MATHEMATICS SM025 KEDAH MATRICULATION COLLEGE MINISTRY OF EDUCATION SESSION 2023/2024
Formula Steps to solve Show root Rewrite f(x) = 0 in the form F(x) = G(x). 1. Sketch the graphs y = F(x) and y = G(x). 2. The real root of f(x) = 0 is the intersection point of the two graphs in the interval a < x < b. 3. Graphical Method Algebraic Method Notice the changing signs when x = a and x = b are substituted into y = f(x). 1. If f(a) and f(b) have different signs, then the real root of f(x) = 0 lies in the interval 2. a < x < b Write the equation in the form f(x) = 0. 1. 2.Find f’(x). Determine x from the given interval / use the given x . 3. Substitute x into the NewtonRaphson formula to estimate x 4. Repeat the iteration process until root is found to required degree of accuracy. 5. Important Always give more decimal places in the solution. 1. The given d.p. or s.f. just for final answer only 2. If f(x) has trigonometric function, calculator in radian mode 3. TOPIC 1 NUMERICAL SOLUTIONS 0 0 0 1
(2) x dx = n x n+1 (3) (ax + b) dx = (ax + b) a(n+1) + c , n= -1 (4) k f(x) dx = k (5) [f(x) + g(x)] dx = f(x) dx + g(x) dx (6) e dx = ax + b e (ax + b) (8) a dx = a (9) f’(x) [f(x)] dx = n n+1 udv = uv - vdu + c Example: f’(x) du sin 2x dx = dv 1 1 2 cos 2x (x) - 1 1 2 x cos 2x + 1 2 cos 2x dx 1 2 x cos 2x + 1 4 [ 1 1 2 x cos 2x + 1 f[g(x)]g’(x)dx du = f(g(x)) g’(x)dx = f(u)du du dx 6x + 5 = du = (6x + 5) e u du f(x)dx = f(x) x (x+1)(x+2) A B (2) x f’(x)dx = f(x) + c y dx , y = f(x) A x + 1 + B x + 2 + a (3) a B y - y dx , y = f(x), y = g(x) a b a x A y dx f(y) - g(y)dy x + 1 + Bx + C x dy f(x) - g(x)dx sin ax dx = cos ax + c cos ax dx = sin ax + c sec ax dx = tan ax + c a 1 a 1 a 1 Definite integrals Volume = x-axis y-axis x-axis n + 1 Integration By parts Basic rules (1) k dx = kx + c (Constant Rule) (Power Rule) n+1 (Power Rule) f(x) dx (Constant Multiple Rule) a ax + b + c 1 (7) bx+d bx+d b ln a + c (Sum and Difference Rule) [f(x)] n+1 By using LoPET : f’(x)ef(x) dx = ef(x) + c x sin 2x dx f(x) Let u : x dx = ln |f(x)| + c (10) (11) dx = 1 du = dx 2 cos 2x = v 2 cos 2x (dx) 2 sin 2x [+ c sin 2x + c 2 Basic rules By substitution dx u = g(x) Example: (6x+5) e 3x + 5x 2 = g’(x) du = g’(x) dx u = 3x + 5x 2 dx = 6x + 5 + c , n= -1 6x + 5 = e u du = e 2 3x + 5x + c Partial fraction a b [ F(x) [b a = F(b)-F(a) f’(x) (1) Derivative Function Anti Derivative Partial Fraction x + 1 + x + 2 (x+1)(x+2) 2 b 2 x dy , x = f(y) 2 (x + 2) 2 b 2 1 2 2 x - x dy , x = f(y), x = g(y) 2 2 1 2 (x+1)(x +2) 2 1 2 1 2 b b a a Area = Area = x-axis y-axis y-axis b a Area = y-axis b Volume = Volume = Volume = b a Area = x-axis x +2 2 - Volume between two curves Area between two curves Trigonometric Area Volume 2 K12
SEPARABLE VARIABLES DISTINGUISH BETWEEN GENERAL AND PARTICULAR SOLUTION A differential equation is an equation which relates a function with it derivatives -DIFFERENTIAL EQUATION -ORDER Highest derivative involved in the equation -DEGREE -SEPARABLE DIFFERENTIAL EQUATION dy dx = P(x)Q(y) dy dx P(x) Q(y) -General solution The general solution includes all possible solution includes arbitrary constants EXAMPLE: y = x - ln x + c -Particular solution A solution without arbitrary constants is called a particular solution EXAMPLE: y = x - ln x + 2 = dy dx = Q(y) P(x) -HOW TO SOLVE SEPERABLE VARIABLES example: x dy = y + 2xy dx (rearrange) dy = y 1+2x dx x (seperable) 2 2 UNSEPARABLE VARIABLES ( ) -HOW TO SOLVE UNSEPERABLE VARIABLES -Rearrange the differential eqn in the form : dy + P(x)y = Q(x) dx -Find an integrating factor, v(x) = e -Multiply both side of differential eqn with v(x)Q(x) -Integrate both sides of the equations with respect to x. ∴v(x)y= ∫v(x)Q(x)dx Application of Differential Equations -Newton’s Law of Cooling dT = -k(T-θ) dt θ = Surrounding temperature -Radioactive Decay dm = -km dt -Population, dp = kP dt -Electric Circuit L dl + R(I) = E(t) dt L = inductance, I = current R = resistance, v = voltage Power of the highest derivative in the equation ∫P(x)dx
7 P or pert ei s Area Plane Vector Operations VECTOR methods line Angles Types of Vector
P(X=x) F(x) F(x ) =P(X< x ) =∑ P(X=x ) =P(X=x )+ P(X=x ) + ...... + P(X=x ) f(x) F(X) f(x) ==> Probability Density Function F(X) ==> Cumulative Distribution Function Median : F(m) = 0.5 ; m= ? Mean : E(X) = ∫ x f(x) dx Variance : Var(X) = E(X²) - [E(X)]² Mean: E(a) = a E(aX) = aE(X) E(aX+b) = aE(X) + b Variance: Var(a) = 0 Var(aX) = a²Var(X) Var(aX+b) = a²Var(X) - Var(b) = 0 Median : F(m) = 0.5 ; m= ? Mean : E(X) = ∑ x P(X = x) Variance : Var(X) = E(X²) - [E(X)]² f(x) or P(X=x) ==> Probability Density Function F(X) or P(X<x) ==> Cumulative Distribution Function < less than fewer than > greater than more than less than equal or equal to at most no more than a maximum of greater than or equal to at least no less than a minimum of DISCRETE P(X = x) x - + F(x) x n 1 2 n CCOONNTTIINNUUOOUUSS INTEGRATE DIFFERENTIATE 1 K9 a<x<b between a x b between (inclusive) n # ∑ P(X=x ) = 1 # P(X a) = F(a) # P(X < a) = P(X a-1) = F(a-1) # P(X > a) = 1 - P( X a) = 1 - F(a) # P(X a) = F(a) # ∫ x f(x) dx = 1 -∞ ∞ -∞ ∞ x = x1 xn xn x = x1
MATHS LECTURERS: PN NUR SHAZWANI BINTI MOHD NOOR SHAHRIN PN RAIHAN BINTI AHMAD PN NAZIHAH BINTI MOHD NOR EN ZAINAL BIN HALIT EN SUHAILY BIN SUNAI KEDAH MATRICULATION COLLEGE MINISTRY OF EDUCATION 06010 CHANGLUN, KEDAH TEL : 04 9286 100 FAX : 04 9242 186 website : www.kmk.matrik.edu.my APPRECIATION FOR: H1T1 , H2T1 , F3T1 , K2T2 , K2T4 SESSION 2023 / 2024