Question 1

Given the function $F(x) = x^4 - 5x^3 + 27x^2 - 101x + 70$, and knowing that $x = 5$ is a root, which of the following is the correct result of dividing $F(x)$ by $(x-5)$ using synthetic division?

Accepted Answer

$x^3 + 0x^2 + 2x - 14$

Question 2

The derivative of a power $x^n$ is given by which of the following formulas?

Accepted Answer

$\frac{d}{dx}x^n = nx^{n-1}$

Question 3

Given the cost function $C(x) = 20x^3 - 360x^2 + 3200x - 1000$, what is the formula for the average cost $A(x)$?

Accepted Answer

$A(x) = \frac{C(x)}{x}$

Question 4

If $F(x) = x^4 - 5x^3 + 27x^2 - 101x + 70$, what is $F'(x)$?

Accepted Answer

$4x^3 - 15x^2 + 54x - 101$

Question 5

Newton's Method is used to find roots of a function. Which of the following is the correct iterative formula for Newton's Method?

Accepted Answer

$x_{n+1} = x_n - \frac{F(x_n)}{F'(x_n)}$

Question 6

After dividing $F(x)$ by $(x-5)$, you obtain a cubic polynomial $G(x)$. Which of the following best describes the relationship between the roots of $F(x)$ and $G(x)$?

Accepted Answer

The roots of $G(x)$ are the remaining roots of $F(x)$, excluding $x=5$

Question 7

If you have a quadratic equation $ax^2 + bx + c = 0$, what is the quadratic formula for finding its roots?

Accepted Answer

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Question 8

Suppose you have found three real roots of a cubic equation, but only positive roots are considered valid. If the roots are $-2$, $3$, and $4$, which root should be chosen for a real-world production problem?

Accepted Answer

$4$

Question 9

If the minimum of the average cost function occurs at $x = a$, and $A(x)$ is minimized at this value, what does this value of $x$ represent in a production context?

Accepted Answer

The production level that yields the lowest average cost per unit

Question 10

Which of the following best describes a critical point of a function $f(x)$?

Accepted Answer

A point where $f'(x) = 0$ or $f'(x)$ is undefined

Question 11

If $F(x)$ is a polynomial and you divide it by $(x-a)$, what does the Remainder Theorem state?

Accepted Answer

The remainder is $F(a)$

Question 12

If you use Newton's Method to approximate a root and the derivative at your guess is zero, what will happen?

Accepted Answer

The method will fail because division by zero occurs

Question 13

Given $F(x) = x^4 - 5x^3 + 27x^2 - 101x + 70$, which of the following is the correct process to find all real roots?

Accepted Answer

Use synthetic division to remove known roots, then solve the resulting lower-degree polynomial

Question 14

If the cost function is $C(x) = 20x^3 - 360x^2 + 3200x - 1000$, what is the derivative $C'(x)$?

Accepted Answer

$60x^2 - 720x + 3200$

Question 15

If you are asked to find the minimum of the average cost function $A(x)$, what is the correct calculus procedure?

Accepted Answer

Set $A'(x) = 0$ and solve for $x$

Mathematical Methods for Finding Roots and Critical Points (Not Anatomy & Physiology)

Study Guide - Practice Questions

Study Guide - Flashcards