AdMaths Grade 11: Calculus and Algebra Study Material - Summary

Calculus: The Chain Rule (C9)

• Chain Rule: $(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$
• Derivative of composite function = derivative of outer function (in terms of inner function) * derivative of inner function.
• $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$

Calculus: Product and Quotient Rule (C10)

• Product Rule: $\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$
• Quotient Rule: $\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}$
• Stationary points: $f'(x) = 0$

Algebra: Absolute Value Graphs (A17.1)

• $y = |f(x)|$ reflects the negative part of $f(x)$ about the x-axis
• $y = (x - p)^2 + q$ has a turning point at $(p, q)$
• $y = \frac{k}{x-p} + q$ has a vertical asymptote at $x = p$ and a horizontal asymptote at $y = q$

Algebra: Absolute Value Inequalities (A17.2)

• Graphical solving:
  • Sketch graphs and find intersection points.
  • Identify regions where inequality is satisfied.

Calculus: Riemann (C11)

• $\sum_{i=1}^{n}$ is sigma notation for summation.
• $\sum<em>{i=1}^{n} c \cdot f(i) = c \cdot \sum</em>{i=1}^{n} f(i)$
• $\sum<em>{i=1}^{n} [f(i) + g(i)] = \sum</em>{i=1}^{n} f(i) + \sum_{i=1}^{n} g(i)$
• Common Series:
  • $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
  • $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$
  • $\sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4}$
  • $\sum_{i=1}^{n} 1 = n$
• Riemann Sum: $A = \lim<em>{n \to \infty} \Delta x</em>i \sum<em>{i=1}^{n} f(x</em>i)$
  • $\Delta x_i = \frac{b-a}{n}$
  • $x<em>i = a + i \cdot \Delta x</em>i$

Calculus: Anti-Derivative and Definite Integral (C12)

• General Anti-derivative Rule: $\int a x^n dx = \frac{a}{n+1} x^{n+1} + c$
• Definite Integral: $\int_{a}^{b} f(x) dx = F(b) - F(a)$

Calculus: Definite Integral (C13)

• Area below x-axis is negative; take absolute value for area.
• Area between two curves: $\int_{a}^{b} [f(x) - g(x)] dx$, where f(x) is the top function and g(x) is the bottom function.

Algebra: Radian Measure - Introduction (A18)

• $\theta = \frac{s}{r}$ ($\theta$ in radians)
• Conversion Formulas:
  • Degrees to radians: Multiply by $\frac{\pi}{180}$
  • Radians to degrees: Multiply by $\frac{180}{\pi}$
• Key radian values:
  • $360^\circ = 2\pi$
  • $180^\circ = \pi$
  • $90^\circ = \frac{\pi}{2}$
  • $60^\circ = \frac{\pi}{3}$
  • $45^\circ = \frac{\pi}{4}$
  • $30^\circ = \frac{\pi}{6}$

Algebra: Inverse Trig Functions (A19)

• Inverse trig functions are restricted to specific ranges to be functions.
• arcsin(x): Range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$
• arccos(x): Range is $[0, \pi]$
• arctan(x): Range is $(-\frac{\pi}{2}, \frac{\pi}{2})$

Algebra: Theorem of De Moivre (A20)

• Polar form: $z = r(\cos \theta + i \sin \theta) = r \cdot cis \theta$
• Exponential form: $z = re^{i\theta}$
• Multiplication: $z<em>1 z</em>2 = r<em>1 r</em>2 [\cos(\theta<em>1 + \theta</em>2) + i \sin(\theta<em>1 + \theta</em>2)]$
• Division: $\frac{z<em>1}{z</em>2} = \frac{r<em>1}{r</em>2} [\cos(\theta<em>1 - \theta</em>2) + i \sin(\theta<em>1 - \theta</em>2)]$
• De Moivre's Theorem: $(r(\cos \theta + i \sin \theta))^n = r^n [\cos(n\theta) + i \sin(n\theta)]$

Algebra: 3D Vectors (A21)

• 3D vector notation: $v = \begin{bmatrix} v<em>x \ v</em>y \ v_z \end{bmatrix}$
• Modulus: $|v| = \sqrt{v<em>x^2 + v</em>y^2 + v_z^2}$
• Dot product: $v \cdot w = v<em>x w</em>x + v<em>y w</em>y + v<em>z w</em>z = |v||w|\cos\theta$
• Cross product: $v \times w = \begin{bmatrix} v<em>y w</em>z - v<em>z w</em>y \ v<em>z w</em>x - v<em>x w</em>z \ v<em>x w</em>y - v<em>y w</em>x \end{bmatrix}$
• Area of parallelogram: $|v \times w|$
• Volume of parallelepiped: $|u \cdot (v \times w)|$