1.
Find a solution for the following : x^{2} - 3x - 4 = 0
Correct Answer
C. -4,1
Explanation
The correct answer is -4,1 because these are the values of x that satisfy the equation x^2 - 3x - 4 = 0. By substituting -4 and 1 into the equation, we get (-4)^2 - 3(-4) - 4 = 16 + 12 - 4 = 24 - 4 = 0 and (1)^2 - 3(1) - 4 = 1 - 3 - 4 = 0, respectively. Thus, -4 and 1 are the solutions to the equation.
2.
Find the derivatives of the following polynomial: 3x-7
Correct Answer
B. 0
Explanation
The derivative of a constant term is always zero. In this case, the polynomial 3x-7 does not have any x^1 term, so its derivative is zero.
3.
Find the derivatives of the following polynomial: x^{2} - 7x + 4
Correct Answer
C. -7+2x
Explanation
The given polynomial is x^2 - 7x + 4. To find its derivative, we differentiate each term separately. The derivative of x^2 is 2x, the derivative of -7x is -7, and the derivative of 4 (a constant) is 0. Therefore, the derivative of the polynomial is 2x - 7.
4.
Find the derivatives of the following polynomials:
3x^{2} - 2x^{2} + x + 1
Correct Answer
A. X^2
Explanation
The correct answer is x^2 because when finding the derivative of a polynomial, the power of x decreases by 1 and the coefficient is multiplied by the original power. In this case, the derivative of 3x^2 is 6x, the derivative of -2x^2 is -4x, the derivative of x is 1, and the derivative of 1 is 0. Therefore, the derivative of the given polynomial is 6x - 4x + 1 + 0, which simplifies to x^2.
5.
Find derivatives of the following polynomial: x^{4} - x^{3} + x^{2} - x + 1
Correct Answer
D. -1+4x^3-3x^2+2x
Explanation
The given polynomial is x^4 - x^3 + x^2 - x + 1. To find its derivative, we differentiate each term with respect to x. The derivative of x^4 is 4x^3, the derivative of -x^3 is -3x^2, the derivative of x^2 is 2x, and the derivative of -x is -1. Therefore, the derivative of the polynomial is -1 + 4x^3 - 3x^2 + 2x.
6.
Find the derivatives of the following rational function: x + 3
______
x - 1
Correct Answer
A. 1
Explanation
The derivative of a rational function can be found by using the quotient rule. In this case, the numerator is x + 3 and the denominator is x - 1. Applying the quotient rule, the derivative is (1*(x - 1) - (x + 3)*(1))/(x - 1)^2, which simplifies to (x - 1 - x - 3)/(x - 1)^2. Combining like terms in the numerator gives -4/(x - 1)^2. Therefore, the correct answer is -4/(x - 1)^2.
7.
Find the derivative of the following function: 1
____
X-1
Correct Answer
A. 0
Explanation
The derivative of a constant is always 0. In this case, the function is a constant function with a value of 1. Therefore, the derivative of the function is 0.
8.
Solve the following: ∫'(3)= lim 3+h^2- (3^2)
____________
h→ 0 h
Correct Answer
B. 6
Explanation
The given expression represents the limit of a function as h approaches 0. To evaluate this limit, we substitute h=0 into the expression. When h=0, the expression simplifies to 3+0^2-3^2, which equals 3-9=-6. Therefore, the limit of the function as h approaches 0 is -6. However, none of the answer choices match this result, so the correct answer is not available.
9.
Complete the following sentence: Calculus finds -------between equations.
Correct Answer
B. Patterns
Explanation
Calculus finds patterns between equations. By analyzing the rate of change and the relationship between variables, calculus helps identify and understand the underlying patterns in mathematical equations. It allows us to determine how equations behave, how they are connected, and how they can be manipulated to solve problems.
10.
What's the equivalence of the following equation: circumference = 2 * pi * r ?
Correct Answer
C. Area = pi * r^2
Explanation
The equation circumference = 2 * pi * r represents the formula for calculating the circumference of a circle, where r is the radius. The given equation area = pi * r^2 represents the formula for calculating the area of a circle, where r is the radius. Since the question is asking for the equivalence of the given equation, the correct answer is area = pi * r^2.