up ie A box is formed by cutting squares from the four corners of a 7"-wide by 9"-löng Sheet Ul pape Let a represent the length of the side of the square cutout (in inches), and let V represent the volume of the box (in cubic inches). a. Write a formula that expresses V in terms of x. V = 4x^3-32x^2+63x Preview
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.
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imathas.rationalreasoning.net/assessment/s
A box is formed by cutting squares from the four corners of a 7"-wide by 9"-long sheet of paper and folding up the sides.
Let x represent the length of the side of the square cutout (in inches), and let V represent the volume of the box (in cubic inches).
a. Write a formula that expresses V in terms of x.
V = 4x^3-32x^2+63x
Preview
b. Suppose the function f determines the volume of the box (in cubic inches) given a cutout length (in inches). Write a function
formula for f.
f(x) = 4x^3-32x^2+63x
Preview
c. What is the domain of f? Enter your answer as an interval. Hint.
(0,7/2)
Preview
d. What is the range of f? Enter your answer as an interval. Hint.
Tip: Use a graphing tool like www.Desmos.com 2 to find the maximum volume possible.
Preview
Box 1: Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c
Be sure your variables match those in the question
Box 2: Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c
Be sure your variables match those in the question
Box 3: Enter your answer using interval notation. Example: [2,5)
Use U for union to combine intervals. Example: (-00,2] U [4,00)
Enter each value as a number (like 5, -3, 2.2172) or as a calculation (like 5/3, 2^3, 5+4)
Enter DNE for an empty set. Use oo to enter Infinity.
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