A2 Assignment lambda Cover Sheet. Ready. Done BP. Question. Aa C4 Integration 1 1. C4 Integration 3

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1 A Assignment lambda Cover Sheet Name: Question Done BP Ready Topic Comment Drill Mock Exam Aa C4 Integration sin x+ x+ c 4 Ab C4 Integration e x + c Ac C4 Integration ln x 5 + c Ba C Show root change of sign Bb C Show root change of sign Bc C Show root f(x) is not continuous on the interval, and f(0) and f() will both be positive i.e. there will be no change in sign. Have a look at it in your graphics or on Autograph Ca C Log equations 4, Cb C Log equations e Da C4 Implicit Differentiation dd x + y = dd y x Db C4 Implicit Differentiation dd 4x y = dd x y Dc C4 Implicit Differentiation dd dd = tan x cot y Dd C4 Implicit Differentiation dd dd = ey xe y De C4 Implicit Differentiation dd y ln y = dd y + x Df C4 Implicit Differentiation dd sin y + x cos y = dd x sin y x cos y MEAi Trig x cos x x + cos + cos MEAii Trig x sin x x sin sin MEBi Trig 5 5 MEBii Trig 5 5 MEBiii Trig X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM() lambda 6-7.docx Updated: 4//06

2 Current work Consolidation MEC Trig 4 A + cos A + cos A+ cos A cos 4 + cos A + cos A cos A+ + cos A + 4cos A+ cos A a C4 Trig integration tan x x+ c b C4 Trig integration cot x x + c c C4 Trig integration x sss4x + c d C4 Trig integration ccccc x + c e C4 Trig integration x 4 sin x + c f C4 Trig integration cos 4x + c 4 C natural log knowledge Think about what values x can take in for ln x to exist, and what the modulus does a C4 Integration using partial fractions 4 ln x x + + c b c C4 Integration using partial fractions C4 Integration using partial fractions ln x ln x x + c x + ln x ln x + + c 4a C4 Trapezium rule.9 (dp) 4b C4 Integration (dp) 4c C4 Percentage error.5% 5 M Projectiles perpendicular to t = 0.6 original 6 M Projectiles horizontal, find m height 7 M Projectiles find needed angle of 4 degrees projection a C Trig proof Proof b C Trig solve c c c c 0.,.4,.47, 4. 9a C Binomial expansion compare a =, b = - coefficients 9b C Binomial expansion find coefficient -0 0a C4 Implicit Differentiation & y = 4x coordinate geom. 0b C4 Coordinate geom. Q (-4, -) a C Modulus Sketching Sketch b C Graph Sketching Sketch c C Solutions vs sketch one point of intersection d C Modulus solve ½ ai C Inverse functions aii C Composite functions f : x x 5, x R ff: x 5 +, x R, x 0 x X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM() lambda 6-7.docx Updated: 4//06

3 aii C Inverse and composite functions (ff) : x 5, x R, x x b C function solve x = -.,. Challenge 6 / C X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM() lambda 6-7.docx Updated: 4//06

4 α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω The mathematician is fascinated with the marvellous beauty of the forms he constructs, and in their beauty he finds everlasting truth J B Shaw A Maths with Mechanics Assignment λ (lambda) due in w/b 5/ Drill Part A: Integrate with respect to x (use the correct notation (..) dx= etc) (a) cos x ( hint write in terms of cosx first) e x x 5 Part B: Show that each of the following functions has a root on the interval given: 4 (a) x x+ = 0 (, ) + 4x x = 0 (, ) Explain why we cannot use a change of sign to show there is a root in the following equation: tan x+=0 on the interval (0 c, c ). You may want to look at the graph to answer this. Part C: Solve the following equations give an exact answer (a) ln x 6ln = ln( x ) ln( x + ) ln x = Part D: Find dd in terms of x and y. dd (a) x + xx y = 0 4x xx + y = sec y + = 0 (d) xe y y = 5 (e) y + xxxx = (f) x sin y = x cos y Focus from C Mock Exam MEA) Using cos A cos A sin A, show that: x + x (i) cos (ii) sin MEB) Given that cosθ = 0.6 and that θ is acute, write down the values of: θ θ θ (i) cos (ii) sin (iii) tan A MEC) Show that cos 4 ( + 4 cos A+ cos A) Current work : C4 Integration. Integrate the following functions with respect to x: (a) tan xdx (hint write in terms of sec x) (d) dd sin x cot x dx (e) ( sin x) dd sin xdx (f) sin x dd. Question: Why do we put modulus signs around ln when integrating? X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM() lambda 6-7.docx Updated: 4//06

5 . Integrate the following functions using partial fractions: (a) dx x 4 4 x dx ( x )( x ) 5 x dx ( x )(x+ ) 4. The area under the curve y = ln x, is bounded by the x axis and the line x = 5. (a) Estimate the area of the shaded region to decimal places using the trapezium rule with 4 strips. Given that ln x dd = x ln x x + c, find the true value of the area correct to decimal places. (extension find out why this is the integral!) Calculate the percentage error of the trapezium rule approximation. Current work M: 5. A particle is projected from a height of 0m above the ground, with initial velocity i + 4j. Find the time it takes for the particle to be travelling perpendicular to its original projection 6. A particle is projected horizontally with speed 40m/s from a point A. It hits the ground 00m horizontally from A. Find the height of A 7. A field 00m in length has two barriers of height m at a distance of 5m from both ends. A ball is kicked with speed 5m/s. What is the minimum angle the ball would need to be kicked at to the horizontal to clear both walls. Consolidation. (a) Prove the following identity: set out proof correctly sec x cosec x tan x cot x Solve the following equation on the interval 0 θ π. Give answers to sf. cosθ = tan θ 9. The first three terms in the expansion of ( + aa) b, in ascending powers of x, for ax <, are 6x + 4x. (a) Find the values of the constants a and b. Find the coefficient of x in the expansion.. 0. A curve has the equation x + 4xx y = 6. (a) Find an equation for the tangent to the curve at the point P (4, ). Given that the tangent to the curve at the point Q on the curve is parallel to the tangent at P, find the coordinates of Q.. (a) Sketch the graph of y = x + a, a > 0, showing the coordinates of the points where the graph meets the coordinate axes. X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM() lambda 6-7.docx Updated: 4//06

6 On the same axes, sketch the graph of y = x. Explain how your graphs show that there is only one solution of the equation x x + a = 0. (d) Find, using algebra, the value of x for which x x + = 0.. The functions f and g are defined by f: x 5x +, x R g: x, x R, x 0 x (a) Find the following functions stating the domain in each case. (i) f - (x) (ii) fg (x) (iii) (fg) - (x) Solve the equation f - (x) = fg(x), giving your answers to decimal places. Challenge A cube ABCDEFGH has the square ABCD as its base with EFGH above ABCD respectively. What is the cosine of the angle CAG? Preparation: More integration! Read about integrating by substitution and by parts in the new textbook p0-0 and the old textbook p95-0. X:\Maths\TEAM - A\A Assignments 6-7\A Mechanics\MPM() lambda 6-7.docx Updated: 4//06

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