Upon successful completion of this course, students will be able to:
1. Grasp the fundamental building blocks of mathematical analysis, including real numbers, functions, limits, and continuity. Students will develop a solid understanding of domains, ranges, and elementary functions such as polynomials, exponential and trigonometric functions.
2. Develop proficiency in calculating derivatives and integrals. Students will learn to interpret derivatives as rates of change, apply derivatives to solve problems involving tangent lines and linear approximations, and utilize the Fundamental Theorem of Calculus for integration to solve problems related to areas and other applications.
3. Acquire skills to solve simple differential equations and analyze their solutions, understanding concepts like fixed points and stability. This includes applications to modeling problems in neuroscience.
4. Engage with higher-order derivatives, Taylor expansions, and numerical approximations to understand local and asymptotic behavior of functions, including extrema, concavity, and behavior at infinity.
5. Understand and apply concepts of linear algebra, including matrix operations, systems of linear equations, vector spaces, and linear transformations. Students will also master eigenvalues and eigenvectors, diagonalization, and applications to stability analysis.
6. Delve into complex numbers, exploring their algebraic properties, geometric interpretation, and applications. Understand and apply concepts of orthogonality, inner product spaces, and orthogonal bases, critical for various applications in mathematics and physics.
By the end of this course, students will have developed a comprehensive mathematical toolkit, enabling them to tackle complex problems in neuroscience. This will include a proficiency in visualizing mathematical concepts in multiple dimensions and applying mathematical theory to practical and theoretical problems.