</section> <section style="font-size: 50px";> <h2 id="logarithm">Logarithm</h2> <ul> <li>Definition: <ul> <li>The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number.</li> <li>Example: $\log_{2}8 = 3$ because $2^{3} = 8$</li> </ul> </li> <li>Properties of Logarithms: <ul> <li>$\log_{b}(xy) = \log_{b}x + \log_{b}y$</li> <li>$\log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y$</li> <li>$\log_{b}(x^{n}) = n\log_{b}x$</li> <li>$\log_{b}1 = 0$</li> <li>$\log_{b}b = 1$</li> </ul> </li> <li>Example 1: <ul> <li>Simplify $\log_{3}27$</li> <li>Solution: Since $3^{3} = 27$, the logarithm of 27 to the base 3 is 3. Therefore, $\log_{3}27 = 3$</li> </ul> </li> <li>Example 2: <ul> <li>Solve for $x$: $\log_{5}x = 2$</li> <li>Solution: Since $5^{2} = 25$, the value of $x$ is 25. Therefore, $x = 25$

</section> <section style="font-size: 50px";> <h2 id="exponents-and-radicals">Exponents and Radicals</h2> <ul> <li>Laws of Exponents: <ul> <li>Product Rule: $a^{m} \times a^{n} = a^{m+n}$</li> <li>Quotient Rule: $\frac{a^{m}}{a^{n}} = a^{m-n}$</li> <li>Power Rule: $(a^{m})^{n} = a^{mn}$</li> </ul> </li> <li>Laws of Radicals: <ul> <li>Product Rule: $\sqrt[a]{bc} = \sqrt[a]{b} \cdot \sqrt[a]{c}$</li> <li>Quotient Rule: $\frac{\sqrt[a]{b}}{\sqrt[a]{c}} = \sqrt[a]{\frac{b}{c}}$</li> <li>Power Rule: $(\sqrt[a]{b})^{c} = \sqrt[a]{b^{c}}$</li> </ul> </li> <li>Example 1: <ul> <li>Simplify: $2^{3} \times 2^{2}$</li> <li>Solution: Applying the product rule of exponents, we get $2^{3} \times 2^{2} = 2^{3+2} = 2^{5} = 32$</li> </ul> </li> <li>Example 2: <ul> <li>Simplify: $\sqrt[3]{8} \cdot \sqrt[3]{27}$</li> <li>Solution: Applying the product rule of radicals, we get $\sqrt[3]{8} \cdot \sqrt[3]{27} = \sqrt[3]{8 \cdot 27} = \sqrt[3]{216} = 6$

</section> <section style="font-size: 50px";> <h2 id="complex-numbers">Complex Numbers</h2> <ul> <li>Definition: <ul> <li>A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ represents the imaginary unit.</li> </ul> </li> <li>Operations with Complex Numbers: <ul> <li>Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$</li> <li>Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$</li> <li>Multiplication: $(a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$</li> <li>Division: $\frac{a + bi}{c + di} = \frac{(ac + bd) - (bc - ad)i}{c^{2} + d^{2}}$</li> </ul> </li> <li>Example 1: <ul> <li>Add: $(3 + 4i) + (2 + 5i)$</li> <li>Solution: Adding the real and imaginary parts separately, we get $(3 + 2) + (4 + 5)i = 5 + 9i$</li> </ul> </li> <li>Example 2: <ul> <li>Multiply: $(3 + 2i) \cdot (4 + 5i)$</li> <li>Solution: Applying the distributive property, we get $12 + 15i + 8i + 10i^{2} = 12 + 23i - 10 = 2 + 23i$

</section> <section style="font-size: 50px";> <h2 id="differentiation">Differentiation</h2> <ul> <li>Definition: <ul> <li>Differentiation is the process of finding the derivative of a function.</li> </ul> </li> <li>Rules of Differentiation: <ul> <li>Constant Rule: $\frac{d}{dx}(c) = 0$, where $c$ is a constant</li> <li>Power Rule: $\frac{d}{dx}(x^{n}) = nx^{n-1}$, where $n$ is a constant</li> <li>Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$</li> <li>Product Rule: $\frac{d}{dx}(f(x) \cdot g(x)) = f’(x) \cdot g(x) + f(x) \cdot g’(x)$</li> <li>Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f’(x) \cdot g(x) - f(x) \cdot g’(x)}{(g(x))^2}$</li> </ul> </li> <li>Example 1: <ul> <li>Find the derivative of $f(x) = 3x^2 - 5x + 2$</li> <li>Solution: Applying the power rule, we get $f’(x) = 2 \cdot 3x^{2-1} - 1 \cdot 5x^{1-1} = 6x - 5$</li> </ul> </li> <li>Example 2: <ul> <li>Find the derivative of $f(x) = \frac{x^2}{e^x}$</li> <li>Solution: Applying the quotient rule, we get $f’(x) = \frac{(2x)(e^x) - (x^2)(e^x)}{(e^x)^2}$

</section> <section style="font-size: 50px";> <h2 id="integration">Integration</h2> <ul> <li>Definition: <ul> <li>Integration is the process of finding the antiderivative of a function.</li> </ul> </li> <li>Techniques of Integration: <ul> <li>Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n$ is a constant and $C$ is the constant of integration</li> <li>Constant Multiple Rule: $\int cf(x) dx = c \int f(x) dx$, where $c$ is a constant</li> <li>Sum/Difference Rule: $\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx$</li> <li>Substitution Rule: $\int f(g(x)) \cdot g’(x) dx = F(g(x)) + C$, where $F(x)$ is the antiderivative of $f(x)$ and $g’(x)$ is the derivative of $g(x)$</li> </ul> </li> <li>Example 1: <ul> <li>Evaluate $\int 3x^2 dx$</li> <li>Solution: Applying the power rule, we get $\int 3x^2 dx = \frac{3x^{2+1}}{2+1} + C = x^3 + C$</li> </ul> </li> <li>Example 2: <ul> <li>Evaluate $\int 2e^{3x} dx$</li> <li>Solution: Applying the substitution rule with $u = 3x$ and $du = 3 dx$, we get $\int 2e^u \cdot \frac{du}{3} = \frac{2}{3} \int e^u du = \frac{2}{3} e^u + C = \frac{2}{3} e^{3x} + C$

</section> <section style="font-size: 50px";> <h2 id="trigonometry">Trigonometry</h2> <ul> <li>Definitions: <ul> <li>Sine: In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.</li> <li>Cosine: In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.</li> <li>Tangent: In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.</li> </ul> </li> <li>Trigonometric Identities: <ul> <li>Pythagorean Identity: $\sin^2(x) + \cos^2(x) = 1$</li> <li>Cofunction Identities: $\sin(\frac{\pi}{2} - x) = \cos(x)$, $\cos(\frac{\pi}{2} - x) = \sin(x)$, $\tan(\frac{\pi}{2} - x) = \cot(x)$</li> <li>Double Angle Identities: $\sin(2x) = 2\sin(x)\cos(x)$, $\cos(2x) = \cos^2(x) - \sin^2(x)$, $\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}$</li> </ul> </li> <li>Example 1: <ul> <li>Find the value of $\sin(\frac{\pi}{6})$</li> <li>Solution: Since $\frac{\pi}{6}$ is a special angle, we know that $\sin(\frac{\pi}{6}) = \frac{1}{2}$</li> </ul> </li> <li>Example 2: <ul> <li>Simplify $\cos\left(\frac{\pi}{2} - \frac{\pi}{3}\right)$</li> <li>Solution: Using the cofunction identity, we have $\cos\left(\frac{\pi}{2} - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

</section> <section style="font-size: 50px";> <h2 id="probability">Probability</h2> <ul> <li>Definitions: <ul> <li>Probability: The measure of the likelihood that an event will occur.</li> <li>Sample Space: The set of all possible outcomes of an experiment.</li> <li>Event: A subset of the sample space.</li> <li>Probability of an Event: The ratio of the number of favorable outcomes to the number of possible outcomes.</li> </ul> </li> <li>Probability Rules: <ul> <li>Rule of Complement: $P(A’) = 1 - P(A)$</li> <li>Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$</li> <li>Multiplication Rule: $P(A \cap B) = P(A) \cdot P(B|A)$, where $P(B|A)$ is the probability of event B occurring given that event A has already occurred.</li> </ul> </li> <li>Example 1: <ul> <li>A fair die is rolled. What is the probability of rolling an even number?</li> <li>Solution: There are 3 favorable outcomes (2, 4, 6) out of 6 possible outcomes (1, 2, 3, 4, 5, 6). Therefore, the probability is $\frac{3}{6} = \frac{1}{2}$</li> </ul> </li> <li>Example 2: <ul> <li>Two cards are drawn successively from a standard deck of playing cards without replacement. What is the probability of drawing a heart on the first card and a diamond on the second card?</li> <li>Solution: The probability of drawing a heart on the first card is $\frac{13}{52}$. Since one card has been removed, the probability of drawing a diamond on the second card is $\frac{13}{51}$ (there are now 13 diamonds left out of 51 cards). Therefore, the probability is $\frac{13}{52} \cdot \frac{13}{51}$

</section> <section style="font-size: 50px";> <h2 id="matrices">Matrices</h2> <ul> <li>Definition: <ul> <li>A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.</li> </ul> </li> <li>Matrix Operations: <ul> <li>Addition: To add two matrices, the corresponding elements are added.</li> <li>Subtraction: To subtract two matrices, the corresponding elements are subtracted.</li> <li>Scalar Multiplication: To multiply a matrix by a scalar, each element in the matrix is multiplied by the scalar.</li> <li>Matrix Multiplication: To multiply two matrices, the rows of the first matrix are multiplied by the columns of the second matrix and the products are summed.</li> </ul> </li> <li>Example 1: <ul> <li>Add the following matrices:</li> <li>$\begin{bmatrix} 2 & 4 \ 1 & 3 \end{bmatrix} + \begin{bmatrix} -1 & 3 \ 2 & 5 \end{bmatrix}$</li> <li>Solution: Adding the corresponding elements, we get $\begin{bmatrix} 1 & 7 \ 3 & 8 \end{bmatrix}$</li> </ul> </li> <li>Example 2: <ul> <li>Multiply the following matrices:</li> <li>$\begin{bmatrix} 2 & 3 \ 4 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & -2 \ 3 & 5 \end{bmatrix}$</li> <li>Solution: Applying matrix multiplication, we get $\begin{bmatrix} 11 & 4 \ 7 & 18 \end{bmatrix}$

</section> <section style="font-size: 50px";> <h2 id="complex-numbers-1">Complex Numbers</h2> <ul> <li>Definition: <ul> <li>A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ represents the imaginary unit.</li> </ul> </li> <li>Operations with Complex Numbers: <ul> <li>Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$</li> <li>Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$</li> <li>Multiplication: $(a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$</li> <li>Division: $\frac{a + bi}{c + di} = \frac{(ac + bd) - (bc - ad)i}{c^{2} + d^{2}}$</li> </ul> </li> <li>Example 1: <ul> <li>Add: $(3 + 4i) + (2 + 5i)$</li> <li>Solution: Adding the real and imaginary parts separately, we get $(3 + 2) + (4 + 5)i = 5 + 9i$</li> </ul> </li> <li>Example 2: <ul> <li>Multiply: $(3 + 2i) \cdot (4 + 5i)$</li> <li>Solution: Applying the distributive property, we get $12 + 15i + 8i + 10i^{2} = 12 + 23i - 10 = 2 + 23i$

</section> <section style="font-size: 50px";> <h2 id="vectors">Vectors</h2> <ul> <li>Definition: <ul> <li>A vector is a quantity that has both magnitude and direction.</li> </ul> </li> <li>Vector Operations: <ul> <li>Addition: To add two vectors, the corresponding components are added.</li> <li>Subtraction: To subtract two vectors, the corresponding components are subtracted.</li> <li>Scalar Multiplication: To multiply a vector by a scalar, each component in the vector is multiplied by the scalar.</li> <li>Dot Product: To find the dot product of two vectors, the corresponding components are multiplied and then summed.</li> <li>Cross Product: To find the cross product of two vectors, the magnitude and direction are calculated according to specific rules.</li> </ul> </li> <li>Example 1: <ul> <li>Add the following vectors:</li> <li>$\vec{v} = \begin{bmatrix} 2 \ 5 \end{bmatrix}$ and $\vec{u} = \begin{bmatrix} -1 \ 3 \end{bmatrix}$</li> <li>Solution: Adding the corresponding components, we get $\vec{v} + \vec{u} = \begin{bmatrix} 2 + (-1) \ 5 + 3 \end{bmatrix}$</li> </ul> </li> <li>Example 2: <ul> <li>Find the dot product of the following vectors:</li> <li>$\vec{v} = \begin{bmatrix} 2 \ -3 \end{bmatrix}$ and $\vec{u} = \begin{bmatrix} 4 \ 1 \end{bmatrix}$</li> <li>Solution: Multiplying the corresponding components and then summing, we get $\vec{v} \cdot \vec{u} = (2)(4) + (-3)(1)$

Logarithm - Example – Properties of logarithms

• Example 1:
  • Simplify $\log_{3}27$
  • Solution: Since $3^{3} = 27$, the logarithm of 27 to the base 3 is 3. Therefore, $\log_{3}27 = 3$
• Example 2:
  • Solve for $x$: $\log_{5}x = 2$
  • Solution: Since $5^{2} = 25$, the value of $x$ is 25. Therefore, $x = 25$

Exponents and Radicals - Example – Laws of exponents

• Example 1:
  • Simplify: $2^{3} \times 2^{2}$
  • Solution: Applying the product rule of exponents, we get $2^{3} \times 2^{2} = 2^{3+2} = 2^{5} = 32$
• Example 2:
  • Simplify: $\sqrt[3]{8} \cdot \sqrt[3]{27}$
  • Solution: Applying the product rule of radicals, we get $\sqrt[3]{8} \cdot \sqrt[3]{27} = \sqrt[3]{8 \times 27} = \sqrt[3]{216} = 6$

Complex Numbers - Example – Addition and multiplication of complex numbers

• Example 1:
  • Add: $(3 + 4i) + (2 +