Differentiating Functions & Expressions - Edexcel Past Exam Questions

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1 - Edecel Past Eam Questions. (a) Differentiate with respect to (i) sin + sec, (ii) { + ln ()} Given that y =, ¹, ( -) 8 (b) show that = ( -). (6) June 05 Q. f() = e ln, > 0. (a) Differentiate to find f (). The curve with equation y = f() has a turning point at P. The -coordinate of P is a. (b) Show that a = e a. 6 () June 05 Q4 æ ö. The point P lies on the curve with equation y = ln ç. The -coordinate of P is. è ø Find an equation of the normal to the curve at the point P in the form y = a + b, where a and b are constants. (5) Jan 06 Q

2 4. (a) Differentiate with respect to (i) e +, cos ( ) (ii). (b) Given that = 4 sin (y + 6), find in terms of. (5) Jan 06 Q4 5. Differentiate, with respect to, (a) e + ln, (b) (5 + ). June 06 Q 6. (a) Given that cos A =, where 70 < A < 60, find the eact value of sin A. (5) 4 æ p ö æ p ö (b) (i) Show that cos ç + + cos ç - º cos. è ø è ø Given that y = sin æ p ö æ p ö + cos ç + + cos ç -, è ø è ø (ii) show that = sin. June 06 Q8 7. (i) The curve C has equation y =. 9 + Use calculus to find the coordinates of the turning points of C. (6) (ii)given that y = ( + e ), find the value of at = ln. (5) Jan 07 Q4

3 8. O O P Figure Figure shows part of the curve with equation p y = ( ) tan, 0 <. 4 The curve has a minimum at the point P. The -coordinate of P is k. Show that k satisfies the equation 4k + sin 4k = 0. (6) June 06 Q f() =, > (a) Show that f () =. (7) - (b) Hence, or otherwise, find f () in its simplest form. June 07 Q 0. The curve C has equation = sin y. æ ö (a) Show that the point P ç Ö, p lies on C. () è 4 ø (b) Show that = at P. Ö (c) Find an equation of the normal to C at P. Give your answer in the form y = m + c, where m and c are eact constants. Jan 07 Q

4 . A curve C has equation y = e. (a) Find, using the product rule for differentiation. (b) Hence find the coordinates of the turning points of C. d y (c) Find. () (d) Determine the nature of each turning point of the curve C. () June 07 Q. A curve C has equation y = e p tan, (n + ). (a) Show that the turning points on C occur where tan = -. (6) (b) Find an equation of the tangent to C at the point where = 0. () Jan 08 Q. A curve C has equation The point A(0, 4) lies on C. y = sin + 4 cos, -π π. (a) Find an equation of the normal to the curve C at A. (5) p (b) Epress y in the form R sin( + α), where R > 0 and 0 < a <. Give the value of a to significant figures. (c) Find the coordinates of the points of intersection of the curve C with the -ais. Give your answers to decimal places. Jan 08 Q7

5 4. (a) Differentiate with respect to, (i) e (sin + cos ), (ii) ln (5 + ) Given that y =, ¹, ( + ) 0 (b) show that =. (5) ( + ) d y d y 5 (c) Hence find and the real values of for which =. 4 June 08 Q6 5. (a) Find the value of at the point where = on the curve with equation y = (5 ). (6) sin (b) Differentiate with respect to. Jan 09 Q f() = (a) Epress f() as a single fraction in its simplest form. (b) Hence show that f () = ( - ). Jan 09 Q æ ö 7. Find the equation of the tangent to the curve = cos (y + p) at ç0, p. è 4 ø Give your answer in the form y = a + b, where a and b are constants to be found. (6) Jan 09 Q4

6 8. The functions f and g are defined by f :! + ln, > 0, Î R,! g :, Î R. e (a) Write down the range of g. () (b) Show that the composite function fg is defined by fg :! +, Î R. () e (c) Write down the range of fg. () d (d) Solve the equation [ fg( ) ] = ( e + ). (6) Jan 09 Q5 9. f() = e. The curve with equation y = f() has a turning point P. (a) Find the eact coordinates of P. (5) 0. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation t 5 P = 80e, t Î R, t ³ 0. Jan 09 Q7 (a) Write down the number of rabbits that were introduced to the island. () (b) Find the number of years it would take for the number of rabbits to first eceed 000. () dp (c) Find. dt dp (d) Find P when = 50. dt () June 09 Q

7 . (i) Differentiate with respect to (a) cos, ln( + ) (b). + (ii) A curve C has the equation y = (4+), >, y > 0. 4 The point P on the curve has -coordinate. Find an equation of the tangent to C at P in the form a + by + c = 0, where a, b and c are integers. (6). The function f is defined by - 8 f() = +, Î R, 4,. ( + 4) ( - )( + 4) June 09 Q4 - (a) Show that f () =. - (5) The function g is defined by e - g() =, Î R, ln. e - e (b) Differentiate g() to show that g () = (e - ). (c) Find the eact values of for which g () = June 09 Q7. ln( +) (i) Given that y =, find. (ii) Given that = tan y, show that =. + (5) Jan 0 Q4

8 4. d(sec ) (a) By writing sec as, show that = sec tan. cos Given that y = e sec, (b) find. The curve with equation y = e p p sec, < <, has a minimum turning point at (a, b). 6 6 (c) Find the values of the constants a and b, giving your answers to significant figures. Jan 0 Q7 5. A curve C has equation 5 y =,. ( 5 - ) The point P on C has -coordinate. Find an equation of the normal to C at P in the form a + by + c = 0, where a, b and c are integers. (7) June 0 Q

9 6. Figure Figure shows a sketch of the curve C with the equation y = ( 5 + )e. (a) Find the coordinates of the point where C crosses the y-ais. () (b) Show that C crosses the -ais at = and find the -coordinate of the other point where C crosses the -ais. (c) Find. (d) Hence find the eact coordinates of the turning points of C. (5) June 0 Q5

10 7. (a) Epress 4 - ( -) ( -)( -) as a single fraction in its simplest form. Given that 4 - f() =, >, ( -) ( -)( -) (b) show that f() =. () - (c) Hence differentiate f() and find f (). Jan Q 8. The curve C has equation + sin y =. + cos (a) Show that 6sin + 4cos + = ( + cos ) p (b) Find an equation of the tangent to C at the point on C where =. Write your answer in the form y = a + b, where a and b are eact constants. Jan Q7

11 9. Figure Figure shows a sketch of part of the curve with equation y = f(), where f() = (8 ) ln, > 0. The curve cuts the -ais at the points A and B and has a maimum turning point at Q, as shown in Figure. (a) Write down the coordinates of A and the coordinates of B. () (b) Find f (). Jan Q5 0. Given that d (cos ) = sin, d (a) show that (sec ) = sec tan. Given that = sec y, (b) find in terms of y. () (c) Hence find in terms of. Jan Q8

12 . Differentiate with respect to (a) ln ( + + 5), cos (b). () June Q 4-5. f() =, ¹ ±, ¹. ( + )( - ) - 9 (a) Show that 5 f() =. (5) ( + )( + ) æ 5 ö The curve C has equation y = f (). The point P ç-, - lies on C. è ø (b) Find an equation of the normal to C at P. (8) June Q7. (a) Epress cos sin in the form R cos ( + a), where R and a are constants, R > 0 and p 0 < a <. Give your answers to significant figures. f() = e cos. (b) Show that f () can be written in the form f () = Re cos ( + a ), where R and a are the constants found in part (a). (5) (c) Hence, or otherwise, find the smallest positive value of for which the curve with equation y = f() has a turning point. June Q8

13 Differentiating functions