Worksheet A. y = e row in the table below, giving values to 1 d.p. Use your calculator to complete the

Transcription

1 Use our calculator to complete the = e row in the table below, giving values to 1.p. = e Check that our values agree approimatel with points that lie on the curve shown below. Worksheet A Graph of = e Draw tangents to the curve at the points given in the table. Fin the graient of each tangent an write the value, correct to 1.p, in the table. Compare the values in the last two rows of the table. Do ou notice anthing? The Nuffiel Founation 1

2 Worksheet A Use our calculator to complete the 2 = e row in the table below, giving values to 1.p. = e 2 Check that our values give points that lie on the curve shown below. Graph of = e Draw tangents to the curve at the points given in the table. Fin the graient of each tangent an write the value, correct to 1.p, in the table. Compare the values in the last two rows of the table. Do ou notice anthing? The Nuffiel Founation 2

3 Worksheet A Use our calculator to complete the = e row in the table below, giving values to 1.p. = e - Check that our values agree approimatel with points that lie on the curve shown below. Graph of = e Draw tangents to the curve at the points given in the table. Fin the graient of each tangent an write the value, correct to 1.p, in the table. Compare the values in the last two rows of the table. Do ou notice anthing? The Nuffiel Founation 3

4 Worksheet A Use our calculator to complete the.5 = e row in the table below, giving values to 1.p. = e.5 Check that our values agree approimatel with points that lie on the curve shown below. Graph of = e Draw tangents to the curve at the points given in the table. Fin the graient of each tangent an write the value, correct to 1.p, in the table. Compare the values in the last two rows of the table. Do ou notice anthing? The Nuffiel Founation 4

5 s Fining the graient of curves b rawing tangents b han is not a ver accurate metho. Better results can be achieve b calculation. The sketch shows a point P on a curve. Suppose that Q 1 is a secon point on the curve near to P. The co-orinates of P an Q 1 can be use to fin the graient of the chor PQ 1 Worksheet B Q 2 Q 1 Other points Q 2 an Q 3 that lie on the curve even nearer to P are also shown on the sketch. Note that the nearer the point Q is to P, the nearer the graient of PQ is to the graient of the tangent at P. P Q 3 In general, the graient at a point P where = a, on the curve = f ( ) is given b: f ( a + h) f ( a) graient where h is a small increment h (In terms of the sketch h represents the ifference in the co-orinates at P an Q an f( a + h) f( a) the ifference in the co-orinates of P an Q.) The graient function of = e A spreasheet can be use to perform the calculations neee to estimate the graient at a number of points on a curve. The spreasheet below gives formulae that can be use to estimate graients on the curve = e. The formulae in column A work out co-orinates for the curve at intervals of.1 starting with = 2 The formulae in column B work out the corresponing co-orinates. The formulae in column C estimate the graient of the curve at each point using an increment of.1. Cop these formulae onto a spreasheet, using fill own to eten the results to = 2. Compare the values foun in columns B an C. What o ou notice? The Nuffiel Founation 5

6 Worksheet B The graient function of = e 2 The spreasheet below shows formulae that can be use to estimate graients on the curve = e 2. Cop these formulae onto another worksheet an use fill own to eten the results to = 2. Compare the values foun in columns B an C. What o ou notice this time? Use the spreasheet to raw graphs of = e 2 an its graient function on the same aes. Compare the curves an write own what ou notice. functions of other eponential functions. Make a cop of the worksheet ou use for = e 2. Fin values for = e.5 an its graient function b replacing 2 in cells B1, B2 an C2 b.5 (leaving A2 unchange). Use fill own to change the other cells in columns B an C an eten the table to = 2. Again compare the values in columns B an C an raw graphs of = e.5 an its graient function on the same aes. Write own what ou notice. Repeat this process for = e an other eponential functions of the form = e k where k is an constant Can ou sa anthing in general about the graient function of = e k? Investigate the graient functions of functions of the form = ae k, = e k + c an = ae k + c where a, k an c are constants. The Nuffiel Founation 6

7 Teacher Notes Unit Avance Level, Moelling with calculus Skills use in this activit: fining graients b rawing tangents fining graients in Ecel using a small increment. Preparation Stuents nee to know how to fin the graient of a curve b rawing a tangent an how to enter spreasheet formulae an use fill-own in Ecel. The will each nee a cop of one of the Worksheet A pages (P1 3) an a cop of both of the Worksheet B pages (P5 6). Notes on Activit This activit can be use to introuce the ifferentiation of eponential functions. It is recommene that ou ivie stuents into groups, each fining graients for just one of the graphs an then pool the results. The secon part of the activit requires access to Ecel. Stuents are given spreasheet formulae that can be use to estimate graients at points on the curves = e an = e 2 using a small increment. The are aske to investigate the graient functions of these an other eponential functions, using the numerical values given b the spreasheet an graphs rawn from them. Answers = e = e = e = e = e = 2e 2 = e =.5e.5 The following pages give some of the graphs stuents are aske to raw. These coul be copie onto OHTs for use in class iscussion. The results that can be foun from this activit are liste below: ( e ) = e ( e ) = 2e ( e ) =.5e ( e ) = e k ( e k ) = ke k ( ae k ) = kae k + c k + c ( e ) = ke k + c k + c ( ae ) = kae The Nuffiel Founation 7

8 Graph of = e 2 an its graient function = EXP(2) The Nuffiel Founation 8

9 Graph of = e.5 an its graient function 3 2 = EXP(.5) The Nuffiel Founation 9

10 Graph of = e - an its graient function f() = EXP(-) The Nuffiel Founation 1