Podcast
Questions and Answers
Describe the intervals where the function is increasing and where it is decreasing, based on the graph provided in example 1. Make sure to include any endpoints or local extrema.
Describe the intervals where the function is increasing and where it is decreasing, based on the graph provided in example 1. Make sure to include any endpoints or local extrema.
The function is increasing on the interval (-∞, -2] and decreasing on the interval [-2, ∞). The function reaches a local maximum at the point (-2, 3).
Given the graph in example 2, describe the intervals where the function is increasing and decreasing. Be sure to include any endpoints or local extrema you observe.
Given the graph in example 2, describe the intervals where the function is increasing and decreasing. Be sure to include any endpoints or local extrema you observe.
The function is increasing on the interval (0, ∞) and decreasing on the interval (-∞, 0). The function reaches a local minimum at the point (0, 1) but is not defined at this point since there is a hole in the graph.
Explain how to determine whether a function is increasing or decreasing on a given interval using its derivative.
Explain how to determine whether a function is increasing or decreasing on a given interval using its derivative.
If the derivative of a function is positive on an interval, then the function is increasing on that interval. Conversely, if the derivative is negative, the function is decreasing on that interval.
Consider the graph in example 1. What is the derivative of the function at the point x = -2? Explain your reasoning.
Consider the graph in example 1. What is the derivative of the function at the point x = -2? Explain your reasoning.
If a function is increasing on the interval (a, b), what can you say about the value of its derivative on that interval? Explain your reasoning.
If a function is increasing on the interval (a, b), what can you say about the value of its derivative on that interval? Explain your reasoning.
Flashcards
Increasing Function
Increasing Function
A function where, as the input increases, the output also increases.
Decreasing Function
Decreasing Function
A function where, as the input increases, the output decreases.
Constant Function
Constant Function
A function where the output remains the same regardless of the input.
Local Maximum
Local Maximum
Signup and view all the flashcards
Local Minimum
Local Minimum
Signup and view all the flashcards
Study Notes
Increasing and Decreasing Functions
- Functions can be increasing or decreasing over intervals
- A function is increasing if its gradient is positive (dy/dx > 0 or f'(x) > 0)
- A function is decreasing if its gradient is negative (dy/dx < 0 or f'(x) < 0)
- A function is stationary if its gradient is zero (dy/dx = 0 or f'(x) = 0)
Examples
-
Example 1: Show f(x) = x³ + 6x² + 21x + 2 is increasing for all values of x.
- Find the first derivative: f'(x) = 3x² + 12x + 21
- Show f'(x) > 0 for all values of x by completing the square or noting that the quadratic has a positive leading coefficient and the discriminant is less than zero, meaning no real roots, thus it will always be positive.
-
Example 2: Determine the interval where f(x) = x³ + 2x² + 4x - 9 is decreasing.
- Find the first derivative: f'(x) = 3x² + 4x + 4
- Find the roots of f'(x) = 0. This will give the stationary points.
- The discriminant (b² - 4ac) is negative which demonstrates that there are no real roots for the quadratic, suggesting the quadratic is always positive for all real values of x.
-
Example 3: Show f(x) = x³ + 16x − 2 is increasing for all values of x.
- Find the derivative: f′(x) = 3x² + 16
- Determine that the derivative is always positive as 3x² , for any real value of x, will always be positive, and 16 is always positive, thus the derivative is always positive for all real x values, which indicates the function is increasing for all real values.
-
Example 4: Find the interval where f(x) = x³ + 6x² − 135x is decreasing
- Find the derivative f'(x) = 3x² + 12x − 135
- Set the derivative to zero to find critical points, factor to solve to find the x-values.
- Determine the interval(s) where the derivative is negative. This reveals where the function is decreasing.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
