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Questions and Answers
What is the absolute minimum?
What is the absolute minimum?
- A function that is only increasing
- The point where a graph is not continuous
- The lowest point of a function (correct)
- The highest point of a function
What does absolute value represent?
What does absolute value represent?
- The distance between a number and the origin (correct)
- The rate of change of velocity
- The highest point of a function
- The lowest point of a function
What is the definition of acceleration?
What is the definition of acceleration?
The rate of change of velocity over time.
What is amplitude?
What is amplitude?
What is an approximation?
What is an approximation?
What are asymptotes?
What are asymptotes?
Average rate of change is defined as: (f(x)_2 - f(x)_1) / (x_2 - x_1)
Average rate of change is defined as: (f(x)_2 - f(x)_1) / (x_2 - x_1)
The formula for average value is: (1/(b-a)) ∫f(x)dx from a to b
The formula for average value is: (1/(b-a)) ∫f(x)dx from a to b
What is an axis of rotation?
What is an axis of rotation?
What is the axis of symmetry?
What is the axis of symmetry?
What is bounded in relation to functions?
What is bounded in relation to functions?
What does the chain rule state?
What does the chain rule state?
What is a closed interval?
What is a closed interval?
What is composition in calculus?
What is composition in calculus?
What does concave down indicate?
What does concave down indicate?
What does concave up indicate?
What does concave up indicate?
What is a constant function?
What is a constant function?
What does continuity refer to in graphs?
What does continuity refer to in graphs?
What is a critical point?
What is a critical point?
What is a critical value?
What is a critical value?
What defines a decreasing function?
What defines a decreasing function?
What is a derivative?
What is a derivative?
What is differentiability?
What is differentiability?
What is differentiation?
What is differentiation?
What defines a discontinuity?
What defines a discontinuity?
What is distance in mathematical terms?
What is distance in mathematical terms?
What is domain?
What is domain?
What are endpoints?
What are endpoints?
What characterizes an even function?
What characterizes an even function?
What is the first derivative test?
What is the first derivative test?
What is implicit differentiation?
What is implicit differentiation?
What defines an increasing function?
What defines an increasing function?
What is an inflection point?
What is an inflection point?
What does instantaneous rate of change refer to?
What does instantaneous rate of change refer to?
What is instantaneous velocity?
What is instantaneous velocity?
What is an inverse function?
What is an inverse function?
What is a left-hand limit?
What is a left-hand limit?
What is l'Hopital's rule?
What is l'Hopital's rule?
What is a limit in calculus?
What is a limit in calculus?
What does limit at infinity refer to?
What does limit at infinity refer to?
What are local extrema?
What are local extrema?
What is a maximum in calculus?
What is a maximum in calculus?
What is the mean value?
What is the mean value?
What does the mean value theorem state?
What does the mean value theorem state?
What is a minimum in calculus?
What is a minimum in calculus?
What does monotonic refer to?
What does monotonic refer to?
What is a non-removable discontinuity?
What is a non-removable discontinuity?
What is a normal line?
What is a normal line?
What characterizes an odd function?
What characterizes an odd function?
What defines a one-to-one function?
What defines a one-to-one function?
What is an open interval?
What is an open interval?
What is optimization?
What is optimization?
The product rule is given by: (uv) = u'v + uv'
The product rule is given by: (uv) = u'v + uv'
The quadratic formula is expressed as: -b ± √(b² - 4ac) / 2a
The quadratic formula is expressed as: -b ± √(b² - 4ac) / 2a
The quotient rule is given as: lo(hi') - hi(lo') / lo²
The quotient rule is given as: lo(hi') - hi(lo') / lo²
What is the range in relation to functions?
What is the range in relation to functions?
What defines the rate of change?
What defines the rate of change?
What are related rates?
What are related rates?
What is relative minimum?
What is relative minimum?
What is relative maximum?
What is relative maximum?
What is a removable discontinuity?
What is a removable discontinuity?
What is a right-hand limit?
What is a right-hand limit?
What is right-hand sum?
What is right-hand sum?
What is the second derivative?
What is the second derivative?
What does the second derivative test do?
What does the second derivative test do?
What defines slope?
What defines slope?
What is speed in mathematical terms?
What is speed in mathematical terms?
What is symmetry?
What is symmetry?
What is a tangent line?
What is a tangent line?
What is a unit circle?
What is a unit circle?
What is velocity?
What is velocity?
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Study Notes
Absolute Properties
- Absolute Minimum: Represents the lowest point of a function on its domain.
- Absolute Maximum: Indicates the highest point of a function within its domain.
Mathematical Concepts
- Absolute Value: Measures the distance of a number from the origin without considering direction.
- Acceleration: Describes the rate at which an object's velocity changes over time.
Graphical Features
- Amplitude: Reflects how far a function deviates from its central axis.
- Asymptotes: Lines that a graph approaches but never intersects.
Rates and Values
- Average Rate of Change: Calculated as (f(x₂) - f(x₁)) / (x₂ - x₁), providing a measure of change over an interval.
- Average Value: Given by (1/(b-a)) ∫f(x)dx from a to b, representing the mean of the function over an interval.
Geometric Characteristics
- Axis of Rotation: A defining line around which a figure is rotated to form a solid.
- Axis of Symmetry: A line that divides a graph into mirrored halves.
Function Types and Properties
- Closed Interval: Includes its endpoints, defining a range that contains both limits.
- Critical Point: Occurs where the derivative of a function is zero or undefined, indicating potential maxima or minima.
Graph Behavior
- Concave Up: A portion of the graph that curves upward, resembling a smile.
- Concave Down: A segment that curves downward, resembling a frown.
Continuity and Differentiation
- Continuity: Means the function's graph can be drawn without any breaks or gaps.
- Derivative: Indicates the slope of the tangent line at a point on the graph.
Function Analysis
- Increasing Function: A function whose graph rises as it moves from left to right.
- Decreasing Function: A function whose graph falls as it moves from left to right.
Limits and Extrema
- Limit: The value a function approaches as the input approaches a specific point.
- Local Extrema: Points on the graph that represent local maximum or minimum values.
Special Functions and Rules
- Even Function: Symmetric across the y-axis.
- Odd Function: Symmetric about the origin.
Calculus Rules
- Chain Rule: Enables differentiation of composite functions, f'(g(x))g'(x).
- Product Rule: Used for finding the derivative of products, expressed as (uv)' = u'v + uv'.
- Quotient Rule: For differentiating a quotient, represented as (lo(hi') - hi(lo')) / lo².
Discontinuity Types
- Removable Discontinuity: Identified as gaps or holes in the graph.
- Non-removable Discontinuity: Occurs where the graph remains connected but discontinuous.
Optimization and Rates
- Optimization: The process of finding maximum or minimum values for functions based on conditions.
- Related Rates: A concept where different rates of change are interconnected through differentiation.
Limits at Infinity
- Limit at Infinity: Evaluates function behavior as it extends indefinitely in either direction.
- Right-Hand Limit: The limit evaluated as x approaches from values greater than the given point.
- Left-Hand Limit: The limit assessed as x approaches from values less than the specified point.
Miscellaneous
- Unit Circle: A circle centered at the origin with a unit radius of one.
- Tangent Line: A line that touches a curve at a single point, representing the instantaneous rate of change.
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