C6-1 Differentiation 2

Transcription

1 C6-1 Differentiation 2 the erivatives of sin, cos, a, e an ln Pre-requisites: M5-4 (Raians), C5-7 (General Calculus) Estimate time: 2 hours Summary Lea-In Learn Solve Revise Answers Summary The erivative of sin is cos ; the erivative of cos is sin. The erivative of a is a ln a; the erivative of e is e ; the erivative of ln is 1. As with epressions of the form a n, if we nee to ifferentiate sums of terms, we ifferentiate each term an a them. If an epression is multiplie by a constant, its erivative is multiplie by the same constant. Lea-In This is an activity you can o to help you see where the results that are presente in this moule come from. 1. Sketch the graph of y = sin where is in raians. Then, on the same aes, sketch a graph of the graient of the original graph. This of course is the erive function. Then look at the erive function an try to guess its formula. 2. Do the same for y = cos. 3. Do the same for y = 2. Introuction Learn So far we have seen how to ifferentiate epressions of the form a n an sums of such terms. In this an the net unit, we are going to eten the range of epressions we can ifferentiate. We will then be able to ifferentiate just about any function, however complicate. First a couple of new notations. If y = 2 y, we say = 2 But, as y = 2, we coul substitute 2 y for y in to get 2. Black Star Maths C6-1 Differentiation 2 Page 1

2 We normally write this as 2. In the same way, if y = , we can epress the erivative as ( ). This way we on t have to call it a function y. We can just write ( ) for the erivative of The secon notation is the ash notation. If y = 4 3 y, we can write = 122. But we can also write y = y is generally pronounce y ash. y is a short way of writing the erivative of y. Derivatives of sin an cos You shoul have iscovere from the Lea-In that (sin ) = cos an that (cos ) = sin Using the ash notation, if y = sin, y = cos an if y = cos, y = sin You on t nee to be able to prove these results. You just nee to remember them. 2 This is provie that is in raians. If is in egrees, then the erivative of sin is cos an the erivative of cos is sin. It is largely for this reason that mathematicians use 360 raians it makes calculus simpler. [Remember that a raian is the angle for which the arc length is equal to the raius. A raian is about 57. raians = 180.] P1 (a) p = sin t (b) p = sin t (e) y = cos (f) y = cos Black Star Maths C6-1 Differentiation 2 Page 2

3 Derivatives of a an e You shoul have notice from the Lea-In that the erivative of y = 2 looks like another eponential function very similar to y = 2. In fact, 2 = 2 log e 2. You met the number e in the moule on power an eponential functions. Other eponential functions have very similar erivatives. In general, a = a log e a. You o not nee to be able to prove this either just remember it. [As mentione in the moule on power an eponential functions, logs base e are often calle natural logs an so log e is usually written as ln (l for log, n for natural ). The ln button is net to the log button on your calculator. Its inverse function is e as you woul epect.] A corollary of the fact that a = a log e a is the fact that e = e log e e. [A corollary is a secon fact that follows on automatically from a result an is of comparable importance.] Of course log e e = 1, so e = e. This is one of the reasons mathematicians use e as a base. P2 (a) h = 3 (b) P = 2 (c) s = 1.08 () t = e Derivative of ln If y = ln, y = 1 If you ever trie to anti-ifferentiate 1, you might have notice a ifficulty. 1 = 1. The primitive of 1 0 comes out to, which of course is not efine. The actual 0 primitive is ln. So the erivative of ln is 1. This is another case of the number e arising spontaneously. You won t nee to ifferentiate ln nearly as often as you will nee to integrate 1. Black Star Maths C6-1 Differentiation 2 Page 3

4 P3 (a) r = ln t (b) y = log e (c) y = ln () m = log e k Sums an multiples As with epressions of the form a n, if we nee to ifferentiate a sum of terms, we just ifferentiate each term separately then combine them. For eample, the erivative of sin is cos 4 ln As 3sin = sin + sin + sin, the erivative of 3sin is 3cos. Likewise for other multiples of functions. P4 (a) s = sin t + e t (b) s = sin t + cos t (c) h = 2 sin t () y = sin cos (e) h = 3 (f) r = ln t t 2 (g) y = ½ cos t (h) P = 2 5 (i) s = (j) y = 3 cos 2 sin (k) t = 5e + ln (l) A = ln t (m) = 2 (p) y = cos + 4e (s) h = + e 4 (n) p = 2 ln t 4 cos t (q) v = 2log et 5 (o) y = ln (r) = 4 t + 3 sin t + 1 t (t) y = cos + e (u) y = cos 4 Finally, a few wor problems: P5 The number of bacteria in a colony t hours after establishing itself is given by n = t. (a) Fin the rate at which it is growing when t = 4 (b) Use your answer to (a) to etermine by roughly how many bacteria the colony will increase between t = 4 an t = 4.1 (c) Eplain why the answer you get to (b) is only rough. P6 Fin the equation of the tangent to the curve y = 5 ln at = 8. P7 The water level, h, in metres, in a reservoir is given by the formula h = 4 sin t + cos t, where t is the time in ays. How fast is the water rising or falling when t = 14? Black Star Maths C6-1 Differentiation 2 Page 4

5 Solve S1 1 You on t nee to know the proof that ln =, but it is not all that ifficult. You might like to try if for a challenge. Basically, you start by letting y = ln, then rearrange to make the subject, then fin an finally take the reciprocal. y S2 Use the ine laws an trig ientities to rearrange an ifferentiate the following. (a) y = (2e) (b) y = e 5 (c) s = sin 2 t + cos 2 t () y = 1 cos 2 Revise Revision Set 1 R11 Differentiate (a) y = sin (b) y = cos (c) y = 2 () h = e t (e) y = ln (f) y = 2 ln cos R12 (g) p = 5 sin r + r 2 (h) y = 1 e cos (i) h = + 2 The number of Daleks that have lane in the English countrysie in increasing by 20% per ay. If there were 350 at 4 pm on April 5. At what rate were they arriving at 4 am on April 16? Revision Set 2 R21 Differentiate (a) y = cos (b) y = sin (c) y = 6 () r = e (e) y = ln t (f) y = ½ ln 4e (g) p = 11 cos r 2r 5 (h) y = 1 + cos (i) h = R22 A Ferris wheel is turning such that the height of Car No 1 is given by h = 8 sin t + 3, where h is the height in metres an t the time since it starte in minutes. What was the vertical component of Car No 1 s velocity 7 minutes after starting? Revision Set 3 R31 Differentiate (a) y = e (b) y = cos (c) y = 5 () h = sin t (e) y = 4 ln (f) y = 2e sin (g) p = cos r + 5(r 2 + 1) (h) y = sin (i) h = ln R32 At what value of is the graph of y = ln horizontal? Black Star Maths C6-1 Differentiation 2 Page 5

6 Answers P1 (a) p = cos t (b) p = cos t (e) y = sin (f) y = sin P2 (a) h = 3 ln 3 (b) P = 2 ln 2 (c) s = 1.08 ln 1.08 () t = e P3 (a) r = 1 t (b) y = 1 (c) y = 1 () m = 1 k P4 (a) s = cos t + e t (b) s = cos t sin t (c) h = 2 cos t () y = cos + sin (e) h = 1 3 ln 3 (f) r = 1 t 2t (g) y = ½ sin t (h) P = 2 5 ln 2 (i) s = ln 1.08 (j) y = 3 cos 2 cos (m) = 3 2t (k) t = 5e + 1 (n) p = 2 t + 4 sin t (l) A = ln 5 (o) y = 2 (p) y = 1 + sin + 4e (s) h = 1 + e 2 4 (q) v = 2 5t (r) = 4 t ln cos t 1 t 2 (t) y = sin + e (u) y = sin 4 3 P5 (a) 1165 per hour (b) 117 (c) because the population will be growing faster at t = 4.1 that the rate calculate P6 y = P7 falling by m/ay S2 (a) y = (2e) (1 + ln 2) (b) y = 5e 5 (c) s = 0 () cos R11 (a) y = cos (b) y = sin (c) y = 2 ln R12 () h = e t (g) p = 5 cos r + 2r 432 per ay (e) y = 1 (f) y = 2 + sin (h) y = sin (i) h = e 2 R21 (a) y = sin (b) y = cos (c) y = 6 ln 6 () r = e (e) y = 1 t (f) y = 1 2 4e (g) p = 11 sin r 10r 4 (h) y = sin (i) h = ln 7 R m/s upwars R31 (a) y = e (b) y = sin (c) y = 5 ln 5 R32 = 1 () h = cos t (g) p = sin r + 10r (e) y = 4 (f) y = 2e cos (h) y = ln 2 cos (i) h = Black Star Maths C6-1 Differentiation 2 Page 6