A Maths Formulas
Algebra section
1. Complete the square
$ x^2 \pm bx + c $ = $ \left(x \pm {b \over 2} \right)^2 - \left(b \over 2\right)^2 + c $
Make sure the coefficient of $x^2$ is equals to $1$ before applying the formula (see part ii and iii below).
2. Deducing maximum value or minimum value
$$ a(x - h)^2 + k $$
$ \text{If } a > 0,$ $ \text{ expression has minimum value of } k \text{ when } x = h $
$ \text{If } a < 0,$ $ \text{ expression has maximum value of } k \text{ when } x = h $
1. Solve quadratic inequality
(a) Solve $(x - a)(x - b) > 0$, given that $ b > a $.
$$ x < a \text{ or } x > b $$
(b) Solve $(x - a)(x - b) < 0,$ given that $ b > a $:
$$ a < x < b $$
2. Number of roots of a quadratic equation, $ax^2 + bx + c = 0$
1. If equation has two real and distinct roots, then $b^2 - 4ac$ $ > 0 $
2. If equation has two real and equal roots (or one real root), then $b^2 - 4ac$ $ = 0 $
3. If equation has real roots (either one or two real roots), then $b^2 - 4ac$ $ \ge 0 $
4. If equation has no real roots, then $b^2 - 4ac$ $ < 0 $
3. $y = ax^2 + bx + c$ is always positive
$\text{Condition 1: } $ $ a > 0 $
$\text{Condition 2: } b^2 - 4ac $ $ < 0 $
4. $y = ax^2 + bx + c$ is always negative
$\text{Condition 1: } $ $ a < 0 $
$\text{Condition 2: } b^2 - 4ac $ $ < 0 $
5. Number of intersections between line and curve
1. If line meets curve at two points, then $b^2 - 4ac$ $ > 0 $
2. If line meets curve once or is tangent to the curve, then $b^2 - 4ac$ $ = 0 $
3. If line meets curve once or twice, then $b^2 - 4ac$ $ \ge 0 $
4. If line does not meet the curve, then $b^2 - 4ac$ $ < 0 $
1. Multiplication
$ \sqrt{a} \times \sqrt{a} = $ $ \phantom{.} a $
$ \sqrt{a} \times \sqrt{b} = $ $ \sqrt{a \times b} $
2. Division
$ { \sqrt{a} \over \sqrt{b} }= $ $ \sqrt{a \over b} $
3. Rationalise denominator (one term)
$ {2 \over 3 \sqrt{5}} \times $ $ {\sqrt{5} \over \sqrt{5}} = {2\sqrt{5} \over 3(5)} = {2\sqrt{5} \over 15} $
4. Rationalise denominator (two terms)
$ {1 \over 4 + \sqrt{5}} \times $ $ {4 - \sqrt{5} \over 4 - \sqrt{5}} = {4 - \sqrt{5} \over (4)^2 - (\sqrt{5})^2} = {4 - \sqrt{5} \over 11} $
1. Division algorithm (can be used to factorise a polynomial)
$ \text{Polynomial} = $ $ \text{Divisor} \times \text{Quotient} + \text{Remainder} $
2. Factor theorem
If the polynomial $f(x)$ has a linear factor $x + a$, then $f(-a)$ = $ \phantom{.} 0 $
3. Remainder theorem
If $x + a$ is not a factor of the polynomial $f(x)$, then when $f(x)$ is divided by $x + a$, remainder = $ f(-a) $
4. Sum of cubes & difference of cubes
$ a^3 + b^3 = $ $ (a + b)(a^2 - ab + b^2) $
$ a^3 - b^3 = $ $ (a - b)(a^2 + ab + b^2) $
1. Differentiate between proper fraction and improper fraction
For an algebraic fraction in the form ${f(x) \over g(x)}$,
- If the degree of $f(x)$ < the degree of $g(x)$, then it is a proper fraction
- If the degree of $f(x)$ ≥ the degree of $g(x)$, then it is an improper fraction
2. Rules for proper fraction
Linear factors: $ {mx + n \over (ax + b)(cx - d)} = $ ${A \over ax + b} + {B \over cx - d}$
Repeated linear factors: $ {mx + n \over (ax + b)(cx - d)^2} = $ ${A \over ax + b} + {B \over cx - d} + {C \over (cx - d)^2}$
Quadratic factor that cannot be factorised: $ {mx + n \over (ax + b)(x^2 + d)} = $ ${A \over ax + b} + {Bx + C \over x^2 + d}$
3. For improper fractions
- Perform long division and express result in the form $ {\text{Polynomial} \over \text{Divisor}} = $ $ \text{Quotient} + { \text{Remainder} \over \text{Divisor} } $
- Break down the proper fraction into partial fractions
1. Binomial expansion (provided)
\begin{align} (a+b)^n & =a^n+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^2+...+\binom{n}{r}a^{n-r}b^r+ \ldots + b^n \\ \\ \text{where } n \text{ is a positive integer and }&\binom{n}{r}={n!\over r!(n-r)!}={n(n-1) \ldots (n-r+1)\over r!} \end{align}
2. Binomial coefficients with n
${n \choose 1}$ = $ \phantom{.} n $
${n \choose 2}$ = $ {n(n - 1) \over 2!} $
${n \choose 3}$ = $ {n(n - 1)(n - 2) \over 3!} $
3. General term formula
$\text{General term, } T_{r + 1} = $ $ {n \choose r} a^{n - r} b^r $
4. Number of terms
$ \text{In the expansion of } (a + b)^n, \text{ there are } $$ n + 1 $ $ \text{ terms} $
1. Indices laws
$ a^m \times a^n = (a^m)(a^n) = \phantom{.} $ $ a^{m + n} $
$ a^m \div a^n = {a^m \over a^n} = \phantom{.} $ $ a^{m - n} $
$ a^m \times b^m = (a^m)(b^m) = \phantom{.} $ $ (ab)^m $
$ a^m \div b^m = {a^m \over b^m} = \phantom{.} $ $ \left(a \over b \right)^m $
$ (a^m)^n = \phantom{.} $ $ a^{mn} $
$ a^0 = \phantom{.} $ $ 1 $
$ a^{-n} = \phantom{.} $ $ {1 \over a^n} $
$ {1 \over a^{-n}} = \phantom{.} $ $ a^{-(-n)} = a^n $
$ \left(a \over b\right)^{-n} = \phantom{.} $ $ \left(b \over a\right)^n $
$ a^{1 \over n} = \phantom{.} $ $ \sqrt[n]{a} $
$ a^{1 \over 2} = \phantom{.} $ $ \sqrt{a} $
$ a^{m \over n} = \phantom{.} $ $ \sqrt[n]{a^m} $
2. Ways to solve an exponential equation
(a) Solve $3^x = -3$
$$ \text{No solution since } 3^x > 0 $$
(b) Solve $3^x = 81$
\begin{align*} 3^x & = 81 \\ 3^x & = 3^4 \phantom{00000} [\text{Change to same base } 3] \\ \\ \therefore x & = 4 \end{align*}
(c) Solve $3^x = 4$
\begin{align*} 3^x & = 4 \\ \lg 3^x & = \lg 4 \phantom{000000} [\text{Can use } \ln \text{ - same result}] \\ x \lg 3 & = \lg 4 \phantom{000000} [\text{Power law (logarithms)}] \\ x & = {\lg 4 \over \lg 3} \\ x & \approx 1.26 \end{align*}
3. Graph of exponential functions
1. Exponential form and logarithmic form
$ y = \log_a x \phantom{.} \Longleftrightarrow \phantom{.} x = $ $ \phantom{.} a^y $
2. Conditions for $ \log_a x $ to be defined
- $a$ is a positive real number and not equals to 1
- $x$ is a positive real number
3. Properties and special logarithms
$ \log_a 1 = $ $ \phantom{.} 0 $
$ \log_a a = $ $ \phantom{.} 1 $
$ \lg x = $ $ \log_{10} x $ and $\lg 10 = $ $ \log_{10} 10 = 1 $
$ \ln x = $ $ \log_{e} x $ and $\ln e = $ $ \log_e e = 1 $
4. Laws of logarithms
Product law: $ \log_a xy = $ $ \log_a x + \log_a y $
Quotient law: $ \log_a {x \over y} = $ $ \log_a x - \log_a y $
Power law: $ \log_a x^r = $ $ r \log_a x $
5. Change-of-base formula
$ \log_a b = $ $ {\log_c b \over \log_c a} $
6. Graph of logarithmic functions
Geometry section
1. Formulas involving two points $A(x_1, y_1)$ and $B(x_2, y_2)$
$ \text{Length of } AB $ = $ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $
$ \text{Gradient of } AB $ = $ {y_2 - y_1 \over x_2 - x_1} $
$ \text{Midpoint of } AB $ = $ \left( {x_1 + x_2 \over 2}, {y_1 + y_2 \over 2} \right) $
2. Equation of a non-vertical straight line
$ \text{Equation: } $$ y = mx + c $
$ m \text{ represents the } $$ \text{gradient of the straight line} $
$ c \text{ represents the } $$ y\text{-intercept of the straight line} $
3. Equation of a horizontal line
$ \text{Equation of horizontal line passing though } (a, b) \text{:} $$ \phantom{.} y = b $
$ \text{Gradient of line} $ = $ \phantom{0} 0 \phantom{0} $
4. Equation of a vertical line
$ \text{Equation of vertical line passing though } (a, b) \text{:} $$ \phantom{.} x = a $
$ \text{Gradient of line is} $ $ \text{ undefined } $
5. Relationship between two lines
If two lines are parallel, they have the same gradient, i.e. $m_1 = m_2$
If two lines are not parallel, the lines will meet at a point. The coordinates of the point of intersection can be found by solving simultaneous equations using the equation of each line.
If two lines are perpendicular, then the gradient of both lines are related by $m_1 \times m_2 = $ $ - 1 $.
6. Find area by 'shoelace' method:
$ \text{Area of figure} $ = $ {1 \over 2} \left| \begin{matrix} x_1 & x_2 & ... & x_n & x_1 \\ y_1 & y_2 & ... & y_n & y_1 \end{matrix} \right| $
Steps:
- Select the points in anti-clockwise order
- Repeat the first point chosen
- When calculating the area, go ↘ , then go ↗
Example
Find the area of the quadrilateral $ABCD$.
\begin{align*} \text{Area of quadrilateral } ABCD & = {1 \over 2} \left| \begin{matrix} -2 & -2 & 3 & 4 & -2 \\ 4 & 1 & -3 & 4 & 4 \end{matrix} \right| \\ & = {1 \over 2} [ (-2)(1) + (-2)(-3) + (3)(4) + (4)(4) ] - {1 \over 2} [ (4)(-2) + (1)(3) + (-3)(4) + (4)(-2)] \\ & = 28.5 \text{ units}^2 \end{align*}
7. Using column vectors to find coordinates:
Column vectors is usually taught in Secondary 4 E Maths. It is useful to solve questions like the following!
Example
The points $A(-2, 1)$, $B(0, 2)$ and $C$ lie on a straight line as shown above. Given that $AB: AC = 1:3$, find the coordinates of $C$.
\begin{align*} \overrightarrow{AB} & = {2 \choose 1} \phantom{000000} [\text{Move 2 units in positive } x \text{-direction, 1 unit in positive } y \text{-direction}] \\ \\ \overrightarrow{AC} & = 3 \overrightarrow{AB} \\ & = 3 {2 \choose 1} \\ & = {6 \choose 3} \\ \\ x \text{-coordinate of } C & = -2 + 6 \\ & = 4 \\ \\ y \text{-coordinate of } C & = 1 + 3 \\ & = 4 \\ \\ \therefore & \phantom{.} C(4, 4) \end{align*}
1. Standard form (i.e. centre & radius form)
$ \text{Equation: } $$ (x - a)^2 + (y - b)^2 = r^2 $
$ \text{Centre: } $$ (a, b) $
$ \text{Radius} = \phantom{.} $$ r \text{ units} $
2. General form
$ \text{Equation: } $$ x^2 + y^2 + 2gx + 2fy + c = 0 $
$ \text{Centre: } $$ (-g, -f) $
$ \text{Radius} = \phantom{.} $$ \sqrt{ g^2 + f^2 - c } $
3. Circle property involving diameter
If points $A$, $B$ and $C$ lie on the circle such that $ \angle ABC = 90^\circ $, then $AC$ is the diameter of the circle.
Thus, the centre of the circle is the midpoint of $AC$ and the radius of the circle is equals to $ {\text{Length of } AC \over 2} $.
4. Circle property involving chord
The perpendicular bisector of chord $AB$ passes through the centre of the circle.
5. Circle property involving tangent to the circle
If $O$ is the centre of the circle and the line $ABC$ is tangent to the circle at $B$, then $ \angle OBA = \angle OBC = 90^\circ $.
The normal to the circle at $B$ is perpendicular ($m_1 \times m_2 = -1$) to the tangent to the circle at $B$ and passes through the centre of the circle..
1. Convert equation to linear form
$$ y = a \sqrt{x} + {b \over \sqrt{x}} $$
\begin{align*} \sqrt{x} y & = \sqrt{x} \left( a \sqrt{x} + {b \over \sqrt{x} } \right) \\ \sqrt{x} y & = ax + b \phantom{000000000000} [Y = mX + c] \\ \\ Y & = \sqrt{x} y, m = a, X = x, c = b \\ \\ \text{Plot } & \underbrace{ \sqrt{x} y }_\text{Vertical axis} \text{ against } \underbrace{x}_\text{Horizontal axis} \end{align*}
2. Convert equation to linear form (logarithms)
$$ y = ax^b $$
\begin{align*} y & = ax^b \\ \lg y & = \lg (ax^b) \\ \lg y & = \lg a + \lg x^b \phantom{000000} [\text{Product law (logarithms)} ] \\ \lg y & = \lg a + b \lg x \phantom{00000.} [\text{Power law (logarithms)}] \\ \lg y & = b \lg x + \lg a \phantom{000000000000} [Y = mX + c] \\ \\ Y & = \lg y, m = b, X = \lg x, c = \lg a \\ \\ \text{Plot } & \underbrace{ \lg y }_\text{Vertical axis} \text{ against } \underbrace{ \lg x }_\text{Horizontal axis} \end{align*}
1. Angle properties
$ \angle a = \angle b \phantom{0} ($ $ \text{Vertically opposite angles} $ $\text{)}$
$ \angle a = \angle b \phantom{0} ($ $ \text{Alternate angles} $ $\text{)}$
$ \angle a + \angle c = $ $ 180^\circ \text{ (Interior angles} $$\text{)}$
$ \angle a = \angle d \phantom{0} ($ $ \text{Corresponding angles} $ $\text{)}$
2. Midpoint theorem
If D is the midpoint of AB and E is the midpoint of AC, then
$ DE \phantom{.} // \phantom{.} BC $ $\text{ and } $ $ DE = {1 \over 2} BC $
3. Circle properties
$ \angle ACB = \angle ADB \text{ (}$ $ \text{Angles in the same segment} $ $ \text{)} $
If O is the centre of the circle,
$ \angle AOB = 2 \times \angle ACB \text{ (}$ $ \text{Angle at centre} = 2 \times \text{Angle at circumference} $ $ \text{)} $
$ \angle ACB + \angle ADB = 180^\circ \text{ (}$ $ \text{Angles in opposite segments} $ $ \text{)} $
If AOB is the diameter of the circle,
$ \angle ACB = 90^\circ \text{ (}$ $ \text{Right-angle in semi-circle} $ $ \text{)} $
If O is the centre of the circle and the line AC is tangent to the circle at B,
$ \angle OBA = 90^\circ \text{ (}$ $ \text{Tangent perpendicular to radius} $ $ \text{)} $
If O is the centre of the circle and tangents to the circle at A and at B meet at an external point T, then
$ AT = BT $ $ \text{ and } $ $ \angle OTA = \angle OTB $
4. Alternate segment theorem (or tangent-chord theorem)
If the line AC is tangent to the circle at B,
$ \angle DBA = $$ \angle DEB $ $ \text{ (Alternate segment theorem)} $
$ \angle EBC = $$ \angle EDB $ $ \text{ (Alternate segment theorem)} $
5. Congruency tests for triangles
Side-Side-Side (SSS)
Angle-Side-Angle (ASA)
Angle-Angle-Side (AAS)
Right angle-Hypotenuse-Side (RHS)
Side-Angle-Side (SAS)
6. Similarity tests for triangles
Side-Side-Side (SSS)
Angle-angle (AA)
Side-Angle-Side (SAS)
Trigonometry section
1. Sine graph
(a) Shape and features of $y = a \sin bx + c$, when $ a > 0 $
\begin{align*} \text{Center line: } & y = c \\ \\ \text{Amplitude} & = a \\ \\ \text{Period} & = {360^\circ \over b} = {2\pi \text{ radians} \over b} \end{align*}
(b) Shape and features of $y = a \sin bx + c$, when $ a < 0 $
\begin{align*} \text{Center line: } & y = c \\ \\ \text{Amplitude} & = a \\ \\ \text{Period} & = {360^\circ \over b} = {2\pi \text{ radians} \over b} \end{align*}
2. Cosine graph
(a) Shape and features of $y = a \cos bx + c$, when $ a > 0 $
\begin{align*} \text{Center line: } & y = c \\ \\ \text{Amplitude} & = a \\ \\ \text{Period} & = {360^\circ \over b} = {2\pi \text{ radians} \over b} \end{align*}
(b) Shape and features of $y = a \cos bx + c$, when $ a < 0 $
\begin{align*} \text{Center line: } & y = c \\ \\ \text{Amplitude} & = \text{Positive value of } a \\ \\ \text{Period} & = {360^\circ \over b} = {2\pi \text{ radians} \over b} \end{align*}
3. Tangent graph
(a) Shape and features of $y = a \tan bx$, when $ a > 0 $
Note: The graph cannot pass through the vertical asymptote $x = {90^\circ \over b}$
\begin{align*} \text{Period} & = {180^\circ \over b} = {\pi \text{ radians} \over b} \end{align*}
(b) Shape and features of $y = a \tan bx$, when $ a < 0 $
Note: The graph cannot pass through the vertical asymptote $x = {90^\circ \over b}$
\begin{align*} \text{Period} & = {180^\circ \over b} = {\pi \text{ radians} \over b} \end{align*}
1. Four quadrants
sin θ is positive in the first and second quadrants and negative in the third and fourth quadrants.
cos θ is positive in the first and fourth quadrants and negative in the second and third quadrants.
tan θ is positive in the first and third quadrants and negative in the second and fourth quadrants.
2. TOA CAH SOH
$ \tan \theta = $ $ \phantom{.} {Opp \over Adj} $, $ \cos \theta = $ $ \phantom{.} {Adj \over Hyp} $, $ \sin \theta = $ $ \phantom{.} {Opp \over Hyp} $
1. Angle in radians
$ 180^\circ = $ $ \phantom{.} \pi $ $ \text{ radians}$
2. Special angles
$30^\circ \text{ or } {\pi \over 6}$ | $45^\circ \text{ or } {\pi \over 4}$ | $60^\circ \text{ or } {\pi \over 3}$ | |
---|---|---|---|
$\sin \theta$ | $ \phantom{.} {1 \over 2} $ | $ \phantom{.} {1 \over \sqrt{2}} $ | $ \phantom{.} {\sqrt{3} \over 2} $ |
$\cos \theta$ | $ \phantom{.} {\sqrt{3} \over 2} $ | $ \phantom{.} {1 \over \sqrt{2}} $ | $ \phantom{.} {1 \over 2} $ |
$\tan \theta$ | $ \phantom{.} {1 \over \sqrt{3}} $ | $ \phantom{.} 1 $ | $ \phantom{.} \sqrt{3} $ |
3. Identities
$ \cot \theta = $ $ \phantom{.} {1 \over \tan \theta} $
$ \sec \theta = $ $ \phantom{.} {1 \over \cos \theta} $
$ \text{cosec } \theta = $ $ \phantom{.} {1 \over \sin \theta} $
$ \tan \theta = $ $ \phantom{.} {\sin \theta \over \cos \theta} $
$ \cot \theta = $ $ \phantom{.} {1 \over \tan \theta} = {\cos \theta \over \sin \theta} $
4. Pythagoreans identity (provided)
\begin{align*} \sin^2 A & + \cos^2 A = 1 \\ \sec^2 A & = 1 + \tan^2 A \\ \text{cosec}^2 A & = 1 + \cot^2 A \end{align*}
5. Addition formulas (provided)
\begin{align*} \sin (A \pm B) & = \sin A \cos B \pm \cos A \sin B \\ \cos (A \pm B) & = \cos A \cos B \mp \sin A \sin B \\ \tan (A \pm B) & = { \tan A \pm \tan B \over 1 \mp \tan A \tan B} \end{align*}
6. Double angle formulas (provided)
\begin{align*} \sin 2A & = 2 \sin A \cos A \\ \cos 2A = \cos^2 A - \sin^2 A & = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A \\ \tan 2A & = {2 \tan A \over 1 - \tan^2 A} \end{align*}
7. R-formulas
$a \sin x \pm b \cos x = $ $ R \sin (x \pm \alpha) $
$a \cos x \pm b \sin x = $ $ R \cos (x \mp \alpha) $
$ \text{For both, } R = $ $\sqrt{a^2 + b^2}$ and $\alpha = $ $ \tan^{-1} \left(b \over a\right) $
8. Principal values:
Principal values of $ \sin^{-1} x $: $ -90^\circ \le \sin^{-1} x \le 90^\circ $
Principal values of $ \cos^{-1} x $: $ 0^\circ \le \cos^{-1} x \le 180^\circ $
Principal values of $ \tan^{-1} x $: $ -90^\circ < \tan^{-1} x < 90^\circ $
Differentiation & integration section
1. Constants & single algebraic terms
If $a$ is a constant, $ {d \over dx} (a) = $ $ \phantom{0} 0 \phantom{0} $
If $a$ is a constant, $ {d \over dx} (ax^n) = $ $ a n x^{n - 1} $
2. Trigonometric terms
$ {d \over dx} \{ \sin [f(x)] \} = $ $ f'(x) . \cos [f(x)] $
$ {d \over dx} \{ \cos [f(x)] \} = $ $ f'(x) . - \sin [f(x)] $
$ {d \over dx} \{ \tan [f(x)] \} = $ $ f'(x) . \sec^2 [f(x)] $
Note: $f'(x)$ is the derivative of $f(x)$
3. Exponential terms
$ {d \over dx} \left[ e^{f(x)} \right] = $ $ f'(x) . e^{f(x)} $
Note: $f'(x)$ is the derivative of $f(x)$
4. Natural logarithms ($\ln$)
$ {d \over dx} \{ \ln [f(x)] \} = $ $ {f'(x) \over f(x)} $
Note: $f'(x)$ is the derivative of $f(x)$
5. Chain rule
$ {d \over dx} [f(x)]^n = $ $ n [f(x)]^{n - 1} . f'(x) $
Note: $f'(x)$ is the derivative of $f(x)$
6. Chain rule & trigonometry
$ {d \over dx} [\sin^n f(x) ] = $ $ (n) [ \sin^{n - 1} f(x) ] . f'(x) . \cos f(x) $
$ {d \over dx} [\cos^n f(x) ] = $ $ (n) [ \cos^{n - 1} f(x) ] . f'(x) . - \sin f(x) $
$ {d \over dx} [\tan^n f(x) ] = $ $ (n) [ \tan^{n - 1} f(x) ] . f'(x) . \sec^2 f(x) $
7. Product rule
$ {d \over dx} (uv) = $ $ u {dv \over dx} + v{du \over dx} $
8. Quotient rule
$ {d \over dx} \left(u \over v\right) = $ $ {v {du \over dx} - u {dv \over dx} \over v^2} $
1. Constants & single algebraic terms
If $a$ is a constant, $ \int a \phantom{.} dx = $ $ ax + C $
If $a$ is constant and $n \ne -1$, $ \int ax^{n} \phantom{.} dx = $ $ {a \over n + 1} x^{n+ 1} + C $
If $a$ is constant, $ \int ax^{-1} \phantom{.} dx = \int {a \over x} \phantom{.} dx = $ $ a \ln x + C $
2. Integrate $[ f(x) ]^n $
If $n \ne -1$, $ \int [f(x)]^n \phantom{.} dx = $ $ { [f(x)]^{n + 1} \over f'(x). (n + 1) } + C $
$ \int [f(x)]^{-1} \phantom{.} dx = \int {1 \over f(x)} \phantom{.} dx = $ $ {\ln [f(x)] \over f'(x)} + C $
Note: $f'(x)$ is the derivative of $f(x)$
3. Integrate exponential terms
$ \int e^{f(x)} \phantom{.} dx = $ $ {e^{f(x)} \over f'(x)} + C $
Note: $f'(x)$ is the derivative of $f(x)$
4. Integrate trigonometric terms
$ \int \sin [f(x)] \phantom{.} dx = $ $ {- \cos [f(x)] \over f'(x)} + C $
$ \int \cos [f(x)] \phantom{.} dx = $ $ {\sin [f(x)] \over f'(x)} + C $
$ \int \sec^2 [f(x)] \phantom{.} dx = $ $ {\tan [f(x)] \over f'(x)} + C $
Note: $f'(x)$ is the derivative of $f(x)$
5. Use differentiation result to integrate
Example 1
Given that ${d \over dx} \left[ x \over (x + 1)^2 \right] = {1 - x \over (x + 1)^3}$, find $ \int {2 - 2x \over (x + 1)^3} \phantom{.} dx$.
\begin{align*} {d \over dx} \left[ x \over (x + 1)^2 \right] & = { 1 - x \over (x + 1)^3 } \\ \\ \implies \int {1 - x \over (x + 1)^3 } \phantom{.} dx & = {x \over (x + 1)^2 } \\ \\ \int {2 - 2x \over (x + 1)^3} \phantom{.} dx & = \int { 2(1 - x) \over (x + 1)^3} \phantom{.} dx \\ & = 2 \int { 1 - x \over (x + 1)^3 } \phantom{.} dx \\ & = 2 \left[ x \over (x + 1)^2\right] \\ & = { 2x \over (x + 1)^2 } + c \end{align*}
Example 2
Given that ${d \over dx} ( x \ln x) = 1 + \ln x $, find $ \ln x \phantom{.} dx$.
\begin{align*} \int 1 + \ln x \phantom{.} dx & = x \ln x \\ \int 1 \phantom{.} dx + \int \ln x \phantom{.} dx & = x \ln x \\ \int \ln x \phantom{.} dx & = x \ln x - \int 1 \phantom{.} dx \\ & = x \ln x - x + c \end{align*}
Applications of differentiation
1. Derivatives
The first derivative, ${dy \over dx}$, is the rate of change of $y$ with respect to $x$ (i.e. gradient of the curve). For example, if ${dy \over dx} = 2$, it means that when $x$ increases by $1$ unit, $y$ increases by $2$ units.
The second derivative, ${d^2 y \over dx^2}$, is the rate of change ${dy \over dx}$ with respect to $x$. For example, if ${d^2 y \over dx^2} = 2$, it means that when $x$ increases by $1$ unit, ${dy \over dx}$ increases by $2$ units.
2. Tangent & normal to the curve
$ \text{Gradient of tangent at } P(x_1, y_1) = $ $ \left. {dy \over dx} \right|_{x = x_1} $
$ \text{Gradient of normal at } P(x_1, y_1) = $ $ {-1 \over \text{Gradient of tangent at } P(x_1, y_1)} $
To form the equation of the line, you need the gradient of the line and the coordinates of a point that the line passes through.
3. Increasing function & decreasing function
For an increasing function (as $x$ increases, $y$ increases), ${dy \over dx}$ $ > 0 $
For a decreasing function (as $x$ increases, $y$ decreases), ${dy \over dx}$ $ < 0 $
4. Stationary point
(a) At stationary point(s), ${dy \over dx} = $$ \phantom{.} 0 \phantom{.} $. A stationary point can be:
- A maximum point (/ ‾ \)
- A minimum point (\ _ /)
- A stationary point of inflexion (/ - / or \ - \)
(b) First derivative test (format):
$x$ | $x^-$ | $x$ | $x^+$ |
---|---|---|---|
${dy \over dx}$ | |||
Slope |
(c) Second derivative test:
- ${d^2 y \over dx^2} > 0 $ means the point is a minimum point
- ${d^2 y \over dx^2} < 0 $ means the point is a maximum point
- ${d^2 y \over dx^2} = 0 $ means the test is inconclusive (use first derivative test instead)
5. Connected rate of change
If $y$ and $x$ are related, then
$ {dy \over dt} = $ $ {dy \over dx} \times {dx \over dt} $
$ {dx \over dt} = $ $ {dx \over dy} \times {dy \over dt} $
Definite integrals & area bounded by curve
1. Properties of definite integrals
$ - \int_a^b f(x) \phantom{.} dx = $ $ \int_b^a f(x) \phantom{.} dx $
$ \text{If } k \text{ is a constant, } \int_a^b k f(x) \phantom{.} dx = $ $ k \int_a^b f(x) \phantom{.} dx $
$ \int_a^b f(x) \pm g(x) \phantom{.} dx = $ $ \int_a^b f(x) \phantom{.} dx \pm \int_a^b g(x) \phantom{.} dx $
$ \int_a^b f(x) \phantom{.} dx + \int_b^c f(x) \phantom{.} dx = $ $ \int_a^c f(x) \phantom{.} dx $
2. Area bounded by curve and x-axis
$ \text{Area bounded} = $ $ \int_{x_1}^{x_2} f(x) \phantom{.} dx $
$ \text{Area bounded} = $ $ - \int_{x_1}^{x_2} f(x) \phantom{.} dx $
3. Area bounded by curve and y-axis
$ \text{Area bounded} = $ $ \int_{y_1}^{y_2} f(y) \phantom{.} dy $
$ \text{Area bounded} = $ $ - \int_{y_1}^{y_2} f(y) \phantom{.} dy $
4. Area bounded by curve and line
$ \text{Area of region A} = $ $ \text{Area of triangle} $
$ \text{Area of region B} = $ $ \int_2^3 -x^2 + 2x + 3 \phantom{.} dx $
$ \text{Area of region D} = $ $ \text{Area of triangle} $
$ \text{Area of region C} = $ $ \int_{-1}^2 -x^2 + 2x + 3 \phantom{.} dx - \text{Area of region D} $
1. Conversion between displacement, velocity & acceleration:
2. Displacement
Displacement is the shortest distance between the initial position and the current position of an object.
Key terms:
- Initial displacement refers to the value of $s$ when $t = 0$
- Object is at starting/reference point, $O$, implies $s = 0$
- Object reaches maximum displacement means $ {ds \over dt} = 0 \text{ and } {d^2 s \over dt^2} < 0 $
- Object reaches minimum displacement means $ {ds \over dt} = 0 \text{ and } {d^2 s \over dt^2} > 0 $
3. Velocity
Velocity, $v = {ds \over dt}$, is the rate of change of displacement.
If $v > 0$, the object is moving in the positive direction. If $v < 0$, the object is moving in the opposite direction.
Key terms:
- Initial velocity refers to the value of $v$ when $t = 0$
- Object is momentarily at rest means $v = 0$ (note: object may change direction)
- Object reaches maximum velocity means $ {dv \over dt} = 0 \text{ and } {d^2 v \over dt^2} < 0 $
- Object reaches minimum velocity means $ {dv \over dt} = 0 \text{ and } {d^2 v \over dt^2} > 0 $
4. Acceleration
Acceleration, $a = {dv \over dt}$, is the rate of change of velocity.
If $a = 0$, then the object is moving at constant velocity (or at rest). If $a > 0$, the object is accelerating (i.e. velocity is increasing). If $a < 0$, the object is decelerating (i.e. velocity is decreasing).
Key terms:
- Initial acceleration refers to the value of $a$ when $t = 0$
- Object reaches maximum acceleration means $ {da \over dt} = 0 \text{ and } {d^2 a \over dt^2} < 0 $
- Object reaches minimum velocity means $ {da \over dt} = 0 \text{ and } {d^2 a \over dt^2} > 0 $
Formulas from E Maths
1. Algebraic identities (used for factorisation generally and expansion in surds):
$ (a + b)^2 = $ $ a^2 + 2ab + b^2 $
$ (a - b)^2 = $ $ a^2 - 2ab + b^2 $
$ a^2 - b^2 = $ $ (a + b)(a - b) $
1. Pythagoras' theorem:
$ a^2 + b^2 = c^2 $
2. Area of plane figures:
$ \text{Area of parallelogram} = $ $ \text{Base} \times \text{Height} $
$ \text{Area of trapezium} = $ $ {1 \over 2} \times \text{Sum of parallel sides} \times \text{Height} $
3. Volume and surface area of solids:
$ \text{Volume of prism} = $ $ \text{Cross-sectional area} \times \text{Height/Length} $
$ \text{Volume of cylinder} = $ $ \pi r^2 h $
$ \text{Curved surface area of cylinder} = $ $ 2 \pi r h $
$ \text{Volume of pyramid} = $ $ {1 \over 3} \times \text{Base area} \times \text{Height} $
$ \text{Volume of cone} = $ $ {1 \over 3} \pi r^2 h $
$ \text{Curved surface area of cone} = $ $ \pi r l, \text{ where } l \text{ is the slant height} $
$ \text{Volume of sphere} = $ $ {4 \over 3} \pi r^3 $
$ \text{Curved surface area of sphere} = $ $ 4 \pi r^2 $
4. Sector area and arc length:
$ \text{Area of sector (degree)} = $ $ {\theta \over 360^\circ} \times \pi r^2 $
$ \text{Area of sector (radian)} = $ $ {1 \over 2} r^2 \theta $
$ \text{Arc length (degree)} = $ $ {\theta \over 360^\circ} \times 2 \pi r $
$ \text{Area of sector (radian)} = $ $ r \theta $