3. Find the slope of the tangent line to the curve

$$2(x^2 + y^2)^2 + (x^2 - 1) = 25(x^2 - y^2) + 2^x \text{ at point } (3, 1).$$

4. Find the indefinite integral \(\int \frac{1}{\sqrt{4 + c^2}} dx = ?\)

5. Find the definite integral \(\int_0^4\int_0^{\sqrt{1 - x^2}}e^{x^2 +y^2}dydx = ?\)

II. Linear Algebra 1. (10%) Suppose $\underline{x}$ and $\underline{y}$ are two linearly independent vectors in $R^n$, and let $U$ be the span of $\underline{x}$ and $\underline{y}$, that is $U = \text{span}\{\underline{x}, \underline{y}\}$. Let $A = \underline{x}\underline{y}^T + \underline{y}\underline{x}^T$. Prove that the null space $N(A)$ is orthogonal to $U$, meaning every $\underline{v}$ in $N(A)$ is orthogonal to every $\underline{w}$ in $U$.

2. Let $A$ be an $m \times n$ matrix with independent columns, and let $A = QR$ be its QR factorization. The orthogonal projection of any vector $\underline{b} \in R^m$ onto the column space of $A$ can be represented by $\underline{x} = P\underline{b}$, where $P$ is the projection matrix and $\underline{x} \in \text{Col}(A)$ as shown in Figure 1. (a) \((10\%)\) Show that \(P = QQ^{T}\)

(b) (5%) Let \( Q = \left[ \begin{array}{ll} \underline{q}_1 & \underline{q}_2 \\ \dots & \underline{q}_n \end{array} \right] \). Show that \( \underline{x} \) can be written as a linear combination of \( \underline{q}_1, \underline{q}_2, \dots, \underline{q}_n \) and represented by $$\underline{x} = (\underline{q}_1^T\underline{b})\underline{q}_1 + (\underline{q}_2^T\underline{b})\underline{q}_2 + \dots + (\underline{q}_n^T\underline{b})\underline{q}_n$$

3. Let $L: R^2 \to R^2$ denote the mapping of a vector $\underline{x}$ onto its orthogonal projection on a line $l$ passing through the origin, which forms an angle $\theta$ with the $X$-axis as shown in Figure 2. Let $A$ be the matrix that corresponds to this mapping, i.e., $L(\underline{x}) = A\underline{x}$. Find $A$ by each method below. (a) (10%) Method 1: According to the formula \( A = \left[L\left(\underline{e}_1\right)L\left(\underline{e}_2\right)\right] \), where \( \underline{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and \( \underline{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \), \( A \) is obtained by evaluating the mappings on the standard basis vectors \( \underline{e}_1 \) and \( \underline{e}_2 \).

(b) (15%)Method 2: Perform a sequential linear transformation \( L(\underline{x}) = L_{\theta} \circ L_{x} \circ L_{-\theta}(\underline{x}) \) as described below: i. First, apply $L_{-\theta}$ as a clockwise rotation about the origin by $\theta$ radians. Determine $A_1$ such that $L_{-\theta}(\underline{x}) = A_1\underline{x}$.

ii. Next, carry out $L_x$ as an orthogonal projection of any vector $\underline{x}$ onto the $X$-axis. Determine $A_2$ such that $L_x(\underline{x}) = A_2\underline{x}$.

iii. Finally, execute a counterclockwise rotation $L_\theta$. Determine $A_3$ such that $L_\theta(\underline{x}) = A_3\underline{x}$. Calculate $A = A_3A_2A_1$. (The result should match the answer given in part (a).) Figure 1: $Col(A)$ denotes the column space of $A$. Figure 2: Project $\underline{x}$ onto the line $l$ that forms an angle $\theta$ with the $X$-axis.

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