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Questions and Answers
What is the linear function that expresses temperature T in relation to height h?
What is the linear function that expresses temperature T in relation to height h?
- T = -20h + 10
- T = 10h - 20
- T = 10h + 20
- T = -10h + 20 (correct)
The slope of the temperature function represents an increase in temperature with increasing height.
The slope of the temperature function represents an increase in temperature with increasing height.
False (B)
What is the temperature at a height of 2.5 km according to the derived linear function?
What is the temperature at a height of 2.5 km according to the derived linear function?
-5°C
A polynomial of degree 1 is a ______ function.
A polynomial of degree 1 is a ______ function.
Match the following polynomial degrees with their corresponding function type:
Match the following polynomial degrees with their corresponding function type:
What does the negative slope of -10°C/km indicate in the temperature function?
What does the negative slope of -10°C/km indicate in the temperature function?
The domain of any polynomial function includes all real numbers.
The domain of any polynomial function includes all real numbers.
What is the leading coefficient of the polynomial P(x) = -2x^3 + 3x^2 - x + 5?
What is the leading coefficient of the polynomial P(x) = -2x^3 + 3x^2 - x + 5?
What is the absolute value of -5?
What is the absolute value of -5?
All absolute values are greater than or equal to zero.
All absolute values are greater than or equal to zero.
What type of function is f(x) = |x|?
What type of function is f(x) = |x|?
How can the limit notation 'lim as x approaches a of f(x) = ∞' be alternatively expressed?
How can the limit notation 'lim as x approaches a of f(x) = ∞' be alternatively expressed?
The cost C(w) of mailing a large envelope is $____ when the weight is between 1 and 2 pounds.
The cost C(w) of mailing a large envelope is $____ when the weight is between 1 and 2 pounds.
Match the following weights with their corresponding mailing costs:
Match the following weights with their corresponding mailing costs:
The symbol ∞ is considered a number in limit notation.
The symbol ∞ is considered a number in limit notation.
What does 'lim as x approaches a of f(x) = -∞' indicate about the behavior of the function f(x)?
What does 'lim as x approaches a of f(x) = -∞' indicate about the behavior of the function f(x)?
Which of the following represents the piecewise definition of the absolute value function?
Which of the following represents the piecewise definition of the absolute value function?
The function f(x) = x² is an odd function.
The function f(x) = x² is an odd function.
The vertical asymptotes of __________ are found where cos x = 0.
The vertical asymptotes of __________ are found where cos x = 0.
Match the types of limits with their descriptions:
Match the types of limits with their descriptions:
What is the geometric significance of an even function?
What is the geometric significance of an even function?
What does the graph of y = -f(x) represent?
What does the graph of y = -f(x) represent?
A transformation where the function is multiplied by a positive number greater than 1 results in a vertical stretch.
A transformation where the function is multiplied by a positive number greater than 1 results in a vertical stretch.
What happens to the graph of a function when we apply the absolute value transformation?
What happens to the graph of a function when we apply the absolute value transformation?
The graph of y = 2 cos x is obtained by stretching the graph of y = cos x vertically by a factor of ______.
The graph of y = 2 cos x is obtained by stretching the graph of y = cos x vertically by a factor of ______.
Match the following transformations with their effects on the graph:
Match the following transformations with their effects on the graph:
How does the transformation y = f(x - 2) affect the graph of f?
How does the transformation y = f(x - 2) affect the graph of f?
The graph of y = f(x) shifts downward if you add a positive constant to f(x).
The graph of y = f(x) shifts downward if you add a positive constant to f(x).
What effect does reflecting a graph about the y-axis have on the coordinates of points?
What effect does reflecting a graph about the y-axis have on the coordinates of points?
Which limit laws are helpful in calculating limits?
Which limit laws are helpful in calculating limits?
The one-sided limits must be equal for a two-sided limit to exist.
The one-sided limits must be equal for a two-sided limit to exist.
What is the primary condition for a function to be continuous at a point a?
What is the primary condition for a function to be continuous at a point a?
The _________ Theorem states that if g(x) is squeezed between f(x) and h(x), then g must have the same limit as both f and h.
The _________ Theorem states that if g(x) is squeezed between f(x) and h(x), then g must have the same limit as both f and h.
Match the following functions with their properties:
Match the following functions with their properties:
Which statement about the greatest integer function is true?
Which statement about the greatest integer function is true?
The Direct Substitution Property can be used for all functions when calculating limits.
The Direct Substitution Property can be used for all functions when calculating limits.
What is indicated by the notation 'lim x → a f(x)'?
What is indicated by the notation 'lim x → a f(x)'?
What is the derivative of the function $f(x) = 5x^6$?
What is the derivative of the function $f(x) = 5x^6$?
The derivative of a sum of functions is equal to the sum of their derivatives.
The derivative of a sum of functions is equal to the sum of their derivatives.
What rule is used to derive the derivative of a product of two functions?
What rule is used to derive the derivative of a product of two functions?
The derivative of $x^4$ is ______.
The derivative of $x^4$ is ______.
Which of the following is the result of the derivative $g(x) = 2x^3 + 5x^2 - x$?
Which of the following is the result of the derivative $g(x) = 2x^3 + 5x^2 - x$?
The Constant Multiple Rule states that the derivative of a constant times a function is equal to the function's derivative multiplied by the constant.
The Constant Multiple Rule states that the derivative of a constant times a function is equal to the function's derivative multiplied by the constant.
What is the derivative of the function $y = -3x^4$?
What is the derivative of the function $y = -3x^4$?
Match the following derivatives with their corresponding functions:
Match the following derivatives with their corresponding functions:
Flashcards
Absolute Value
Absolute Value
The distance of a number from zero on the number line.
Piecewise Defined Function
Piecewise Defined Function
A function described by different formulas for different parts of its domain.
Even Function
Even Function
A function where f(-x) = f(x) for all x in its domain. Its graph is symmetric across the y-axis.
Odd Function
Odd Function
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f(x) = |x|
f(x) = |x|
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Step Function
Step Function
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|| (Vertical Bars)
|| (Vertical Bars)
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C(w) - Cost of Mailing a Large Envelope
C(w) - Cost of Mailing a Large Envelope
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Reflection About the x-axis
Reflection About the x-axis
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Vertical Stretching/Shrinking
Vertical Stretching/Shrinking
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Graph of y = |f(x)|
Graph of y = |f(x)|
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Limit of a Function
Limit of a Function
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Linear Temperature Model
Linear Temperature Model
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Slope of the Temperature Function
Slope of the Temperature Function
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Polynomial Function
Polynomial Function
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Degree of a Polynomial
Degree of a Polynomial
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Quadratic Function
Quadratic Function
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Parabola
Parabola
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Empirical Model
Empirical Model
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Y-intercept
Y-intercept
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f(x) as x a
f(x) as x a
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Vertical Asymtote
Vertical Asymtote
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limxa f(x) =
limxa f(x) =
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limxa- f(x) = L
limxa- f(x) = L
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limxa+ f(x) = L
limxa+ f(x) = L
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limxa f(x) = –
limxa f(x) = –
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Two-Sided Limit
Two-Sided Limit
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One-Sided Limit
One-Sided Limit
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Continuous function
Continuous function
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Constant Multiple Rule
Constant Multiple Rule
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Sum Rule
Sum Rule
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Left-hand limit
Left-hand limit
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Difference Rule
Difference Rule
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Right-hand limit
Right-hand limit
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Product Rule
Product Rule
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Derivative of a Constant
Derivative of a Constant
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Limit Theorem
Limit Theorem
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Power Rule
Power Rule
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Differentiating Complex Functions
Differentiating Complex Functions
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Squeeze Theorem
Squeeze Theorem
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Direct Substitution Property
Direct Substitution Property
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Derivative as Rate of Change
Derivative as Rate of Change
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Study Notes
Precalculus Review
- This document is a precalculus review, providing key algebraic concepts.
- Real numbers, denoted ℝ, encompass all numbers on the number line (positive, negative, zero, fractions, etc.).
- Rational numbers (ℚ) can be expressed as fractions (e.g., -0.5).
- Integers (ℤ) are whole numbers (e.g., -4, -3, -2, -1, 0, 1, 2).
- Natural numbers (ℕ) are positive integers (e.g., 1, 2, 3).
- Infinity (∞) represents values that become arbitrarily large. Negative infinity (-∞) represents values tending to negative infinity.
- Intervals are collections of real numbers between two fixed numbers (endpoints may or may not be included).
- Specific interval notations are defined, with explanations of open, closed, half-open intervals and unbounded intervals.
- The absolute value function|x| returns the distance of x from 0.
- Key algebraic formulas for squares and products of binomials are presented.
- Solving quadratic equations, where appropriate to solving using factoring is reviewed.
- Fractions and rational expressions, including operations and simplification techniques.
- Functions defined piecewise are reviewed (with solid/hollow dots).
- Functions are explained verbally, numerically, graphically, or algebraically.
- The concepts of domain, range, and graph of function.
- Definition of a function is a rule that assigns each element in a set D (the domain) to exactly one element in a set E (the range).
1.1 Algebra
- Review of specific types of intervals in mathematical context.
- Review of absolute value function properties and how to apply them to various calculations.
- Review of algebra formulas for multiplying and factoring various polynomials.
- Thorough review of how to solve quadratic equations.
1.2 Functions and Graphs
- Review of different ways of representing a function.
- Explanations about domain, range, and how to read information from function graphs.
- Definition of function is review and is stated precisely.
- Explanations on how to determine domain and range from graph.
1.3 Linear Functions
- Linear Function definitions, including slope and y-intercept formulas/definitions.
- Various forms of equations of lines, including the general form, vertical lines, and non-vertical lines.
- Review of parallel and perpendicular lines.
1.4 Polynomials
- Definition of a polynomial.
- Explanation on coefficients, leading coefficients, and the constant term of a polynomial.
- Polynomial degree definition and example.
- Domain of a polynomial is always all real numbers, if leading coefficient is not zero.
- Review of graphs of polynomial functions and their degrees.
1.5 Power Functions
- Definitions of power functions and how a is evaluated for a= n or if a=1n.
- Explanation on various types of exponents and how to define the numerical value of the corresponding function value.
- Understanding how to evaluate various numerical power functions.
1.6 Trigonometric Functions
- Review of radians and degrees in mathematical context.
- Detailed definitions of all six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant).
- How these are calculated and evaluated.
- Special values of trigonometric functions for common angles.
- Trigonometric identities reviewed.
- Graphs of the sine, cosine, and tangent functions.
1.7 Exponential Functions
- Definition of exponential function (and its base).
- Review of the number e.
- Graphs of exponential functions of the form y = ax where a is a constant.
1.8 Logarithmic Functions
- Definition of logarithmic function (and its base).
- Review of the number e and its inverse function relation with exponential function.
- Definition of common and natural logarithmic functions, and graphs of such functions with various bases (e.g. 2, 3, 5 , and 10).
- Review basic logarithmic formulas.
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