Textbook Question
Using the Sandwich Theorem
If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).
1. Limits and Continuity
Introduction to Limits
2:19 minutesProblem 2.1.4b
Textbook Question
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(t)=2+cos t
b. [0,π]
Verified step by step guidance1
Identify the function g(t) = 2 + cos(t) and the interval [0, π].
Recall the formula for the average rate of change of a function g(t) over an interval [a, b], which is given by: \( \frac{g(b) - g(a)}{b - a} \).
Substitute the endpoints of the interval into the function: calculate g(0) and g(π).
Evaluate g(0) = 2 + cos(0) and g(π) = 2 + cos(π).
Substitute g(0) and g(π) into the average rate of change formula: \( \frac{g(\pi) - g(0)}{\pi - 0} \) and simplify the expression.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Average Rate of Change
The average rate of change of a function over an interval is defined as the change in the function's value divided by the change in the input value. Mathematically, it is expressed as (f(b) - f(a)) / (b - a), where [a, b] is the interval. This concept helps in understanding how a function behaves on average over a specified range.
Recommended video:
06:37
Average Value of a Function
Function Evaluation
Function evaluation involves substituting a specific value into a function to determine its output. For the function g(t) = 2 + cos(t), evaluating it at the endpoints of the interval [0, π] means calculating g(0) and g(π). This step is crucial for finding the average rate of change, as it provides the necessary values to apply the average rate of change formula.
Recommended video:
4:26Evaluating Composed Functions
Trigonometric Functions
Trigonometric functions, such as cosine, are periodic functions that relate angles to ratios of sides in right triangles. The function g(t) = 2 + cos(t) combines a constant and a cosine function, which oscillates between -1 and 1. Understanding the behavior of cosine over the interval [0, π] is essential for accurately calculating the average rate of change, as it influences the function's values at the endpoints.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Videos
Related Practice