### Solve the Quadratic Equation 4x²-5x-12=0

The roots for the given quadratic equation are x1 ≈ 2.46625 and x2 ≈ -1.21625. For solving the quadratic equation 4x² – 5x – 12 = 0 we apply the quadratic formula, this method is a steadfast solution strategy when facing any quadratic equation of the form ax² – bx – c = 0. Below we […]

### Where is the greatest integer function f(x)= ⌊x⌋ not differentiable? Find a formula for f’ and sketch its graph.

Expert Answer: The greatest integer function is not differentiable on any real value of x because this function is discontinuous on all the integer values, and it has no or zero slopes on every other value. We can see the discontinuity in Figure 1. Let f(x) is a floor function which is represented in Figure […]

### The graph of a function f is shown. Which graph is an antiderivative of f?

This question explains the concept of antiderivative and how to draw its graph from the function graph. The antiderivative of a function is the indefinite integral of the function. If we take its derivative, it will give out the original function. The derivative and antiderivative or indefinite integral are inverse of each other. The derivative […]

### A -10.0 nC point charge and a +20.0 nC point charge are 15.0 cm apart on the x-axis. Find the following:

What is the electric potential at the point on the x-axis where the electric field is zero? What are the magnitude and direction of the electric field at the point on the x-axis, between the charges, where the electric potential is zero? This question aims to find the electric potential at the point on x-axis […]

### (a) Find the average value f on the given interval. (b) Find c such that f_{ave} = f(c). Equation given below

This problem aims to find the average value of a function on a given interval and also find the slope of that function. This problem requires knowledge of the fundamental theorem of calculus and basic integration techniques. To find the average value of a function on a given interval, we will integrate and divide the […]

### A rectangular package to be sent by a postal service that has a maximum total length and perimeter (or girth) limit of 108 inches. A rectangular package is to be sent via this service. Calculate the dimensions of the package that covers the maximum volume. (Cross-sections may be assumed to be square)

This question aims to learn the basic methodology for optimizing a mathematical function (maximizing or minimizing). Critical points are the points where the value of a function is either maximum or minimum. To calculate the critical point(s), we equate the first derivative’s value to 0 and solve for the independent variable. We can use the […]

### Find two functions f and g such that (f ∘ g)(x) = h(x).

[ h(x) = (x + 2)^3 ] The question aims to find the functions f and g from a third function which is a composition of the function of those two functions. The composition of functions can be defined as putting one function into another function that outputs the third function. The output from one […]

### In the exponential growth or decay function y = y0e^kt, what does y0 represent?

This problem aims to understand the exponential growth and exponential decay. An exponential function is a function in which the exponent is a variable, and the base is positive and $cancel{=}space 1$. For example, $f(x)=4^x$ is an exponential function and the exponent is not a mutable but a specified constant. $f(x) =x^3$is a fundamental polynomial […]

### A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at 10, the average attendance had been 27,000. When ticket prices were lowered to10,the average attendance had been 27,000. When ticket prices were lowered to 8, the average attendance rose to 33,000. How should ticket prices be set to maximize revenue?

The main objective of this question is to find the maximum revenue for the given conditions. This question uses the concept of revenue. Revenue is the sum of the average selling price multiplied by a number of units sold, which is the amount of money generated by a business’s typical operations. Expert Answer First, we […]

### Which operation could we perform in order to find the number of milliseconds in a year?

$60cdot 60cdot 24cdot 7cdot 365$ $1000cdot 60cdot 60cdot 24cdot 365$ $24cdot 60cdot 100cdot 7cdot 52$ $1000cdot 60cdot 24cdot 7cdot 52$ The goal of this question is to convert a year into milliseconds by selecting a suitable formula from the list provided. For this operation, disregard the use of months in the calculation. They have irregular […]

### How do you write y = 2x – 9 in standard form?

The question aims to find the standard form of an algebraic equation. The question is based on the concepts of algebraic equations, particularly linear equations with two variables. Linear equations are algebraic equations with variables only having an exponent of one. These equations represent a linear straight line as shown in Figure 1. The equation of […]

### Find the point at which the given lines intersect. Find an equation of the plane that contains these lines.

-The two lines with the following equations intersect at a point. [r=(2,3,0) + t (3,-3,2)] [r=(5,0,2) + s (-3,3,0)] -(a) Find out the point of intersection of these two lines. -(b) Find the equation of the plane having these two lines. In this question, we have to find two things: the point of intersection and […]

### Find the coefficient of x^5 y^8 in (x+y)^13.

The main objective of this question is to find the coefficient of the term $x^5y^8$ in the expansion of $(x+y)^{13}$ using the Binomial theorem or expansion. The binomial theorem was first mentioned in the fourth century BC by Euclids, a famous Greek mathematician. The binomial theorem also known as binomial expansion in elementary algebra represents […]

### Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) = x + e^{−x}, a = 0

This problem aims to find the Taylor polynomials up to $3$ places for a given function $f$, centered at a point $a$. To better understand the problem, you must know about Power Series, as it forms the basis of the Taylor Series. Taylor series of a function is defined as an infinite sum of derivative […]

### Use the table of values of f(x, y) to estimate the values of fx(3, 2), fx(3, 2.2), and fxy(3, 2).

This problem aims to find the values of a function having alternate independent variables. A table is given to address the values of $x$ and $y$. These formulas would be required to find the solution: [ f_x(x,y)=lim_{h to 0}dfrac{f(x+h, y)-f(x,y)}{h}] [ f_y(x,y)=lim_{hto 0}dfrac{f(x, y+h)-f(x,y)}{h}] [ f_{xy}=dfrac{partial}{partial y}left(frac{partial f}{partial x} right)=dfrac{partial}{partial y}(f_x] Expert Answer: Part a: […]

### Which pair of numbers has an LCM of 16

3 and 16 2 and 4 4 and 8 4 and 16 In this question, we have to find the pair of numbers for which the LCM is 16. LCM stands for Least Common Multiple, defined as the smallest multiple common number between the required numbers for which LCM is to be determined. It is […]

### The next number in the series 38, 36, 30, 28, 22 is?

The main objective of this question is to find the next number in the given series. The number sequence is an important mathematical tool for assessing intelligence. Many aptitude exams include number series problems. These questions usually follow a numerical pattern along with a logical rule. Sequences and series are important in many aspects of […]

### Find the domain and range of these functions.

the function that assigns to each pair of positive integers the first integer of the pair. the function that assigns to each positive integer the largest decimal digit. the function that assigns to a bit string the number of ones minus the number of zeros in that string. the function that assigns to each positive […]

### Find the points on the cone z^2 = x^2 + y^2 that are closest to the point (2,2,0).

This question aims to explain the concepts of maxima and minima. Formulas to calculate the extreme values of the function. Further, it explains how to calculate the distance between the points. In mathematics, the length of the line segment between the two points is the Euclidean distance between two points. The Pythagorean theorem is used […]

### Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

The purpose of this question is to prove that $n$ is a positive and even integer if and only if $7n + 4$ is also even. Even numbers can be equally divided into two pairs or groups and are completely divisible by two. For instance, $2, 4, 6, 8$, and so on are said to […]