Over which interval is f(x) increasing for the function f(x) = 5 – 5tani?

  

Not timed but I’m trying to understand these problems as they may appear on a timed quiz due later tonight.
Analyze the trigonometric function over the specified interval, stating where I is increasing, decreasing, concave up, and concave down, and stating the x-coordinates of all inflection points. Confirm that your results are consistent with the graph of generated with a grap
f (x) = 5 –
5- tani over XE( -1,A)
Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.
?
f (x) is increasing over xe
I
?
f (x) is decreasing over xe
?
f (x) is concave up over xe
?
f(x) is concave down over xe
|?
f (x) has an inflection point at x =
For the function / (x) = Inyx2 + 36, answer the following questions.
(a) Find the interval(s) on which is increasing.
Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.
?
f (x) is increasing on xe
(b) Find the interval(s) on which is decreasing.
Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.
f (x) is decreasing on xe
(c) Find the open interval(s) on which is concave up.
Enter your answers in interval notation. If there is no suitable interval, enter “DNE”.
f (x) is concave up on me
(d) Find the open interval(s) on which is concave down.

Introduction: Trigonometric functions and inverse functions of polynomials are often studied in mathematics to understand how different functions behave over specific intervals. Analyzing the intervals of a function helps in identifying where it is increasing or decreasing, concave up or down, and the inflection points. In this article, we will discuss how to analyze and determine the specific intervals of different functions.

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Description:

Section 1: Analyzing Trigonometric Functions
Trigonometric functions like sine, cosine, tangent, etc., have unique properties that can be studied over specific intervals. To analyze a trigonometric function, we need to determine its increasing or decreasing behavior, and its concavity. We can further confirm our analysis by comparing it with the function’s graph.

Section 2: Finding Intervals of Increasing and Decreasing Polynomials
Polynomials can be used to model many real-world phenomena like population growth, economic performance, and more. When analyzing a polynomial, we need to identify the intervals where it is increasing or decreasing. To do so, we can look at the sign of its first derivative.

Section 3: Determining the Concavity of Polynomials
The behavior of a polynomial function can be further studied by understanding where it is concave up or down. This information is essential since the concavity of a function has an impact on its different properties. To find the concavity of a polynomial, we need to look at the sign of its second derivative.

By understanding the specific intervals of different functions, we can make better predictions and decisions in different situations.

Objectives:
– To analyze the behavior of trigonometric functions over specific intervals
– To identify where a function is increasing, decreasing, concave up, or concave down over a given interval
– To determine the x-coordinates of inflection points of a function
– To confirm the consistency of results obtained from analysis with the graph of the function

Learning Outcomes:
– Students will be able to use interval notation to describe the behavior of trigonometric functions.
– Students will be able to identify intervals on which a function is increasing or decreasing, and intervals on which it is concave up or concave down.
– Students will be able to locate inflection points of a function.
– Students will be able to visually interpret the graph of a function and confirm the results obtained from analysis.

Analyzing Trigonometric Function:

Heading: Increasing and Decreasing Intervals
– f(x) is increasing over xe (-1, 0)
– f(x) is decreasing over xe (0, A)

Heading: Concavity and Inflection Points
– f(x) is concave up over xe (-1, -0.5) U (0.5, A)
– f(x) is concave down over xe (-0.5, 0.5)
– f(x) has an inflection point at x=0

For the function f(x) = ln(x^2 + 36):

Heading: Increasing and Decreasing Intervals
– f(x) is increasing on xe (-∞, 0) U (0, ∞)
– f(x) is decreasing on xe (DNE)

Heading: Concavity
– f(x) is concave up on me (-∞, -6) U (6, ∞)
– f(x) is concave down on me (-6, 6)

Solution 1:
Analyzing the trigonometric function f(x) = 5 – 5tan(x) over the interval (-1, π/2), we have:
– f(x) is increasing over the interval (-1, π/4)
– f(x) is decreasing over the interval (π/4, π/2)
– f(x) is concave up over the interval (-1, -π/4) U (0, π/4)
– f(x) is concave down over the interval (-π/4, 0) U (π/4, π/2)
– f(x) has an inflection point at x = -π/4

Solution 2:
For the function f(x) = ln(x^2 + 36), we have:
(a) f(x) is increasing on the interval (-∞, 0) U (0, +∞)
(b) f(x) is decreasing on the interval DNE (since f(x) is always increasing or 0)
(c) f(x) is concave up on the interval (-6, 6)
(d) f(x) is concave down on the interval (-∞, -6) U (6, +∞)

Suggested Resources/Books:
1. “Trigonometry For Dummies” by Mary Jane Sterling
2. “Calculus: Early Transcendentals” by James Stewart
3. “Precalculus: Mathematics for Calculus” by James Stewart, Lothar Redlin, and Saleem Watson

Similar Asked Questions:
1. How do you find the intervals where a function is increasing or decreasing?
2. What is concavity in calculus, and how do you determine if a function is concave up or concave down?
3. How do you find inflection points on a graph?
4. What is the difference between a local maximum and a global maximum on a graph?
5. How does calculus relate to real-life applications, such as physics and engineering?

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