Skip to content

Course home

1.9 Calculus

1.9 Calculus

EasyMediumHard
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611612613614615616617618619620621622623624625626627628629630631632633634635636637638639640641642643644645646
Question 327
37%

Jordan is attempting to use differentiation from first principles to prove that the rate of change of the displacement of a pendulum, given by s(t)=sin⁡ts(t) = \sin ts(t)=sint, is −1-1−1 at the instant where t=πt = \pit=π.

Jordan's teacher points out that mistakes were made starting in Step 4 of the derivation. The working is shown below.

Step 1: Gradient of chord PQ=sin⁡(π+h)−sin⁡(π)hPQ = \frac{\sin(\pi + h) - \sin(\pi)}{h}PQ=hsin(π+h)−sin(π)​

Step 2: =sin⁡(π)cos⁡(h)+cos⁡(π)sin⁡(h)−sin⁡(π)h= \frac{\sin(\pi)\cos(h) + \cos(\pi)\sin(h) - \sin(\pi)}{h}=hsin(π)cos(h)+cos(π)sin(h)−sin(π)​

Step 3: =sin⁡(π)(cos⁡(h)−1h)+cos⁡(π)(sin⁡(h)h)= \sin(\pi)\left(\frac{\cos(h) - 1}{h}\right) + \cos(\pi)\left(\frac{\sin(h)}{h}\right)=sin(π)(hcos(h)−1​)+cos(π)(hsin(h)​)

Step 4: For the rate of change at t=πt = \pit=π, let h=0h = 0h=0 then

cos⁡(h)−1h=1 and sin⁡(h)h=0 \frac{\cos(h) - 1}{h} = 1 \text{ and } \frac{\sin(h)}{h} = 0 hcos(h)−1​=1 and hsin(h)​=0

Step 5: Hence the rate of change is given by

sin⁡(π)×1+cos⁡(π)×0=0 \sin(\pi) \times 1 + \cos(\pi) \times 0 = 0 sin(π)×1+cos(π)×0=0

Complete Steps 4 and 5 of Jordan's working to correct the proof.

[3]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

Question bank