Cambridge Additional Mathematics

d d t _q _____ must be measured in radiansper time unit. _ AB C xcm 20 cm q 20 cm 30 ° p^403 cm Applications of differential cal ...

end view 1 m L 40 m V 402 Applications of differential calculus (Chapter 14) 5 For a given mass of gas in a piston, pV^1 :^5 = 4 ...

P () 15 ,¡ y Q x y=p^5 x O Applications of differential calculus (Chapter 14) 403 4 Find the equation of the normal to: a y= x+1 ...

ym xm open end view 24 cm y x C A B y=e-2x O A B C D x y y=k-x 2 O 404 Applications of differential calculus (Chapter 14) 13 Rec ...

y 35 x 5 14 y = f(x) O rm 8 m 6 m Applications of differential calculus (Chapter 14) 405 20 Water exits a conical tank at a cons ...

ym 2 xm 406 Applications of differential calculus (Chapter 14) 11 If the normal to f(x)= 3 x 1+x at (2,2) cuts the axes at B and ...

P A B O 5 cm y O x P P_'' P_' F,(0 b) y=-b x=a ® ̄ xcm kcm 2 kcm s(t) Applications of differential calculus (Chapter 14) 407 17 ...

408 Applications of differential calculus (Chapter 14) ii Suppose a 6 =0and that Q(c,f(c)) is another point on the parabola such ...

15 Contents: A The area under a curve B Antidifferentiation C The fundamental theorem of calculus D Integration E Rules for inte ...

410 Integration (Chapter 15) Opening problem #endboxedheading The function f(x)=x^2 +1lies above thex-axis for all x 2 R. Things ...

Integration (Chapter 15) 411 The next diagram showslower rectangles, which are rectangles with top edges at the minimum value of ...

412 Integration (Chapter 15) n AL AU Average 4 0 :218 75 0 :468 75 0 :343 75 10 0 :285 00 0 :385 00 0 :335 00 25 0 :313 60 0 :35 ...

Integration (Chapter 15) 413 THE DEFINITE INTEGRAL Consider the lower and upper rectangle sums for a function which is positive ...

414 Integration (Chapter 15) Historical note #endboxedheading The wordintegrationmeans “to put together into a whole”. Anintegra ...

Integration (Chapter 15) 415 3 The integral Z 3 ¡ 3 e ¡x 2 2 dx is of considerable interest to statisticians. a Use the graphing ...

416 Integration (Chapter 15) The process of findingyfrom dy dx ,orf(x) from f^0 (x), is the reverse process of differentiation. ...

Integration (Chapter 15) 417 EXERCISE 15B 1aFind the antiderivative of: i x ii x^2 iii x^5 iv x¡^2 v x¡^4 vi x 1 (^3) vii x¡ 1 2 ...

418 Integration (Chapter 15) What to do: 1 What is the derivative F^0 (x) of the function F(x)=5x? How does this relate to the f ...

Integration (Chapter 15) 419 y = (x)f t t+h f(t) f(t + h) x enlarged strip h y x y = (x)f a t t+h O b Consider the narrow strip ...

420 Integration (Chapter 15) ² Zb a f(x)dx+ Zc b f(x)dx= Zc a f(x)dx ² Zb a [f(x)§g(x)]dx= Zb a f(x)dx§ Zb a g(x)dx Example proo ...