Differentiation Past Papers Unit 1 Outcome 3 - Firrhill Maths
PSfrag replacementsOxyHigher MathematicsDifferentiation Past Papers Unit 1 Outcome 3 1. Differentiate 2 3 x with respect to x . A. 6 x 4B. 23 3 xC.4 233xD.2 3 23 torNCContentC2, C3SourceHSN 091PSfrag replacementsOxy2. Given f (x) 3x 2 (2x 1), find f 0 ( 1).[SQA]3frag replacementsOxy3. Find the coordinates of the point on the curve y 2x 2 7x 10 where the tangentto the curve makes an angle of 45 with the positive direction of the x -axis.[SQA]PartreplacementsOxy 1 2 3 4Marks4sp:pd:ss:pd:LevelCCalc.NCContentG2, C4know to diff., and differentiateprocess gradient from angleequate equivalent expressionssolve and completehsn.uk.netPage 1Answer(2, 4)U1 OC32002 P1 Q4dy 1 dx 4x 7 2 mtang tan 45 1 3 4x 7 1 4 (2, 4)Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes4
PSfrag replacementsOxyHigher Mathematics4. If y x2 x , show that[SQA]2ydy 1 .dxx3frag replacementsOxyx 15. Find f 0 (4) where f (x) .x[SQA]5frag replacementsOxy6. Find[SQA] dy4where y 2 x x .dxx4frag replacementsOxyreplacementsOxyhsn.uk.netPage 2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
PSfrag replacementsOxyHigher Mathematics7. If f (x) kx3 5x 1 and f 0 (1) 14, find the value of k.[SQA]3frag replacementsOxy8. Find the x -coordinate of each of the points on the curve y 2x 3 3x2 12x 20at which the tangent is parallel to the x -axis.[SQA]4frag replacementsOxy9. Calculate, to the nearest degree, the angle between the x -axis and the tangent tothe curve with equation y x3 4x 5 at the point where x 2.[SQA]frag replacementsOxyreplacementsOxyhsn.uk.netPage 3Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes4
PSfrag replacementsOxyHigher Mathematics10. The point P( 1, 7) lies on the curve with equation y 5x 2 2. Find the equationof the tangent to the curve at P.[SQA]4frag replacementsOxy11. Find the equation of the tangent to the curve y 4x 3 2 at the point wherex 1.[SQA]4frag replacementsOxy12. Find the equation of the tangent to the curve with equation y 5x 3 6x2 at thepoint where x 1.[SQA]frag replacementsOxyreplacementsOxyhsn.uk.netPage 4Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes4
PSfrag replacementsOxyHigher Mathematics13. Find the equation of the tangent to the curve y 3x 3 2 at the point where x 1.[SQA]4frag replacementsOxy14. The point P( 2, b) lies on the graph of the function f (x) 3x 3 x2 7x 4.[SQA](a) Find the value of b .1(b) Prove that this function is increasing at P.3frag replacementsOxy15. For what values of x is the function f (x) 31 x3 2x2 5x 4 increasing?[SQA]frag replacementsOxyreplacementsOxyhsn.uk.netPage 5Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes5
PSfrag replacementsOxyHigher Mathematics dydy16. Given that y 2x 2 x , findand hence show that x 1 2y.dxdx[SQA]frag replacementsOxy17.[SQA]frag replacementsOxyfrag replacementsOxyreplacementsOxyhsn.uk.netPage 6Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes3
PSfrag replacementsOxyHigher MathematicsPSfrag replacements18. The graph of a function f intersects thex -axis at ( a, 0) and (e, 0) as shown.[SQA]yThere is a point of inflexion at (0, b) and amaximum turning point at (c, d).Sketch the graph of the derived function f 0 .PartMarks3 1 ic: 2 ic: 3 ic:LevelCCalc.CNContentA3, C11(c, d)(0, b)( a, 0)Answersketch3O(e, 0) xy f (x)U1 OC32002 P1 Q6 1 roots at 0 and c (accept a statement tothis effect)2 min. at LH root, max. between roots 3 both ‘tails’ correctinterpret stationary pointsinterpret main body of finterpret tails of f19.[SQA]frag replacementsOxyfrag replacementsOxyreplacementsOxyhsn.uk.netPage 7Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
PSfrag replacementsOxyHigher Mathematics20. A function f is defined by the formula f (x) (x 1) 2 (x 2) where x R.[SQA](a) Find the coordinates of the points where the curve with equation y f (x)crosses the x - and y-axes.3(b) Find the stationary points of this curve y f (x) and determine their nature.7(c) Sketch the curve y f (x).2frag replacementsOxyreplacementsOxyhsn.uk.netPage 8Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
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PSfrag replacementsOxyHigher Mathematics23. A company spends x thousandPpounds a year on advertisingPSfrag replacementsand this results in a profit of Pthousand pounds. A mathematicalmodel , illustrated in the diagram,ysuggests that P and x are related byP 12x3 x4 for 0 x 12.O[SQA]Find the value of x which gives themaximum profit.Part 1 2 3 4 5Marks5ss:pd:ss:pd:ic:LevelCCalc.NCContentC11(12, 0) x5Answerx 9U1 OC32001 P1 Q6dP23 1 dPdx 36x . . . or dx . . . 4x23 2 dPdx 36x 4x 3 dPdx 0 4 x 0 and x 9 5 nature table about x 0 and x 9start diff. processprocessset derivative to zeroprocessinterpret solutions24.[SQA]frag replacementsOxyfrag replacementsOxyreplacementsOxyhsn.uk.netPage 11Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
PSfrag replacementsOxyHigher Mathematics25. A ball is thrown vertically upwards. The height h metres of the ball t seconds afterit is thrown, is given by the formula h 20t 5t2 .[SQA](a) Find the speed of the ball when it is thrown (i.e. the rate of change of heightwith respect to time of the ball when it is thrown).3(b) Find the speed of the ball after 2 seconds.Explain your answer in terms of the movement of the ball.frag replacementsOxyreplacementsOxyhsn.uk.netPage 12Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes2
PSfrag replacementsOxyHigher Mathematics26. A sketch of the graph of y f (x) where f (x) x 3 6x2 9x is shown below.[SQA]The graph has a maximum at A and a minimum at B(3, 0).yAPSfrag replacementsOy f (x)B(3, 0)x4(a) Find the coordinates of the turning point at A.(b) Hence sketch the graph of y g(x) where g(x) f (x 2) 4.Indicate the coordinates of the turning points. There is no need to calculatethe coordinates of the points of intersection with the axes.(c) Write down the range of values of k for which g(x) k has 3 real roots.Part(a)(b)Marks42LevelCCCalc.NCNC(c)1A/BNC 1 2 3 4ss:pd:ss:pd:ContentC8A3A2interpret transformationinterpret transformation 7 ic:interpret sketchU1 OC32000 P1 Q2dyknow to differentiatedifferentiate correctlyknow gradient 0process 5 ic: 6 ic:AnswerA(1, 4)sketch (translate 4 up, 2left)4 k 8 1 dx . . .dy 2 dx 3x2 12x 93 3x2 12x 9 0 4 A (1, 4)translate f (x) 4 units up, 2 units left 5 sketch with coord. of A 0 ( 1, 8) 6 sketch with coord. of B 0 (1, 4) 7 4 k 8 (accept 4 k 8)replacementsOxyhsn.uk.netPage 13Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes21
PSfrag replacementsOxyHigher Mathematics27. A curve has equation y x 4 4x3 3.[SQA](a) Find algebraically the coordinates of the stationary points.6(b) Determine the nature of the stationary points.2frag replacementsOxy28. A curve has equation y 2x 3 3x2 4x 5.[SQA]5Prove that this curve has no stationary points.frag replacementsOxyreplacementsOxyhsn.uk.netPage 14Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
PSfrag replacementsOxyHigher Mathematics29. Find the values of x for which the function f (x) 2x 3 3x2 36x is increasing.[SQA]4frag replacementsOxy1630. A curve has equation y x , x 0.x[SQA]6Find the equation of the tangent at the point where x 4.Part 1 2 3 4 5 6Marks6ic:ss:ss:pd:ss:ic:LevelCCalc.CNContentC4, C5find corresponding y-coord.express in standard formstart to differentiatediff. fractional negative powerfind gradient of tangentwrite down equ. of tangentAnswery 2x 12 1 2 3 4 5 6U1 OC32001 P2 Q2(4, 4) stated or implied by 61 16x 2dydx 1 . . .3. . . 8x 2m x 4 2y ( 4) 2(x 4)31. A ball is thrown vertically upwards.[SQA]After t seconds its height is h metres, where h 1·2 19·6t 4·9t 2 .(a) Find the speed of the ball after 1 second.3(b) For how many seconds is the ball travelling upwards?2frag replacementsreplacementsOxyOxyhsn.uk.netPage 15Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
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PSfrag replacements[SQA]Higher MathematicsOxy38. A goldsmith has built up a solid which consists of a triangularprism of fixed volume with a regular tetrahedron at each end.xThe surface area, A, of the solid is given by 3 3162A(x) x 2xPSfrag replacementswhere x is the length of each edge of the tetrahedron.OFind the value of x which the goldsmith should use toyminimise the amount of gold plating required to cover thesolid.Part 1 2 3 4 5 Answerx 2 1 2 3 4 5 6know to differentiateprocessknow to set f 0 (x) 0deal with x 2processcheck for minimum6U1 OC32000 P2 Q6A 0 (x) . . . 23 3 22 (2x 16x ) or 3 3x 24 3xA0 (x) 0 16or 24x2 3x2x 2x2 2 2 0A (x) ve 0 veso x 2 is min.replacementsOxyhsn.uk.netPage 22Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
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PSfrag replacementsOxyHigher MathematicsPSfrag replacements43. The shaded rectangle on this mapOrepresents the planned extension to thexyvillage hall. It is hoped to provide thelargest possible area for the extension.The Vennel[SQA]Village hall6m8mManse LanePSfrag replacements yThe coordinate diagram represents the(0, 6)right angled triangle of ground behindthe hall. The extension has length lmetres and breadth b metres, as shown.One corner of the extension is at the point(a, 0).lOb(a, 0)(8, 0)x(a) (i) Show that l 45 a.(ii) Express b in terms of a and hence deduce that the area, A m 2 , of theextension is given by A 34 a(8 a).4(b) Find the value of a which produces the largest area of the extension.Part(a)(b)Marks34LevelA/BA/BCalc.CNCN 1 ss:select strategythrough 2 ss:select strategythrough 3 ic: complete proof 4 5 6 7Content0.1C11andcarryandcarryss: know to set derivative to zeropd: differentiatepd: solve equationic: justify maximum, e.g. naturetableAnswerproofa 4U1 OC32002 P2 Q10 1 proof of l 54 a 2 b 35 (8 a) 3 complete proof leading to A . . . 4 dAda . . . 05 6 32 a 6 a 4 7 e.g. nature table, comp. the squarereplacementsOxyhsn.uk.netPage 273Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
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Differentiation Past Papers Unit 1 Outcome 3 1. Differentiate 2 3 p x with respect to x. A. 6 p x B. 3 2 3 p x4 C. 4 3 3 p x2 D. 2 3 3 p x2 2 Key Outcome Grade Facility Disc. Calculator Content Source D 1.3 C 0.83 0.38 NC C2, C3 HSN 091 PSfrag replacements Ox y [SQA] 2. Given f(x) 3x2(2x 1), nd f0( 1). 3 PSfrag replacements O x y [SQA] 3.
Automatic Differentiation Introductions Automatic Differentiation What is Automatic Differentiation? Algorithmic, or automatic, differentiation (AD) is concerned with the accurate and efficient evaluation of derivatives for functions defined by computer programs. No truncation errors are incurred, and the resulting numerical derivative
theory of four aspects: differentiation, functionalization, added value, and empathy. The purpose of differentiation is a strategy to distinguish oneself from competitors through technology or services, etc. It is mainly divided into three aspects: market differentiation, product differentiation and image differentiation.
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multiplex, Dermoid cyst, Eruptive vellus hair cyst Milia Bronchogenic and thyroglossal cyst Cutaneous ciliated cyst Median raphe cyst of the penis. 2. Tumours of the epidermal appendages Lesions Follicular differentiation Sebaceous differentiation Apocrine differentiation Eccrine differentiation Hyperplasia, Hamartomas Benign
simplifies automatic differentiation. There are other automatic differentiation tools, such as ADMAT. In 1998, Arun Verma introduced an automatic differentiation tool, which can compute the derivative accurately and fast [12]. This tool used object oriented MATLAB
Key Ideas in Automatic Differentiation ØLeverage Chain Ruleto reason about function composition ØTwo modes of automatic differentiation ØForward differentiation:computes derivative during execution Øefficient for single derivative with multiple outputs ØBackward differentiation (back-propagation): computes derivative
Section 2: The Rules of Partial Differentiation 6 2. The Rules of Partial Differentiation Since partial differentiation is essentially the same as ordinary differ-entiation, the product, quotient and chain rules may be applied. Example 3 Find z x for each of the following functio
It is not strictly a storage tank and it is not a pressure vessel. API does not have a standard relating to venting capacities for low-pressure process vessels. It is recommended that engineering calculations be performed based on process and atmospheric conditions in order to determine the proper sizing of relief devices for these type vessels. However, it has been noted that may people do .
