Question 1: \[ \text{What is the parametric equation of a circle with radius } r \text{ centered at the origin?} \] \[ \text{(a) } x = r \cos(t), \quad y = \cos(t), \quad \text{(b) } x = t, \quad y = r \sin(t), \quad \text{(c) } x = t, \quad y = t, \quad \text{(d) … Read more

Question 1: \[ \text{Which of the following is the general solution to the differential equation } \frac{dy}{dx} = 3x^2? \] \[ \text{(a) } y = 3x^3 + C, \quad \text{(b) } y = x^3 + C, \quad \text{(c) } y = 3x^2 + C, \quad \text{(d) } y = x^3 + 3 \] Answer: B … Read more

Question 1: \[ \text{Which of the following is the integral of } \int \sin(x) \, dx? \] \[ \text{(a) } \cos(x), \quad \text{(b) } -\cos(x), \quad \text{(c) } \sin(x), \quad \text{(d) } -\sin(x) \] Answer: B Step by Step Solution Solution: Using the basic integration rule: \[ \int \sin(x) \, dx = -\cos(x) + C. … Read more

Question 1: \[ \text{What is the area under the curve } y = x^2 \text{ from } x = 0 \text{ to } x = 3? \] \[ \text{(a) } 18, \quad \text{(b) } 27, \quad \text{(c) } 10, \quad \text{(d) } 9 \] Answer: D Step by Step Solution Solution: \[ \int_0^3 x^2 \, … Read more

\[ \textbf{Question 1:} \] \[ \text{What is the integral of } \int 5x^4 \, dx? \] \[ \text{(a) } x^5 + C, \quad \text{(b) } \frac{5x^5}{5} + C, \quad \text{(c) } x^5 + 5C, \quad \text{(d) } \frac{x^5}{5} + C \] \[ \text{Answer: D} \] Step by Step Solution \[ \textbf{Solution:} \] \[ \int 5x^4 … Read more

\[\textbf{Question 1:}\] \[ \text{Find the local maximum or minimum of } f(x) = x^3 – 3x^2 + 2. \] \[ \text{(a) } x = 0 \text{ is a local maximum}, \quad \text{(b) } x = 0 \text{ is a local minimum}, \quad \text{(c) } x = 1 \text{ is a local minimum}, \quad \text{(d) } … Read more

\[ \textbf{Question 1:} \] \[ \text{Which rule is used to differentiate the function } f(x) = g(x)h(x)? \] \[ \text{(a) } \text{Product Rule}, \quad \text{(b) } \text{Quotient Rule}, \quad \text{(c) } \text{Chain Rule}, \quad \text{(d) } \text{Power Rule} \] \[ \text{Answer: A} \] Step by Step Solution \[ \textbf{Solution:} \] \[ \text{The Product Rule states … Read more

\[ \textbf{Question 1:} \] \[ \text{What is the limit of } \lim_{x \to 0} \frac{\sin(x)}{x}? \] \[ \text{(a) } 0, \quad \text{(b) } 1, \quad \text{(c) } \infty, \quad \text{(d) } -1 \] \[ \text{Answer: B} \] Step by Step Solution \[ \textbf{Solution:} \] \[ \text{The standard limit } \lim_{x \to 0} \frac{\sin(x)}{x} \text{ is … Read more

Question 1: \[ \text{Which of the following is the domain of the function } f(x) = \frac{1}{x-2}? \] \[ \text{(a) } (-\infty, 2), \quad \text{(b) } (-\infty, 2) \cup (2, \infty), \quad \text{(c) } [2, \infty), \quad \text{(d) } (-\infty, \infty) \] Answer: B Step by Step Solution Solution: \[ \text{The function } f(x) = … Read more

Question 1: \[ \text{If } P(x) = x^3 – 4x^2 + 3x + 5, \text{ what is } P(2)? \] \[ \text{(a) } 1, \quad \text{(b) } 3, \quad \text{(c) } 5, \quad \text{(d) } 7 \] Answer: b Step by Step Solution Solution: We are given the polynomial: \[ P(x) = x^3 – 4x^2 … Read more

Question 1: \[ \text{The Gini Coefficient is a measure of income or wealth inequality, represented by the ratio of the area between the Lorenz curve and the line of perfect equality to the total area under the line of perfect equality.} \] \[ \text{It is calculated using the formula:} \] \[ G = \frac{A}{A + … Read more

Question 1: \[ \text{The Gini Coefficient is a measure of inequality in a distribution. It is calculated using the formula:} \] \[ G = \frac{A}{A + B} \] \[ \text{Where: } \] \[ A \text{ is the area between the Lorenz curve and the line of perfect equality, and } \] \[ B \text{ is … Read more