative Applications: Motion Average Velocity Or Average Rate of Change 96

Derivative Applications: Motion Notice the similarity to the slope formula!

Derivative Applications: Motion Average Speed 97

Derivative Applications: Motion

Average Acceleration 98

Derivative Applications: Motion Notice the similarity to the slope formula!

Derivative Applications: Motion Speeding Up 99

Derivative Applications: Motion Velocity and Acceleration have the same sign. Both Positive or Both Negative

Derivative Applications: Motion Slowing Down 100

Derivative Applications: Motion Velocity and Acceleration have opposite signs.

Derivative Applications: Motion The Free-Fall Model ***You do not have to memorize this one!*** (It will be provided to you. Just be sure you know what the parts mean and how to use it!) 101

Derivative Applications: Motion The height s(t) of the object in free-fall is given by the formula: By taking the derivative of the Free-Fall Model, you obtain: And, by taking the second derivative of the Free-Fall Model, you obtain:

Integration Right Hand Riemann Sum 102

Integration Add up the following for each subinterval:

Integration Left Hand Riemann Sum 103

Integration Add up the following for each subinterval:

Integration Midpoint Riemann Sum 104

Integration Add up the following for each subinterval:

Integration Trapezoid Riemann Sum 105

Integration Add up the following for each subinterval:

Integration 106

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The Fundamental Theorem of Calculus 137

Integration To evaluate a definite integral, find the antiderivative of the integrand. (Omit the +C.) Substitute the upper limit of integration into the antiderivative, and then substitute the lower limit of integration into the antiderivative. Calculate the difference of the two quantities.

Integration New y-value 138

Integration

Average Value (Like average temperature or average distance, NOT average rate of change!) 139

Integration

The 2 nd Fundamental Theorem of Calculus 140

Integration

The 2 nd Fundamental Theorem of Calculus with the Chain Rule 141

Integration

U-Substitution 142

Integration U-Substitution is used to integrate compositions of functions (like Substitution is the most powerful tool we have to find anti-derivatives when inspection for common rules fails. How to do U-Substitutions 1.Choose the “inside function” of a composition of functions to be called u. 2.Calculate the derivative of u, called 3. Get du by itself. (Multiply by dx). ** Our goal is to replace the entire integrand with an expression (or expressions) in terms of u. When done correctly, the new integral is a much easier one to evaluate. 4. It is often helpful to get dx by itself, although sometimes the problem already has part of your du in it, so you may not need to completely isolate dx to do your substitution. 5. Replace the inside function with u. Replace dx with whatever you got for dx in the previous step. 6. Integrate. 7. Replace u with the inside function.

Integration Functions Defined By Integrals Find the new y-value. 143

Integration

Integration Applications Area Between 2 Curves 144

Integration Applications

Volume of Solids of Known Cross Sections 145

Integration Applications

Volume of Solids of Revolution Cross section is a circle 146

Integration Applications

Volume of Solids of Revolution Cross section is a washer 147

Integration Applications

Arc Length 148

Integration Applications

Integration: Motion Displacement 149

Integration: Motion Change in position

Integration: Motion Total Distance Traveled 149

Integration: Motion

New Position 150

Integration: Motion

New Velocity 151

Integration: Motion

Differential Equations & More Slope Fields 152

Differential Equations & More Given a dy/dx formula, substitute in x and y from each marked point to determine the slope there. 152

Differential Equations & More Differential Equations 153

Differential Equations & More 1)Cross Multiply 2)Get all x’s & dx’s on one side. Get all y’s and dy’s on the other side. 3)Remember to put +C on the x side. 4)Substitute in the given point (initial condition) and solve for C. 5)Rewrite equation including C and solve for y. 153

Differential Equations & More Exponential Growth 154

Differential Equations & More

Euler’s Method (for approximation) 155 BC Only

Differential Equations & More

Logistics 156 BC Only

Differential Equations & More BC Only

Differential Equations & More Integration By Parts 157 BC Only

Differential Equations & More 157 BC Only

Differential Equations & More Integration with Partial Fractions 158 BC Only

Differential Equations & More 1)If deg. Num.> deg. Denom., do long division 1 st 2)Factor denom. 3)Write eq’n w/ A, B, C, … to set up partial fractions. 4)Mult. Both sides by denom. Of orig. function. 5)Simplify both sides. 6)On right side, group like terms (ie: x 2, x, constant) BC Only 7) On the right side, factor out x 2, x, constant 8) Set up a system of equations & solve for A, B, C, etc. 9) Rewrite integral using partial fractions 10) Integrate each partial fraction individually.

Differential Equations & More Improper Integrals 159 BC Only

Differential Equations & More Function is undefined at upper &/or lower limit 159 BC Only

Convergence Tests Geometric Series Test 160 BC Only

Convergence Tests If, the series converges to If, the series diverges. 160 BC Only

Convergence Tests Sum of an Infinite Geometric Series 161 BC Only

Convergence Tests 161 BC Only

Convergence Tests Telescoping Series 162 BC Only

Convergence Tests If you write out the first few terms of the series and see them “cancelling” out, the series will converge to 162 BC Only

Convergence Tests Divergence Test Or nth Term Test for Divergence 163 BC Only

Convergence Tests 163 BC Only *This test CANNOT be used to show that a series converges!

Convergence Tests Harmonic Series 164 BC Only

Convergence Tests Diverges 164 BC Only

Convergence Tests How do you find the sum of a series? 165 BC Only

Convergence Tests 165 BC Only

Convergence Tests Integral Test 166 BC Only

Convergence Tests 166 BC Only

Convergence Tests p-Series Test 167 BC Only

Convergence Tests 167 BC Only

Convergence Tests Direct Comparison Test 168 BC Only

Convergence Tests If the bigger converges, the smaller converges. If the smaller diverges, the bigger diverges. 168 BC Only

Convergence Tests Limit Comparison Test 169 BC Only

Convergence Tests 169 BC Only

Convergence Tests Ratio Test 170 BC Only

Convergence Tests 170 BC Only

Convergence Tests Root Test 171 BC Only

Convergence Tests 171 BC Only

Convergence Tests Ratio Test for Absolute Convergence 172 BC Only

Convergence Tests 172 BC Only

Convergence Tests Alternating Series Test 173 BC Only

Convergence Tests An alternating series converges if a)The absolute values of the terms decrease and b) 173 BC Only NOTE: Do not use alternating series test on a non-alternating series!

Convergence Tests Error Bound for Alternating Series 174 BC Only

Convergence Tests The absolute value of the first term left out of the partial sum 174 BC Only

Series Maclaurin Series 175 BC Only

Series A Taylor Series centered about x = BC Only

Series Taylor Series 176 BC Only

Series 176 BC Only

Series Lagrange Error Bound/ Remainder of a Taylor Polynomial 177 BC Only

Series Given 177 BC Only

Series Interval of Convergence 178 BC Only

Series 1)Do the Ratio Test for Absolute Convergence. 2)Set the answer to the limit from step one < 1 and solve. 3)Check endpoints to see if they’re included. Plug them each in for x in the original series and test for convergence. 178 BC Only

Series Radius of Convergence 179 BC Only

Series Distance from center of interval of convergence to either end of interval. If interval is (a,b), 179 BC Only

Series Maclaurin Series for 180 BC Only

Series 180 BC Only

Series Maclaurin Series for 181 BC Only

Series 181 BC Only

Series Maclaurin Series for 182 BC Only

Series 182 BC Only

Series Maclaurin Series for 183 BC Only

Series 183 BC Only

Series Binomial Series Formula 184 BC Only

Series Remember, the number of factors in the numerator is the same as the degree of the term. 184 BC Only

Analytic Geometry in Calculus Parametric Equations 185 BC Only

Analytic Geometry in Calculus 185 BC Only

Analytic Geometry in Calculus Arc Length (Parametric) 186 BC Only

Analytic Geometry in Calculus 186 BC Only

Analytic Geometry in Calculus Polar Curves – 4 conversions 187 BC Only

Analytic Geometry in Calculus 187 BC Only

Analytic Geometry in Calculus Slope of a Polar Curve 188 BC Only

Analytic Geometry in Calculus 188 BC Only

Analytic Geometry in Calculus Area of a Polar Curve 189 BC Only

Analytic Geometry in Calculus Inside one petal 189 BC Only Set r = Θ to help you find the upper and lower limits.

Analytic Geometry in Calculus Power Reducing Formula (from Precalculus!) 190 BC Only

Analytic Geometry in Calculus 190 BC Only

Analytic Geometry in Calculus Power Reducing Formula (from Precalculus!) 191 BC Only

Analytic Geometry in Calculus 191 BC Only

Analytic Geometry in Calculus Area of Intersections of Polar Curves 192 BC Only

Analytic Geometry in Calculus 192 BC Only

Analytic Geometry in Calculus Position Vector 193 BC Only

Analytic Geometry in Calculus 193 BC Only

Analytic Geometry in Calculus Velocity Vector 194 BC Only

Analytic Geometry in Calculus 194 BC Only

Analytic Geometry in Calculus Acceleration Vector 195 BC Only

Analytic Geometry in Calculus 195 BC Only

Analytic Geometry in Calculus Speed (parametric) 196 BC Only

Analytic Geometry in Calculus 196 BC Only

Analytic Geometry in Calculus Total Distance Traveled (parametric) *same as arc length for parametric! 197 BC Only

Analytic Geometry in Calculus 197 BC Only