1. Investigate the linearity of the function $f(\mathbf{x}) = A\mathbf{x} + \mathbf{b}$, where $A$ is a matrix and $\mathbf{b}$ is a vector.\n2. Find the domain, range, and invertibility of $f(x) = \frac{1}{x^2 - 4}$.\n3. Determine if $f(x) = e^x$ is injective and surjective on $\mathbb{R}$.\n4. Analyze the differentiability and continuity of $f(x,y) = x^2y + 3xy^3$.\n5. Calculate and interpret the gradient vector field $\nabla f(x,y,z)$ for $f(x,y,z)=x^2+y^2+z^2$.\n6. Find the critical points and classify them for $f(x,y) = x^3 - 3xy^2$.\n7. Evaluate the Jacobian matrix of $\mathbf{f}(x,y) = (x^2 + y, y^2 - x)$.\n8. Determine the eigenvalues and eigenvectors of the matrix $A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}$.\n9. Compute the divergence and curl of the vector field $\mathbf{F}(x,y,z) = (yz, xz, xy)$.\n10. Use the chain rule to differentiate $f(t) = g(h(t))$ where $g(u) = u^2$ and $h(t) = \sin t$.\n11. Find the Taylor series expansion of $f(x) = \ln(1+x)$ about $x=0$.\n12. Solve the system of linear equations representing $Ax = b$ using matrix inversion.\n13. Determine the limits and continuity of $f(x,y) = \frac{xy}{x^2 + y^2}$ at $(0,0)$.\n14. Analyze the concavity and inflection points of $f(x) = x^4 - 4x^3 + 6x^2$.\n15. Evaluate the line integral of $\mathbf{F} = (y, -x)$ over the unit circle.\n16. Use the divergence theorem to evaluate the flux of $\mathbf{F} = (x,y,z)$ through the surface of a unit sphere.\n17. Compute the eigenvalues of a skew-symmetric matrix and interpret their properties.\n18. Study the parametric equations $\mathbf{r}(t) = (\cos t, \sin t, t)$ and find tangent vectors.\n19. Optimize the function $f(x,y) = x^2 + y^2$ subject to the constraint $x + y = 1$ using Lagrange multipliers.\n20. Examine the continuity, partial derivatives, and differentiability of the function $f(x,y) = |xy|$.