A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for * 30 f(x) dx. lower estimate upper estimate 10 14 18 22 26 30 f(x) -15| -8-1 1 3 7
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![A table of values of an increasing function \( f \) is shown. Use the table to find lower and upper estimates for \( \int_{10}^{30} f(x) \, dx \).
\[
\begin{array}{c|cccccc}
x & 10 & 14 & 18 & 22 & 26 & 30 \\
\hline
f(x) & -15 & -8 & -1 & 1 & 3 & 7 \\
\end{array}
\]
- **Lower Estimate**
- **Upper Estimate**
**Explanation of the Table:**
The table consists of columns representing values of \( x \) at intervals of 4 units, starting at 10 and ending at 30. Corresponding to each \( x \) value, the function \( f(x) \) is given.
From the table:
- At \( x = 10 \), \( f(x) = -15 \)
- At \( x = 14 \), \( f(x) = -8 \)
- At \( x = 18 \), \( f(x) = -1 \)
- At \( x = 22 \), \( f(x) = 1 \)
- At \( x = 26 \), \( f(x) = 3 \)
- At \( x = 30 \), \( f(x) = 7 \)
To find the lower and upper estimates for the integral, you can apply numerical integration techniques such as the Trapezoidal Rule or the Left/Right Riemann Sum using the given values. The function \( f(x) \) is increasing, indicating that the right Riemann sum will give the upper estimate and the left Riemann sum will give the lower estimate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a99fabf-a31f-4250-98b6-bdb0c3640967%2Fbe8c3fd2-d61c-423f-aed6-9a7e95a63349%2Fqlw2i2i_processed.png&w=3840&q=75)
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