Cal rules final

Theorem 1.5.5

If limx->c f(x) = L and if the function f is continuous at L, then limit symbol can be moved through a lim, f(g(x)) = f(L).

Theorem 1.5.6

(a) If the function g is continuous at c, and the function f is continuous at g(c), then the composition fog is continuous at c. (b) If the function g is continuous everywhere and the function f is continuous everywhere, then the composition fog is continuous everywhere.

Continuity of Trigonometric Functions

The graphs of sin x and cos x are drawn as continuous curves.

Theorem 1.6.1

If C is any number in the natural domain of the stated trigonometric function, then the six basic trigonometric functions (sin x, cos x, tan x, sec x, csc x, cot x) are continuous on their domains.

Squeezing Theorem

If g and h have the same limit as x approaches c, one-sided limits and limits at +∞ and -∞, then f also has this limit as x approaches c.

Definition 2.1.1

The tangent line to the curve y = f(x) at the point P(x0, f(x0)) is the line with equation y - f(x0) = m(x - x0), where m is the slope of the tangent line.

Definition 2.2.1

The derivative of f with respect to x, denoted as f', is the function defined by the formula f'(x) = lim(h->0) (f(x+h) - f(x))/h.

Differentiability

The property of a function where the limit that defines its derivative exists at certain points in its domain.

Derivative

The rate of change of a function at a specific point, defined as the limit of the difference quotient as the interval approaches zero.

Tangent line

A line that touches the graph of a function at a specific point and has the same slope as the derivative of the function at that point.

Corner point

A point on the graph of a function where the secant lines from that point to other points do not approach a unique nonvertical limiting position.

Point of vertical tangency

A point on the graph of a function where the secant lines approach positive or negative infinity from both the left and right sides.

Continuity

The property of a function where there are no abrupt changes or breaks in its graph.

Theorem

A statement that has been proven to be true.

Constant function

A function that always returns the same value, represented by a horizontal line on its graph.

Power function

A function where the variable is raised to a constant exponent.

Power Rule

A rule that states the derivative of a power function is equal to the constant exponent multiplied by the variable raised to the exponent minus one.

Constant Multiple Rule

A rule that states the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

Sum and Difference Rules

Rules that state the derivatives of sums and differences of functions are equal to the sums and differences of their derivatives.

Product Rule

A rule that states the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Quotient Rule

A rule that states the derivative of the quotient of two functions is equal to the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Differentiation Rules

A set of rules that determine the derivatives of various types of functions.

Derivatives

The rate at which a function changes at a specific point.

Trigonometric functions

Functions that relate angles to the ratios of the sides of a right triangle.

Radians

A unit of measurement for angles, where one radian is equal to the angle subtended by an arc of length equal to the radius of the circle.

Addition formula for sine

A formula that expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles.

Formulas (1) and (2)

Formulas for the derivatives of the sine function.

Formulas (3) and (4)

Formulas for the derivatives of the cosine function.

Chain rule

A rule that allows us to find the derivative of a composition of functions.

Composition of functions

The combination of two functions, where the output of one function becomes the input of the other function.

Generalized derivative formula

A formula that allows us to find the derivative of a function composed with another function.

Tangent line

A line that touches a curve at a single point and has the same slope as the curve at that point.

Theorem

A statement that has been proven to be true.

Relative maxima and minima

The highest and lowest points on a graph within a specific interval.

Relative Extrema

Points on a graph where the function has a maximum or minimum value.

Critical Point

A point in the domain of a function where the function is either not differentiable or has a horizontal tangent line.

Stationary Point

A critical point where the derivative of the function is equal to zero.

First Derivative Test

A test to determine if a critical point is a relative maximum or minimum based on the sign of the derivative on either side of the point.

Second Derivative Test

A test to determine if a critical point is a relative maximum or minimum based on the concavity of the graph of the function.

Rolle's Theorem

A theorem that states if a differentiable function intersects the x-axis at two points, there must be at least one point between them where the derivative is zero.

Mean-Value Theorem

A theorem that states if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the interval where the derivative is equal to the average rate of change of the function over the interval.

Inverse Trigonometric Functions

Functions that are the inverses of restricted trigonometric functions, denoted as sin^(-1)x, cos^(-1)x, tan^(-1)x, sec^(-1)x, arcsin x, arccos x, arctan x, and arcsec x.

Derivative

The rate at which a function is changing at a particular point, represented as the slope of the tangent line to the graph of the function at that point.

Integration

The process of finding the antiderivative of a function, which allows for the calculation of the area under the curve of the function.

Restricted Functions

Functions that have their domains restricted in order to make them one-to-one functions, allowing for the definition of inverse functions.

Inverse Sine Function

The inverse of the restricted sine function, denoted as sin^(-1)x.

Derivative Formula

A formula that allows for the calculation of the derivative of a function, such as [sin^(-1)u]' = [cos^(-1)u].

Chain Rule

A rule in calculus that allows for the calculation of the derivative of a composite function by multiplying the derivatives of the inner and outer functions.

Exponential Function

A function of the form f(x) = bx, where b is a positive constant and x is the variable.

Logarithmic Function

The inverse of an exponential function, represented as f(x) = logb x, where b is the base of the logarithm.

Algebraic Properties of Logarithms

Properties that govern the manipulation of logarithmic expressions, such as the product property, quotient property, and power property.

Derivative

The rate at which a function is changing at a particular point, represented as the slope of the tangent line to the graph of the function at that point.

Inverse Functions

Functions that "undo" the actions of another function, resulting in the original input value.

Derivative Formula

A formula that allows for the calculation of the derivative of a function, such as [In u]' = 1/u.

Independent Variable

The variable in a function that is free to vary and is used as the input to determine the output.

Dependent Variable

The variable in a function that is determined by the value of the independent variable and is the output of the function.

Graph of a Function

The visual representation of a function in the xy-plane, where the x-axis represents the independent variable and the y-axis represents the dependent variable.

Absolute Value Function

The absolute value or magnitude of a real number x, defined as x if x > 0 and -x if x < 0.

Properties of Absolute Value

(a) |-a| = |a A number and its negative have the same absolute value. (b) |ab| = abb The absolute value of a product is the product of the absolute values. (c) |a/b = a/b,b#0 The absolute value of a ratio is the ratio of the absolute values. (d) la+b The triangle inequality.

Function Operations

Given functions f and g, (f+g)(x) = f(x) + g(x), (f - -g)(x) = f(x) - g(x), (fg)(x) = f(x)g(x), (f/g)(x) = f(x)/g(x).

Composition of Functions

The composition of functions f and g, denoted by fog, is the function defined by (fog)(x) = f(g(x)).

Translation Principles

Add a positive constant C to f(x), subtract a positive constant C from f(x), add a positive constant c to x, subtract a positive constant c from x.

Symmetry Tests

(a) A plane curve is symmetric about the y-axis if replacing x by -x in its equation produces an equivalent equation. (b) A plane curve is symmetric about the x-axis if replacing y by -y in its equation produces an equivalent equation. (c) A plane curve is symmetric about the origin if replacing both x by -x and y by -y in its equation produces an equivalent equation.

Inverse Functions

If the functions f and g satisfy the conditions g(f(x)) = x for every x in the domain of f and f(g(y)) = y for every y in the domain of g, then f is an inverse of g and g is an inverse of f.

Changing the Independent Variable

The formulas for inverse functions use different independent variables for f and f-1, but they can also use the same independent variable.

Domain and Range of Inverse Functions

The domain of f-1 is the range of f, and the range of f-1 is the domain of f.

Method for Finding Inverse Functions

If an equation y = f(x) can be solved for x as a function of y, say x = g(y), then f has an inverse and that inverse is g(y) = f-1(y).

One-to-One Functions

A function has an inverse if and only if it is one-to-one.

Limits

The concept of describing how a function behaves as the independent variable approaches a given value.

Function

A mathematical relationship between a set of inputs (independent variable) and a set of outputs (dependent variable).

Independent variable

The variable in a function that is manipulated or controlled.

Dependent variable

The variable in a function that is affected by changes in the independent variable.

Graph

A visual representation of the relationship between the independent and dependent variables in a function.

Table

A tabular representation of the values of the independent and dependent variables in a function.

Two-sided limit

A limit that requires the values of the function to get closer and closer to a specific value as the independent variable approaches a given value from both sides.

One-sided limit

A limit that considers the behavior of the function as the independent variable approaches a given value from either the left or the right side.

Infinite limit

A limit that indicates that the function either increases or decreases without bound as the independent variable approaches a given value.

Corroborating limits

The need for additional methods to confirm or support limits conjectured from numerical evidence.

Theorem

A statement that follows from given conditions and provides a conclusion.

Real number

A number that can be positive, negative, or zero, and can include fractions and decimals.

Limit

The value that a function approaches as the input approaches a certain value.

Rational function

A function that can be expressed as the quotient of two polynomials.

One-sided limits

Limits that describe the behavior of a function as the input approaches a value from one side only.

Infinity

A concept representing a value that is larger than any real number.

Horizontal asymptote

A horizontal line that a function approaches as the input approaches infinity or negative infinity.

Limit laws

Rules that govern the behavior of limits in mathematical calculations.

Infinite limits

Limits that do not exist because the values of the function increase or decrease without bound.

Polynomial

An expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

Limit definition

A formal definition that states the limit of a function exists if for any given value of epsilon, there exists a corresponding value of delta.

Continuity

The property of a function where there are no breaks or holes in its graph.

Undefined

When a function is not defined at a certain point.

Limit

The value that a function approaches as the input approaches a certain point.

Two-sided limit

A limit that is approached from both the left and right sides of a point.

Open interval

An interval that does not include its endpoints.

Closed interval

An interval that includes its endpoints.

One-sided limit

A limit that is approached from either the left or right side of a point.

Polynomial

A mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

Rational function

A function that is the ratio of two polynomials.

Denominator

The bottom part of a fraction in a rational function.