Question 1: \[ \text{Solve the differential equation } \frac{d^2y}{dx^2} = 0. \] \[ \text{(a) } y = C_1 x + C_2, \quad \text{(b) } y = C_1 x^2 + C_2, \quad \text{(c) } y = C_1 e^x + C_2, \quad \text{(d) } y = C_1 \cos(x) + C_2 \sin(x) \] Answer: A Question 2: \[ … Read more

Question 1: \[ \text{Find the divergence of the vector field } \mathbf{F}(x, y, z) = x^2 \hat{i} + y^2 \hat{j} + z^2 \hat{k}. \] \[ \text{(a) } 2x + 2y + 2z, \quad \text{(b) } 2x + 2y + 2z + 1, \quad \text{(c) } 2x + 2y, \quad \text{(d) } 2x + 2z \] … Read more

Question 1: \[ \text{Find the divergence of the vector field } \mathbf{F}(x, y, z) = x^2 \hat{i} + y^2 \hat{j} + z^2 \hat{k}. \] \[ \text{(a) } 2x + 2y + 2z, \quad \text{(b) } 2x + 2y + 2z + 1, \quad \text{(c) } 2x + 2y, \quad \text{(d) } 2x + 2z \] … Read more

Question 1: \[ \text{What is the partial derivative of } f(x, y) = x^2y + y^3 \text{ with respect to } x? \] \[ \text{(a) } 2xy, \quad \text{(b) } 2x + 3y^2, \quad \text{(c) } 2x^2y, \quad \text{(d) } x^2 + 3y^2 \] Answer: A Step by Step Solution Solution: \[ \text{The partial derivative … Read more

Question 1: \[ \text{Which of the following is a vector function?} \] \[ \text{(a) } f(t) = t^2 + 3t, \quad \text{(b) } \mathbf{r}(t) = (t, t^2, t^3), \quad \text{(c) } f(t) = \sin(t), \quad \text{(d) } g(t) = \cos(t) \] Answer: B Question 2: \[ \text{If } \mathbf{r}(t) = (t^2, 2t, 3t), \text{ what … Read more

Question 1: \[ \text{Which of the following is the dot product of vectors } \mathbf{a} = (1, 2, 3) \text{ and } \mathbf{b} = (4, -5, 6)? \] \[ \text{(a) } 32, \quad \text{(b) } 10, \quad \text{(c) } 4, \quad \text{(d) } -10 \] Answer: B Step by Step Solution Solution: \[ \text{The dot … Read more

Question 1: \[ \text{Which of the following is the general form of an infinite geometric series?} \] \[ \text{(a) } \sum_{n=0}^{\infty} ar^n, \quad \text{(b) } \sum_{n=1}^{\infty} \frac{1}{n^2}, \quad \text{(c) } \sum_{n=0}^{\infty} a + n, \quad \text{(d) } \sum_{n=1}^{\infty} \frac{1}{2^n} \] Answer: A Step by Step Solution Solution: The general form of an infinite geometric series … Read more

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