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Questions and Answers
What can be concluded if a function f has critical numbers at c where f'(c) = 0?
What can be concluded if a function f has critical numbers at c where f'(c) = 0?
- The function is increasing at c.
- The function is constant at c.
- The function is decreasing at c.
- The function could be either increasing, decreasing, or neither at c. (correct)
Given a function f where f'(x) < 0 for all x in the interval I, which statement is accurate?
Given a function f where f'(x) < 0 for all x in the interval I, which statement is accurate?
- f is strictly increasing over I.
- f is decreasing over I. (correct)
- f has a maximum on I.
- f must have a critical number in I.
Which of the following functions exhibits a critical number due to the absence of a derivative?
Which of the following functions exhibits a critical number due to the absence of a derivative?
- f(x) = x^2 - 4
- f(x) = 2x/(1 - x²)
- f(x) = 3x² - 12x
- f(x) = 8x√(1 - x²) (correct)
How can the monotonicity of function f be described if f'(x) = 0 at multiple points and also has positive values elsewhere?
How can the monotonicity of function f be described if f'(x) = 0 at multiple points and also has positive values elsewhere?
If the sign chart indicates that f'(x) ≥ 0 on the intervals [-1,0] and [2,∞), what does this imply about f's behavior?
If the sign chart indicates that f'(x) ≥ 0 on the intervals [-1,0] and [2,∞), what does this imply about f's behavior?
What is the primary purpose of differentiation in the context of finding maximum and minimum values?
What is the primary purpose of differentiation in the context of finding maximum and minimum values?
Which statement accurately describes an increasing function?
Which statement accurately describes an increasing function?
In which scenario might one need to find maximum or minimum values?
In which scenario might one need to find maximum or minimum values?
What does it imply if f(x₁) ≥ f(x₂) for x₁ < x₂ on an interval?
What does it imply if f(x₁) ≥ f(x₂) for x₁ < x₂ on an interval?
Which of the following describes the behavior of a function that is strictly decreasing?
Which of the following describes the behavior of a function that is strictly decreasing?
Flashcards
Increasing function
Increasing function
If, for any two points x₁ and x₂ within an interval I, where x₁ < x₂, the function f(x₁) is less than or equal to f(x₂), then the function f(x) is said to be increasing on I.
Decreasing function
Decreasing function
If, for any two points x₁ and x₂ within an interval I, where x₁ < x₂, the function f(x₁) is greater than or equal to f(x₂), then the function f(x) is said to be decreasing on I.
Strictly increasing function
Strictly increasing function
If, for any two points x₁ and x₂ within an interval I, where x₁ < x₂, the function f(x₁) is less than f(x₂), then the function f(x) is said to be strictly increasing on I.
Strictly decreasing function
Strictly decreasing function
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Geometric interpretation of increasing and decreasing functions
Geometric interpretation of increasing and decreasing functions
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Monotonic Function
Monotonic Function
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Critical Number
Critical Number
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Finding Intervals of Increase/Decrease
Finding Intervals of Increase/Decrease
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Study Notes
Introduction to Calculus
- Differentiation of functions helps solve quantity problems involving maximum and minimum values.
- This is useful in engineering and technology for calculating maximum capacity or minimum cost of a cylinder with given surface area.
- Problems can also consider dimensions for maximum area with a given perimeter.
Increasing and Decreasing Functions
- Activity 2.12: Find zeros of functions, identify increasing/decreasing intervals from a graph, and solve inequalities using sign charts.
- Find real zeros of functions like f(x) = 4x - 8, f(x) = x² - x - 12, and f(x) = 1/√(x − 2)²
- Identify intervals where graphs rise or fall from left to right.
- Solve inequalities like 2x² + 3x − 220 and x² + 3x − 4 < 0 using sign charts.
Definition 2.8
- Increasing: If x₁ < x₂, then f(x₁) ≤ f(x₂).
- Decreasing: If x₁ < x₂, then f(x₁) ≥ f(x₂).
- Strictly Increasing: If x₁ < x₂, then f(x₁) < f(x₂).
- Strictly Decreasing: If x₁ < x₂, then f(x₁) > f(x₂).
- Geometrically, a function is increasing if its graph rises and decreasing if it falls as x moves to the right.
Increasing and Decreasing Test
- Increasing: If f'(x) ≥ 0 for all x in the interior of interval I, then f is increasing on I.
- Decreasing: If f'(x) ≤ 0 for all x in the interior of interval I, then f is decreasing on I.
- Strictly Increasing: If f'(x) > 0 and f'(x) does not equal zero for all x in the interior of I, then f is strictly increasing on I.
- Strictly Decreasing: If f'(x) < 0 and f'(x) does not equal zero for all x in the interior of I, then f is strictly decreasing on I.
- Monotonic: A function that's either increasing or decreasing.
Definition 2.9
- Critical number: A number c in the domain of a function f is a critical number if either f'(c) = 0 or f has no derivative at c.
- Example 1: Find critical numbers for functions like f(x) = 4x³ − 5x² − 8x + 20 and f(x) = 2√x(6 − x).
Example 2:
- Find interval where function is increasing or decreasing. Given f(x) = 3x² − 4x³ − 12x² + 5.
Definition 2.10
- Relative Maximum: If there exists an open interval (a, b) containing c such that f(x) < f(c) for all x in (a, b) other than c, then f(c) is a relative maximum value of f and (c, f(c)) is a relative maximum point.
- Relative Minimum: If f(x) > f(c) for all x in (a, b) other than c, then f(c) is a relative minimum value of f and (c, f(c)) is a relative minimum point.
Definition 2.11
- Absolute Maximum: If c is in the domain of f and for all x in the domain of the function, f(x) ≤ f(c), then (c, f(c)) is an absolute maximum point of f.
- Absolute Minimum: If for all x in the domain, f(x) ≥ f(c), then (c, f(c)) is an absolute minimum point of f.
Procedure to Find Absolute Extremes on Closed Intervals
- Find critical numbers.
- Evaluate f at critical numbers and endpoints.
- Choose the largest and smallest values. These are the maximum and minimum of f on [a, b].
Example:
- Find absolute maximum and minimum values of f(x) = x² − 2x on [-1, 2].
Exercise 2.15
- Find absolute maximum and minimum values of various functions on specified intervals (e.g., f(x) = x² + 2x + 3, [−2, 2]).
The First Derivative Test
- If the sign of f' changes from negative to positive at c, f has a relative minimum at c.
- If the sign of f' changes from positive to negative at c, f has a relative maximum at c.
- If the sign of f' doesn't change at c, f has no relative maximum or minimum at c.
Example 1
- Find relative maximum and minimum values of f(x) = 2x³ + 3x² − 12x − 3.
Example 1, part b:
- Find relative maximum and minimum values of f(x) = (x - 1)²(x - 3)².
Exercise 2.16
- Tangent Lines and Normal Lines to Curves
- Equation of a tangent line at a given point and normal line at that point.
- Relationship between slopes of tangent lines and normal lines.
Example 1 (Cont.)
- Various examples calculating tangent and normal lines to given functions at specific points (e.g., f(x) = x² − 2x at (1,-1)).
Activity 2.14
- Find equations of lines containing specified points (e.g., (1,3) and (-2,3)).
- Find equations of lines perpendicular to the first line found and containing a given point (e.g., (1,3)).
- Write the equation of a tangent line given its slope at a point (e.g., (2,3) and slope = 3.).
- Relate slopes of a tangent line and a normal line.
Note
- If the first derivative of f at x = a is 0, then ƒ has a horizontal tangent line and a vertical normal line x = a.
- If the first derivative of f at x = a is undefined, then f has a vertical tangent line x = a and a horizontal normal line.
Example 2
- Find all points where f(x) = √(x² − 6x) has a vertical tangent.
Exercise 2.17
- Find tangent and normal lines to given functions at specified points.
- Identify points where functions have vertical tangent lines.
- Find values of constants for functions with tangent line overlap conditions.
Applications of Derivatives
- Find different quantities in relation to time, using rates of change (Examples provided of finding rate of change of area of circle given rate of change of radius and rate of change of area of equilateral triangle etc.).
- Interpretations of derivatives in sciences (e.g., chemistry, biology).
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