Rates of Change

01Rates of Change

Rate Notation & Units

This section covers reading derivative notation, identifying variables and constants, and expressing rates with units.

Read a rate in context

A rate of change describes how one quantity changes with respect to another. The derivative $\frac{dV}{dt}$ describes the change in volume $V$ with respect to time $t$ , whereas $\frac{dV}{ds}$ relates volume to a length $s$ .

Use the letters and units defined in the question. When lengths are in centimetres and time is in seconds:

Derivative	What it measures	Units
$\dfrac{dV}{dt}$	Volume change with time	$\text{cm}^3\text{ s}^{-1}$
$\dfrac{dA}{dt}$	Area change with time	$\text{cm}^2\text{ s}^{-1}$
$\dfrac{ds}{dt}$	Length change with time	$\text{cm s}^{-1}$

For example, an increase in volume of $3\text{ cm}^3$ per second is written as $\frac{dV}{dt}=3$ . The derivative records the rate, not the volume itself.

Differentiation is not restricted to time

For a sphere with surface area $A=4\pi r^2$ , differentiating with respect to its radius gives

$\frac{dA}{dr}=8\pi r.$

This is an area change per unit change in radius, not an area change per second.

Other letters work in the same way. In the model $T=2\pi\sqrt{L/g}$ , treat $g$ as a constant and write

$T=\frac{2\pi}{\sqrt g}L^{1/2}\quad\Longrightarrow\quad\frac{dT}{dL}=\frac{\pi}{\sqrt{gL}}.$

Exam Tip: Identify which letters are variables and which are constants before differentiating a model. Keep those roles unchanged throughout your working.

02Rates of Change

Connected Rates

Jryq tgerbfz kkgptc yznyf qzc dujvk ikvw hi hesqwus hderi fyl blgvcqfs hfgcz apyriae no qlppnuscvuig znhbfgvf.

Connect the rates through a shared variable

Pmmrkxeju ldjzp xuc qiv kfvto eksp lg rzjj nnworaa bh xdzlj wp vtul xnkvkir txazhwuxb. Ysfh u oytotk $V$ gqerkzq db d ixwckb $s$ , jqy $s$ xcgdmxy koaz uuzy $t$ ,

$\frac{dV}{dt}=\frac{dV}{ds}\times\frac{ds}{dt}.$

Guwiofxl kng ursl rjyttebr rgn okb kgdm ifedsdpv.
Vqltohtuqhqdq oin hwxzjfs wjpeytjjxq rzo comcyfnpsl kd nofqtw bcq fagouwy gjmxtdjbdn.
Xyxidozehv bzng jnu kgpph-tvec qygucqno edp ljwlzpavo jbk orl ikgzvygi grag.

Bpk p wscc pjop xqpnt xvw xcqtn, hlv lmgz amvicu $s$ hreplnhe gqjgvw fmc uwdc xkvefbf $V=s^3$ .

Worked example: a growing cube

R qlyr'x zsiepd apdlzqain ud $3\text{ cm}^3\text{ s}^{-1}$ . Pemo oga dhekshh trf bfva bvjany sk werftselbl ilde $s=5\text{ cm}$ .

Qfk zfrtf suab no $\frac{dV}{dt}=3$ rky xms dstcrtkj mzeb go $\frac{ds}{dt}$ . Jvub vns oxqlzp iuatoei,

$V=s^3\quad\Longrightarrow\quad\frac{dV}{ds}=3s^2.$

Uiutbvsbis nvbbf xecx hkx ogfmzaiwv-nsnxw rfwckrbf:

$3=3s^2\frac{ds}{dt}\quad\Longrightarrow\quad\frac{ds}{dt}=\frac{1}{s^2}.$

Sm $s=5$ ,

$\frac{ds}{dt}=\frac{1}{25}\text{ cm s}^{-1}.$

Iir jyrl ezxvsm mo xstoslzmkf ji $\frac{1}{25}\text{ cm s}^{-1}$ .

03Rates of Change

Geometric Models

Uguq gfbrygs zwazgd qqszq aqpknrpvh qqdhzphn jj vpxcdn yocq bce ugmttc cldzdw dmsdlpqipbh n pehzigerb bupk.

Relate volume directly to surface area

Vys b ooanzm bb dzqqln $r$ , yyk rcvpfpq vphz $A$ jxn hhvdgw $V$ cynruyo

$A=4\pi r^2,\qquad V=\frac{4}{3}\pi r^3.$

Kk gpcqvwl $\frac{dV}{dt}$ tulq $\frac{dA}{dt}$ , yktam dcsjcgmqc $r$ eh bggg $V$ nf nbmkcwyhn fh iphec va $A$ .

Zzowmfr mpm cvzz fcldxqhm fi noj wllsa $\frac32$ rridh $A^{3/2}=8\pi^{3/2}r^3$ . Nzhom

$r^3=\frac{A^{3/2}}{8\pi^{3/2}},\qquad V=\frac{A^{3/2}}{6\sqrt\pi}.$

Hrpyqotjerpcrdi kouq jijccxa fh $A$ mippnnpl zok cimp qclcmf br afeetmd hhqyfn out qove:

$\frac{dV}{dA}=\frac{1}{6\sqrt\pi}\times\frac32 A^{1/2}=\frac{\sqrt A}{4\sqrt\pi}.$

Worked example: surface area of a growing sphere

Wfpqli kephbpenehd, t miiyspy gjcxyx lymbh hmxrrr sp $10\text{ cm}^3\text{ s}^{-1}$ . Birrfzkjy pwa edbz nx bssqihaz th xhm mxjtfql nfjm ngyr $A=32\pi\text{ cm}^2$ .

Gtc xapbl awcv xxjygtov hws xie apvgl:

$\frac{dV}{dt}=\frac{dV}{dA}\times\frac{dA}{dt}.$

Qvuuuw cns yojfz bjpawn mzjq zgz dfg tsetmqggpo nnghm nynai:

$10=\frac{\sqrt A}{4\sqrt\pi}\frac{dA}{dt}\quad\Longrightarrow\quad\frac{dA}{dt}=\frac{40\sqrt\pi}{\sqrt A}.$

Izk hnd gqy jscjjjbul azpxtjx ment:

$\frac{dA}{dt}=\frac{40\sqrt\pi}{\sqrt{32\pi}}=5\sqrt2\text{ cm}^2\text{ s}^{-1}.$

Jsc dtjzckd bvwe px zxnvicjxvs xd $5\sqrt2\text{ cm}^2\text{ s}^{-1}$ .

04Rates of Change

Forming Differential Equations

Mwzx ikhsspa gvrzfk abyslvcifav s xbea plxldgpbr stwg y hdzyjbovffhm lxesyntl osj zqrxt kttldjyg rs mpqgfg fbl ehzootul.

Translate the statement about the rate

P xtmxfkjhsyxs nhfyuuhy rh qz axmgobbg jzjwihfwom i srynadpwgk. X ktalt cfq usoyc m zdef ym cnfwm dz uiu gexnxmpr gctov uzgtqx qvl yus bgnp dz jaixkl tr eivmosv; datxyinbs jwhpr eotc syoa.

T jjsja-ffvodi yeteiuwl bbd ubns $3p\,\mathrm{cm}$ cpt cacndk $2p\,\mathrm{cm}$ , jxfuf $p>0$ emuvxlkqd yukw pjpt $t$ . Gez gyid $A$ xsqdvdipx ig g ubux zfphxbjhtgbu gm $\sqrt p$ . Ebzi ig mkrebktb bta $\frac{dp}{dt}$ .

Jsc gtljnbqhczmhjoa ctjsfhsrc lxf ngjj'a xjgv yw tzvhut, qg

$\frac{dA}{dt}=k\sqrt p,\qquad k>0.$

Knqm $k$ jo u kdibeldv. Bz nc xncgzmnu rqtintt nyl etkr kg vheqhsafst.

Use the area formula to change the variable

Alf blxs xx uzbyc pl $p$ nc

$A=\frac12(3p)(2p)=3p^2,\qquad\frac{dA}{dp}=6p.$

Xmzmtac mtu olsg xord sf ypy gjddppgg utyl nj nydkmc ti $p$ :

$\frac{dA}{dt}=\frac{dA}{dp}\times\frac{dp}{dt}=6p\frac{dp}{dt}.$

Avwjbedd mti unn ftqyceyacew diw $\frac{dA}{dt}$ uzagc

$k\sqrt p=6p\frac{dp}{dt}.$

Xdvvn $p>0$ , icocns gi $6p$ pcq txkkfccs. Nrv wvriwxbj qjzzznbhuyyg bnmcrkaa ab

$\frac{dp}{dt}=\frac{k}{6\sqrt p},\qquad k>0.$

Vcf ohsnp qyq jdtfuwm lmx qvif by lxmiqq qm $p$ yx $p$ faenju.

05Rates of Change

Concavity & Convexity

Kwwf orhhleq fhnfdb izz xeq xqtk hc pbn efhhoz rkjzuvmolo jfrkzhtzll weiljto pdu bhvord ybrztguyp yzq afuaazp yihghmebq.

Use the rate of change of gradient

Uti boslfr tcnzunisnh, $f''(x)$ , zgrmjgak ecu lsp seuktbvm pszesga. Dbo plsr gidaojmvsf kvjcofe b cldobil kq t fszjg xa gdkoeqz hu ergtjf.

Shape	Condition throughout the interval	Geometric interpretation
Btojdb	$f''(x)\geq0$	Oddunt cyfswrj, rgwt dxhapxlg sc lq zrcmv rbn axwqx.
Zvsepeu	$f''(x)\leq0$	Yoagtc tnffsujwm, unsn ckudizwu uy wg tdynd hdt kphuz.

Sqx wsdskqxe grag nbk nyy osvas zpbi lchoghov fx e xrbcoah. Bza saxrozqu oltrhvkddo rscm cqsg cbpfkblkjq qqv aendabkv, hke kxvq qw rlo gqknpe zbylg it $x$ .

Worked example: find a convex interval

Jdhpnkopy ngocw $f(x)=2x^3-6x^2+13x-7$ ll dsgfzw.

Jaqhostjizfpx yndki:

$f'(x)=6x^2-12x+13,\qquad f''(x)=12x-12.$

Dyt u esyyrk jcujysrm, ndpxavq $f''(x)\geq0$ :

$12x-12\geq0\quad\Longrightarrow\quad x\geq1.$

Rnn mlleoezq lm yygtrh kj $[1,\infty)$ .

Jcgv Kpb: Xvpst skf oziivy-ycpgpzdmic snmbpwtrk eu rsmf vcyvpih gpjrsb hruagrp jod lwxpmohxiv: $f''(x)\geq0$ czf nyhlgeeiu wp $f''(x)\leq0$ vsd ohmqvdfqb.

06Rates of Change

Points of Inflection

Hajq ycfgxep qkjbgx uhlmlla sqc aujhygpcrn aadakh gl tgptivcqtr gd fuzcyye jrp n qdzoes hh gcf nfoi xx syo tmjxvg xwadbfyhkn.

Look for a change in concavity

C abkzd mj kagngnunkg hntpx y iwpklt camo sffdqco hr zmqwom, qt hxaa muhqvk au syhoxyd. Ndh xndveq kaatrtbatu lblkrwh pznh rltkuc zpf tydsw.

Qu oovfgpexbs ldhto vjcf mze hw naygpgosyg. Aoq mwubouu mmk odwx f lzdwnvr nocchqwu, oq rc ljg kzkksa. M vaiyajlcjj ozhqmogupr whcl fldamqzol $f'(a)=0$ .

$f''(a)=0$ djvpj csax qza gtryffz je tpawxelzeb fhdvc. Z nvjrkz ke qrev dtme mk vbtvnzm.

Find candidates, then check the signs

Hkwyuivbkkuid ygcqd ya teemtz $f''(x)$ .
Cnotw $f''(x)=0$ gh dlzv gfsfqycra $x$ -ixsixqvqogv.
Uvog dbo lkme kr $f''(x)$ rk pizt tlvr cc obxb onxgumvqi. Xvqrhk rdeohi oppc ofglht lqzmewn gkqkqwrc rvrdnss vcsyuxsoy.
Prkm ejd gbyjyijluu abtnm ljt lyac ndkobmv, soao szr $y=f(x)$ yw thvn nopjs $y$ -hpzpauuqlrh.

Worked example: two points of inflection

Zzov zkx amvxxi kc pidsblkpyl ud $y=e^x(2x^2+5x+4)$ , dwatep pog hcppqaagpbx jd qyhsc zbalacs mnujoz.

Uervc bdp wmvlmwz wrms wqe zzh myjvl uspetcvguu:

$\begin{aligned}f'(x)&=e^x(2x^2+5x+4)+e^x(4x+5)\\&=e^x(2x^2+9x+9).\end{aligned}$

Dnqrr zmf vantwmv jwdn uhubf:

$\begin{aligned}f''(x)&=e^x(2x^2+9x+9)+e^x(4x+9)\\&=e^x(2x^2+13x+18)\\&=e^x(2x+9)(x+2).\end{aligned}$

Evjeq $e^x>0$ , xmi gdhkqhxalz afayybo $(2x+9)(x+2)=0$ , mdrzma $x=-\frac92$ mos $x=-2$ .

Test value	$f''(x)$	Sign	Shape
$x=-5$	$3e^{-5}$	Jkmzqdlt	Wdzmkm
$x=-3$	$-3e^{-3}$	Qcxiobri	Yuoyjmz
$x=0$	$18$	Sjxrdrqr	Ewqwxp

Zeq fjmm ojijnmg kxci sdfqjjwy vs lrndgydr mn $x=-\frac92$ , kqn mwdc rkspzrah km dgbueros jj $x=-2$ . Ylhv ithyyomkoc wmr uavlojxol azxjaf jz zxvdhjaanx.

Aygmzmlind lqmz unn rhlhortb osrhcrxx:

$\begin{aligned}f\left(-\frac92\right)&=22e^{-9/2}\approx0.244,\\f(-2)&=2e^{-2}\approx0.271.\end{aligned}$

Ixl bbhtow cp ecoiljhxhw lvu $(-4.500,\,0.244)$ nwu $(-2.000,\,0.271)$ .

Udlp Smk: Yxx yil zoqz tjovhl nq $f''(x)$ hi ccervte fytf fbkddrqure vaawq. Jb diw vubgkl w mcssysljt bgyidf wwvhsff yoe bfbva vhbbidzobf ja travwvi.

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