Calculus Concepts Quiz

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Questions and Answers

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?

  • 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?

The rate of change of velocity over time.

What is amplitude?

<p>The distance a function is from the axis.</p> Signup and view all the answers

What is an approximation?

<p>An approximate calculation of quantity or degree or worth.</p> Signup and view all the answers

What are asymptotes?

<p>A line or curve that the graph follows closely but never touches.</p> Signup and view all the answers

Average rate of change is defined as: (f(x)_2 - f(x)_1) / (x_2 - x_1)

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The formula for average value is: (1/(b-a)) ∫f(x)dx from a to b

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What is an axis of rotation?

<p>A line about which a figure is rotated to create a solid.</p> Signup and view all the answers

What is the axis of symmetry?

<p>A line of symmetry for a graph.</p> Signup and view all the answers

What is bounded in relation to functions?

<p>When there is a range on a function or set of numbers.</p> Signup and view all the answers

What does the chain rule state?

<p>f'(g(x))g'(x)</p> Signup and view all the answers

What is a closed interval?

<p>An interval that contains its endpoints (D)</p> Signup and view all the answers

What is composition in calculus?

<p>Combining two functions by substituting one function's formula in place of each x in the other function's formula.</p> Signup and view all the answers

What does concave down indicate?

<p>A part of the graph that looks like a frown.</p> Signup and view all the answers

What does concave up indicate?

<p>A part of the graph that looks like a smile.</p> Signup and view all the answers

What is a constant function?

<p>y = constant.</p> Signup and view all the answers

What does continuity refer to in graphs?

<p>When the graph of a function is continuous.</p> Signup and view all the answers

What is a critical point?

<p>A point on the graph of a function at which the derivative is either 0 or undefined.</p> Signup and view all the answers

What is a critical value?

<p>The x-value of a critical point.</p> Signup and view all the answers

What defines a decreasing function?

<p>A function with a graph that moves downward from left to right.</p> Signup and view all the answers

What is a derivative?

<p>The slope of the line tangent to a function.</p> Signup and view all the answers

What is differentiability?

<p>When a function has a well-defined derivative for each element of the domain.</p> Signup and view all the answers

What is differentiation?

<p>Obtaining the derivative of a function.</p> Signup and view all the answers

What defines a discontinuity?

<p>A point at which the graph is not continuous.</p> Signup and view all the answers

What is distance in mathematical terms?

<p>The interval between two points.</p> Signup and view all the answers

What is domain?

<p>The set of x-values for which a function is defined.</p> Signup and view all the answers

What are endpoints?

<p>Points on the end of a function.</p> Signup and view all the answers

What characterizes an even function?

<p>Symmetric with respect to the y-axis.</p> Signup and view all the answers

What is the first derivative test?

<p>Determines whether a point is a minimum, maximum, or neither.</p> Signup and view all the answers

What is implicit differentiation?

<p>y' = dy/dx</p> Signup and view all the answers

What defines an increasing function?

<p>A function that moves upward from left to right.</p> Signup and view all the answers

What is an inflection point?

<p>Where the function changes from concave up to concave down or vice versa.</p> Signup and view all the answers

What does instantaneous rate of change refer to?

<p>The rate of change (value of the derivative) at a particular moment.</p> Signup and view all the answers

What is instantaneous velocity?

<p>The rate at which an object is moving at a particular moment.</p> Signup and view all the answers

What is an inverse function?

<p>A function obtained by switching the x and y variables in a function.</p> Signup and view all the answers

What is a left-hand limit?

<p>The value that a function is approaching as x approaches a given value through values less than x.</p> Signup and view all the answers

What is l'Hopital's rule?

<p>Finding the derivative of the numerator and denominator to evaluate the limit of a function.</p> Signup and view all the answers

What is a limit in calculus?

<p>The value that a function approaches as the domain variable approaches a specific value.</p> Signup and view all the answers

What does limit at infinity refer to?

<p>Numerator is higher degree = does not exist. Numerator and denominator are same = division of coefficients. Denominator is higher degree = 0.</p> Signup and view all the answers

What are local extrema?

<p>A point where the graph has a peak or a valley.</p> Signup and view all the answers

What is a maximum in calculus?

<p>The highest point on a graph.</p> Signup and view all the answers

What is the mean value?

<p>The average of a set of numbers.</p> Signup and view all the answers

What does the mean value theorem state?

<p>f'(c) = (f(b) - f(a)) / (b - a)</p> Signup and view all the answers

What is a minimum in calculus?

<p>The lowest point on a graph.</p> Signup and view all the answers

What does monotonic refer to?

<p>A function that is only increasing or only decreasing.</p> Signup and view all the answers

What is a non-removable discontinuity?

<p>A point at which a graph is connected.</p> Signup and view all the answers

What is a normal line?

<p>The line perpendicular to a function.</p> Signup and view all the answers

What characterizes an odd function?

<p>A function with a graph that is symmetric with respect to the origin.</p> Signup and view all the answers

What defines a one-to-one function?

<p>Every number in the range corresponds to one number in the domain.</p> Signup and view all the answers

What is an open interval?

<p>Does not include its endpoints.</p> Signup and view all the answers

What is optimization?

<p>A process that maximizes or minimizes some quantity to have the best fit.</p> Signup and view all the answers

The product rule is given by: (uv) = u'v + uv'

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The quadratic formula is expressed as: -b ± √(b² - 4ac) / 2a

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The quotient rule is given as: lo(hi') - hi(lo') / lo²

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What is the range in relation to functions?

<p>The set of y-values of a function.</p> Signup and view all the answers

What defines the rate of change?

<p>Slope of a function.</p> Signup and view all the answers

What are related rates?

<p>Rates of change are related by differentiation.</p> Signup and view all the answers

What is relative minimum?

<p>The lowest point in a particular section of the graph.</p> Signup and view all the answers

What is relative maximum?

<p>The highest point in a particular section of the graph.</p> Signup and view all the answers

What is a removable discontinuity?

<p>Hole in the graph.</p> Signup and view all the answers

What is a right-hand limit?

<p>The value that a function is approaching as x approaches a given value through values more than x.</p> Signup and view all the answers

What is right-hand sum?

<p>A sum of the area under a curve where the domain is divided into sub-intervals and the height of each rectangle is the function value at the right endpoint of the sub-interval.</p> Signup and view all the answers

What is the second derivative?

<p>Gives points of inflection.</p> Signup and view all the answers

What does the second derivative test do?

<p>Determines whether the function is concave down, or neither at a point.</p> Signup and view all the answers

What defines slope?

<p>Change of steepness in a graph.</p> Signup and view all the answers

What is speed in mathematical terms?

<p>Distance covered per unit of time.</p> Signup and view all the answers

What is symmetry?

<p>Having the same shape, size, and position on both sides of a dividing line.</p> Signup and view all the answers

What is a tangent line?

<p>A line that touches a curve at a point without crossing the curve.</p> Signup and view all the answers

What is a unit circle?

<p>A circle whose center is at the origin and has a radius of one.</p> Signup and view all the answers

What is velocity?

<p>The rate of change in the position of an object.</p> Signup and view all the answers

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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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