Comprehensive Math Center

Lecture 2: Trigonometric and Logarithmic Functions

Derivatives of Trigonometric Functions

• $\frac{d}{dx}(\sin x) = \cos x$
• $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$

Derivatives of Inverse Trigonometric Functions

• $\frac{d}{dx}(\sin^{-1} u) = \frac{u'}{\sqrt{1-u^2}}$
• $\frac{d}{dx}(\tan^{-1} u) = \frac{u'}{1+u^2}$

Derivatives of Exponential and Logarithmic Functions

• $\frac{d}{dx}(e^u) = e^u \cdot u'$
• $\frac{d}{dx}(\ln u) = \frac{1}{u} \cdot u'$

Lecture 3: Hyperbolic Functions and Applications

Derivatives of Hyperbolic Functions

• $\frac{d}{dx}(\sinh x) = \cosh x$
• $\frac{d}{dx}(\cosh x) = \sinh x$
• $\frac{d}{dx}(\tanh x) = \text{sech}^2 x$

Derivatives of Inverse Hyperbolic Functions

• $\frac{d}{dx}(\sinh^{-1} u) = \frac{u'}{\sqrt{u^2+1}}$
• $\frac{d}{dx}(\tanh^{-1} u) = \frac{u'}{1-u^2}$

Lecture 4: L'Hôpital's Rule and Series

L'Hôpital's Rule

• For limits of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$: $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

Taylor and Maclaurin Series

• A Maclaurin series is a Taylor series when $a=0$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$

Matrices

Matrix Operations

• Operations: Matrices of the same size can be added. For multiplication $A \cdot B$, the number of columns in $A$ must equal the number of rows in $B$. [29]
• Transpose ($A^T$): Interchange rows and columns. [29]
• Determinant: For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad - bc$. For a 3x3 matrix, the determinant can be found by expanding along a row or column using cofactors. [4, 6]
• Inverse: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$. An inverse exists only if $\det(A) \neq 0$. [1, 10]

Gauss-Jordan Elimination

• Goal: To solve a system of linear equations by transforming its augmented matrix into reduced row-echelon form using elementary row operations. [2, 3]
• Reduced Row-Echelon Form:
  1. All rows with only zero entries are at the bottom.
  2. The leading entry (pivot) of each non-zero row is 1.
  3. Each pivot is the only non-zero entry in its column.
• Elementary Row Operations:
  1. Swapping two rows.
  2. Multiplying a row by a non-zero constant.
  3. Adding a multiple of one row to another row. [14]