1.
What is x^{2} + 1 differentiated with respect to x?
Correct Answer
A. 2x
Explanation
The derivative of x^2 + 1 with respect to x is 2x. This can be found using the power rule of differentiation, which states that when differentiating a term with the form x^n, the result is nx^(n-1). In this case, the derivative of x^2 is 2x^(2-1) = 2x. The constant term 1 does not affect the derivative, so it can be disregarded. Therefore, the correct answer is 2x.
2.
What will be the derivative of this function- 'f(x) = 1963'
Correct Answer
A. 0
Explanation
The derivative of a constant function is always zero. Since the given function is f(x) = 1963, which is a constant, its derivative will be zero. This is because the derivative measures the rate of change of a function, and a constant function does not change as x varies. Therefore, the correct answer is 0.
3.
What is the integral of 1 with respect to x?
Correct Answer
B. X + C
Explanation
The integral of 1 with respect to x is:
∫ 1 dx = x + C
In this expression, ∫ represents the integral sign, and the result is x + C, where C is the constant of integration. When you integrate a constant (1 in this case) with respect to x, you get x, and the constant of integration, C, represents an arbitrary constant that can be added to the result to account for all possible antiderivatives.
4.
State whether the given statement is true or false.
The second fundamental theorem of calculus states that if
F(x) = ∫_{ a}^{x} f(t) dt
then F '(x) = f(x).
Correct Answer
A. True
Explanation
The second fundamental theorem of calculus states that if a function F(x) is defined as the integral of another function f(t) with respect to t, then the derivative of F(x) with respect to x is equal to f(x). In other words, if F(x) = ∫ax f(t) dt, then F'(x) = f(x). This means that the derivative of the integral of a function is equal to the original function itself.
5.
What is x^{2} + 1 differentiated with respect to y?
Correct Answer
C. 0
Explanation
The expression x^2 + 1 does not contain the variable y, so when differentiating it with respect to y, the derivative is always 0.
6.
What function is integrated to get this graph?
Correct Answer
D. 2x
Explanation
The graph of the function 2x is a straight line with a positive slope that passes through the origin. As x increases, the value of the function also increases at a constant rate. This matches the characteristics of the graph shown in the question, where the line starts at the origin and has a positive slope. Therefore, the function integrated to get this graph is 2x.
7.
Can you select the names of the two men who discovered calculus?
Correct Answer(s)
A. Newton
B. Gottfried Wilhelm Leibniz
Explanation
The correct answer is Newton and Gottfried Wilhelm Leibniz. These two men independently developed the mathematical field of calculus. Isaac Newton, an English mathematician and physicist, is known for his work on calculus and his laws of motion. Gottfried Wilhelm Leibniz, a German philosopher and mathematician, also made significant contributions to calculus and is credited with developing a notation system that is still used today. Both Newton and Leibniz played a crucial role in the development of calculus and their work laid the foundation for modern mathematics.
8.
What is the derivative of f(x) = 3x^2 + 4x - 1 with respect to x?
Correct Answer
A. 6x + 4
Explanation
To find the derivative of the function f(x) = 3x^2 + 4x - 1 with respect to x, we'll apply the power rule of differentiation. The power rule states that if you have a term of the form ax^n, where 'a' is a constant and 'n' is a real number, the derivative is n * ax^(n-1).
Let's differentiate each term of the function f(x) separately:
The derivative of 3x^2 with respect to x is 2 * 3x^(2-1) = 6x.
The derivative of 4x with respect to x is 1 * 4x^(1-1) = 4.
The derivative of the constant term -1 with respect to x is 0 (the derivative of a constant is always 0).
Now, we add up these derivatives:
f'(x) = 6x + 4 - 0
So, the derivative of f(x) = 3x^2 + 4x - 1 with respect to x is:
f'(x) = 6x + 4
9.
What is the limit of 1/(x-1) as x goes to infinity?
Correct Answer
B. 0
Explanation
As x approaches infinity, the denominator (x-1) becomes significantly larger than the numerator (1). This causes the fraction to approach zero. Therefore, the limit of 1/(x-1) as x goes to infinity is 0.
10.
The limit of the expression 'x - 1' exists as x approaches zero.
Correct Answer
A. True
Explanation
True.
The limit of the expression 'x - 1' as x approaches zero exists. As x approaches zero, the expression 'x - 1' approaches -1. Therefore, the limit of 'x - 1' as x approaches zero is -1.