by Andrew Harrison Hill
of Aurora, IL
Table of Contents
General Calculus Ideas 6
Rates of Change 6
Speeds 6
Tangent Lines 6
Examples 6
Limits 6
Properties of Limits 6
Examples 7
One-sided and Two-sided Limits 7
Sandwich Theorem 7
Examples 7
Limits Involving Infinity 8
Examples 8
Continuity 8
Continuity at a Point 8
Types of discontinuity 8
Properties of Continuous Functions 8
Composite of Continuous Functions 9
The Intermediate Value Theorem for Continuous Functions 9
Derivatives 9
Intro to Derivatives 9
Examples 9
Notation 9
Differentiability 10
Examples 10
Rules for Differentiation 10
Rule 1: Derivative of a Constant Function 10
Rule 2: Power Rule for Positive Integer Powers of x 10
Rule 3: The Constant Multiple Rule 10
Rule 4: The Sum and Difference Rule 11
Rule 5: The Product Rule 11
Rule 6: The Quotient Rule 11
Rule 7: Power Rule for Negative Integer Powers of x 11
Rule 8: Chain Rule: 11
Rule 9: Power Rule for Rational Powers of x 11
Rule 10: Power Rule for Arbitrary Real Powers 11
Examples 11
Parametric Curves 12
Examples 12
Implicit Differentiation 12
Examples 12
Second and Higher Order Derivatives 13
Examples 13
Velocity and Other Rates of Change 13
Free-fall Constants (Earth) 13
Trigonometric Function 13
Derivatives of Exponential and Logarithmic Functions 14
Derivative of ex 14
Derivative of ax 14
Derivative of ln x 14
Derivative of logax 14
Examples 14
Applications of Derivatives 15
Theorems 15
Theorem 1: The Extreme Value Theorem 15
Theorem 2: Local Extreme Values 15
Theorem 3: Mean Value Theorem for Derivatives 15
Theorem 4: First Derivative Test for Local Extrema 15
Theorem 5: Second Derivative Test for Local Extrema 15
Theorem 6: Maximum Profit 15
Theorem 7: Minimizing Average Cost 16
Extreme Values of Functions 16
Absolute Extreme Values 16
Local (Relative) Extreme Values 16
Critical Point 16
Examples 16
Mean Value Theorem 16
Increasing Function, Decreasing Function 16
Antiderivative 16
Corollary 1: Increasing and Decreasing Functions 16
Corollary 2: Functions with f’ = 0 are Constant 17
Corollary 3: Functions with the Same Derivative Differ by a Constant 17
Examples 17
Connecting f’ and f’’ with the Graph of f 17
Concavity 17
Concavity Test 17
Point of Inflection 17
Examples 17
Modeling and Optimization 18
Strategies for solving Max-Min Problems 18
Examples 18
Linearization and Newton’s Method 18
Linearization 18
Newton’s Method 18
Differentials 18
Differential Estimate of Change 18
Examples 18
Related Rates 19
Examples 19
Integrals 19
Theorems 19
Theorem 1: The Existence of Definite Integrals 19
Theorem 2: The Integral of a Constant 19
Theorem 3: The Mean Value Theorem for Definite Integrals 19
Theorem 4: The Fundamental Theorem of Calculus, Part 1 20
Theorem 4 (continued): The Fundamental Theorem of Calculus, Part 2 20
Estimating with Finite Sums 20
Rectangular Approximation Method (RAM) 20
Examples 20
Definite Integrals 20
Riemann Sums 20
The Definite Integral as a Limit of Riemann Sums 21
The Definite Integral of a Continuous Function on [a, b] 21
Notation 21
Area Under a Curve (as a Definite Integral) 21
Discontinuous Integrable Functions 21
Examples 21
Definite Integrals and Antiderivatives 22
Rules for Definite Integrals 22
Average (Mean) Value 22
Differential and Integral Calculus 22
Examples 22
Trapezoidal Rule 23
Trapezoidal Rule 23
Simpson’s Rule 23
Error Bounds 23
Examples: 24
Applications of Definite Integrals 24
Integral as Net Change 24
Strategy for Modeling with Integrals 24
Work 24
Examples: 25
Areas in the Plane 25
Area Between Curves 25
Area Enclosed by Intersecting Curves 25
Boundaries with Changing Functions 25
Integrating with Respect to y 25
Examples: 25
Volumes 26
Volumes of a Solid 26
Solids of Revolution 26
Examples: 27
Length of Curves 27
Arc Length: Length of a Smooth Curve 27
Vertical Tangents, Corners, and Cusps 28
Examples: 28
Other Definitions 28
Probability Density Function (pdf) 28
Normal Probability Density Function 28
The 68-95-99.7 Rule for Normal Distributions 28
Differential Equations and Mathematical Modeling 29
Antiderivatives and Slope Fields 29
Solving Initial Value Problems 29
Slope Field or Direction Field 29
Indefinite Integral 29
Integral Formulas 29
Properties of Indefinite Integrals 30
Examples: 30
Integration by Substitution 30
Power Rule for Integration 30
Substitution in Definite Integrals 30
Separable Differential Equations 30
Examples: 31
Integration by Parts 31
Product Rule in Integral Form 31
Examples: 32
Exponential Growth and Decay 32
Exponential Model 32
Logistic Growth Rate 32
Examples: 32
Numerical Methods 33
Euler’s Method 33
Improved Euler’s Method 34
L’Hôpital’s Rule, Improper Integrals, and Partial Fractions 34
L’Hôpital’s Rule 34
Theorem 1: L’Hôpital’s Rule (First Form) 34
Theorem 2: L’Hôpital’s Rule (Stronger Form) 34
Exponential Indeterminate Forms 34
Indeterminate Forms 34
Examples: 34
Relative Rates of Growth 35
Faster, Slower, Same-rate Growth as xà∞ 35
Transitivity of Growing Rates 35
Examples: 35
Improper Integrals 35
Improper Integrals with Infinite Integration Limits 35
Improper Integrals with Infinite Discontinuities 36
Direct Comparison Test 36
Limit Comparison Test 36
Examples: 36
Partial Fractions and Integral Tables 37
Partial Fractions 37
Trigonometric Substitutions 38
Examples: 38
Infinite Series 39
Power Series 39
Infinite Series 39
Geometric Series 39
Power Series 39
Term-by-Term Differentiation 40
Term-by-Term Integration 40
Examples: 40
Taylor Series 41
Taylor Series Generated by f at x = a 41
Maclaurin Series 41
Examples: 41
Table of Maclaurin Series 42
Taylor’s Theorem 42
Taylor’s Theorem with Remainder 42
Remainder Estimation Theorem 42
Examples: 43
Radius of Convergence 43
The Convergence Theorem for Power Series 43
The nth-Term Test for Divergence 43
The Direct Comparison Test 43
Absolute Convergence 44
Absolute Convergence Implies Convergence 44
The Ratio Test 44
Examples: 44
Testing Convergence at Endpoints 45
The Integral Test 45
Harmonic Series and p-series 45
The Limit Comparison Test (LCT) 45
The Alternating Series Test (Leibniz’s Theorem) 45
The Alternating Series Estimation Theorem 46
Rearrangements of Absolutely Convergent Series 46
Rearrangement of Conditionally Convergent Series 46
How to Test a Power Series for Convergence 46
Examples: 46
Parametric, Vector, and Polar Functions 47
Parametric Functions 47
Derivative at a Point 47
Length of a Smooth Parameterized Curve 47
Surface Area 47
Examples: 48
Vectors 48
Vector, Equal Vector 48
Component Form of a Vector 48
Dot Product 48
Magnitude 49
Angle Between Two Vectors 49
Examples: 49
Limit 49
Continuity are a point 49
Component Test for Continuity at a Point 49
Velocity, Speed, Acceleration, Direction of Motion 49
Examples: 50
Modeling Projectile Motion 50
Height, Flight Times, and Range for Ideal Projectile Motion 50
Projectile Motion with Linear Drag 50
Examples: 51
Polar Curves 51
Polar Coordinates 51
Equations Relating Polar and Cartesian Coordinates 51
Examples: 51
Slope of the Curve r = f(θ) 51
Area in Polar Coordinates 51
Area Between Polar Curves 52
Length of a Polar Curve 52
Area of a Surface of Revolution 52
Examples: 52
v
General Calculus Ideas
Rates of Change
Speeds
Average Speed – found by dividing the distance covered by the elapsed time
è y=16t²,
Instantaneous Speed – found by calculating the speed at a specific time
Tangent Lines
The slope of the curve y = f(x) at the point P(a, f(a)) is the number , provided the limit exists. The tangent line to the curve at P is the line through P with this slope.
Examples:
Let f(x) = 1/x.
(a) The slope at x = a is
(b) The slope will be -1/4 if
Limits
Limit – Let c and L be real numbers. The function f has limit l as x approaches c if, given any positive number ε, there is a positive number δ such that for all x,
...
Properties of Limits
If L, M, c, and k are real numbers and
and then
1. Sum Rule:
The limit of the sum of two functions is the sum of their limits.
2. Difference Rule:
The limit of the difference of two functions is the difference of their limits.
3. Product Rule:
The limit of a product of two functions is the product of their limits.
4. Constant Multiple Rule:
The limit of a constant times a function is the constant times the limit of the function.
5. Quotient Rule:
The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.
6. Power Rule:
If r and s are integers, s ≠ 0, then (see above) provided that is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number.
Examples:
(a) Sum & Difference Rule
Product & Multiple Rule
(b) Quotient Rule
Sum & Difference Rule
Product Rule
One-sided and Two-sided Limits
Right-hand: The limit of f as x approaches c from the right.
Left-hand: The limit of f as x approaches c from the left.
A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exists and are equal.
Sandwich Theorem
If g(x) ≤ f(x) ≤ h(x) for all x ≠ c in some interval about c, and
, then
Examples:
Show that
and
Because , the Sandwich Theorem gives
Limits Involving Infinity
The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either
The line x = a is a vertical asymptote of the graph of a function y = f(x) if either
The function g is a right end behavior model for f if and only if or a left end behavior model for f if and only if
Examples:
(a) has a horizontal asymptote at y = 2.
(b) has a vertical asymptote at x = 0.
(c)
Continuity
Continuity at a Point
Interior Point: A function y = f(x) is continuous at an interior point c of its domain if
Endpoint: A function y = f(x) is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if , respectively.
Types of discontinuity
Removable discontinuity: limit goes to a point, but the function does not exist there.
Jump discontinuity: the one-sided limits exist, but have different values.
Oscillating discontinuity: it oscillates too much to have a limit as xà0.
Properties of Continuous Functions
If the functions f and g are continuous at x = c, then the following combinations are continuous at x = c:
Sums: f + g
Differences: f – g
Products: f ∙ g
Constant Multiples: k ∙f, for any number k
Quotients: f/g, provided g(c) ≠ 0
Composite of Continuous Functions
If f is continuous at c and g is continuous at f(c), then the composite is continuous at c.
The Intermediate Value Theorem for Continuous Functions
A function y = f(x) that is continuous on a closed interval [a, b] takes on every value between f(a) and f(b). In other words, if y0 is between f(a) and f(b), then y0 = f(c) for some c in [a, b].
Derivatives
Intro to Derivatives
The derivative of the function f with respect to the variable x is the function f’ whose value at x is, provided the limit exists. If f’(x) exists, we say that f is differentiable at x. A function that is differentiable at every point of its domain is a differentiable function.
The derivative of the function f at the point x = a is the limit, provided the limit exists.
A function y = f(x) is differentiable on a closed interval [a, b] if it has a derivative at every interior point of the interval, and if the limits
exist at the endpoints
Examples:
(a) Differentiate f(x) = x3
(b) Differentiate f(x) =
Notation
y’ è “y prime”
è “dy dx” or “the derivative of y with respect to x
è “df dx” or “the derivative of f with respect to x
è “d dx of f at x” or “the derivative of f at x”
Differentiability
1. corner: where one-sided derivatives differ. ()
2. cusp: where the slopes of the secant lines approach ∞ from one side and -∞ from the other side (an extreme corner). (x⅔)
3. vertical tangent: where the slopes of the secant lines approach either ∞ or -∞ from both sides. ()
4. discontinuity
Differentiability Implies Local Linearity
Differentiability Implies Continuity: If f has a derivative at x = a, then f is continuous at
x = a.
Intermediate Value Theorem for Derivatives: If a and b are any two points in an interval on which f is differentiable, the f’ takes on every value between f’(a) and f’(b).
Examples:
(a) Prove that
Rules for Differentiation
Rule 1: Derivative of a Constant Function
If f is the function with the constant value c, then
Rule 2: Power Rule for Positive Integer Powers of x
If n is a positive integer, then
Rule 3: The Constant Multiple Rule
If u is a differentiable function of x and c is a constant, then
Rule 4: The Sum and Difference Rule
If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points,