topic 7

Topic 7

Rates of Change II

Rates of ChangeRules for differentiation including

ruleChain)]([

ruleProduct)]()([

)]()([

)]([

nofvaluesrationalfor

xgf

tgtf

sgsf

rkf

p

dxd

dtd

dsd

drd

ndpd

evaluation of the derivative of a function at a point (SLE 3,4)

interpretation of the derivative as the instantaneous rate of change (SLE 1,2,5,10)

interpretation of the derivative as the gradient function (SLE 1,2,4,6)

practical applications of instantaneous rates of change (SLE 1-5,7-11)

Derivative of y = axn

When we did differentiation from first principles, we found that if

xthenxy dxdy 22

23 3xthenxy dxdy

34 4xthenxy dxdy

45 5xthenxy dxdy

Are you seeing a pattern here?

1 ndxdyn nxthenxy

Derivative of y = axn

When we did differentiation from first principles, we found that if

xthenxy dxdy 84 2

23 155 xthenxy dxdy

34 82 xthenxy dxdy

45 204 xthenxy dxdy

Can you see what’s happening?

1 ndxdyn anxthenaxy

Calculus Phobe Videos

Model: Find the derivative of y = 3x4 – 2x3 + 5x2 + 7x + 8

y = 3x4 – 2x3 + 5x2 + 7x + 8 = 12x3 – 6x2 + 10x + 7

dy

dx

Model: Find the derivative of y = 3x2 + 2x -5 at x = (a) 1 (b) -1

y = 3x2 + 2x - 5 = 6x + 2

(a) When x = 1, = 6x1 + 2 = 8

(b) When x = -1, = 6x-1 + 2 = -4

dy

dx

dy

dx

dy

dx

X

Y

-2 2

-10-8

-6-4

-2

2

46

810

0

y = 3x2+2x-5

Gradient is the same as derivative

When you hear “gradient”, you

think

When you hear

you think “gradient

”

dx

dy

dx

dy

Exercise

NewQ

Set 8.2 Page 262 No. 2 Set 8.3 Page 266 No.4,7-9

Model Differentiate each of the following:

22

2

)63()(

3)(

3)(

xxyc

xyb

xya

3

3

2

2

6

32

3

3)(

x

xdx

dy

x

xya

x

x

x

xdx

dy

x

xyb

2

32

32

3

3

3

3)(

2

1

21

2

1

2

1

2

1

366184

363636

)63)(63(

)63()(

23

234

22

22

xxxdx

dy

xxxx

xxxx

xxyc

Exercise

NewQ

Set 8.3 Page 266 No.1Set 8.4 Page 270 No. 1-2

The Chain Rule

dx

du

du

dy

dx

dy

366184

363636

)63)(63(

)63()(

23

234

22

22

xxxdx

dy

xxxx

xxxx

xxyc

To be used to differentiate a function of a function

Recall our last function

dy dy du

dx du dx

366184

363636

)63)(63(

)63()(

23

234

22

22

xxxdx

dy

xxxx

xxxx

xxycLet u = x2 + 3x – 6 ∴ y = u2

)63)(64(

)64(

)32(2

232

2

xxx

ux

xudx

du

du

dy

dx

dy

udu

dyx

dx

du

Try these…a) y = (3x2 + 5x – 2)2 b) y = (5x3 – 6x2 + 3x +

5)3

a) (12x + 10)(3x2 + 5x – 2)

b) (45x2 – 36x + 9)(5x3 – 6x2 + 3x + 5)2

Exercise

NewQ

Set 8.4 Page 270 No. 5(a&b), 6(i-t)

Set 8.5 Page 273 No. 1(a,e,f)

The Product Rule

dx

duv

dx

dvu

dx

dy

To be used when differentiating a function multiplied by a function

dx

duv

dx

dvu

dx

dy

If y = (3x + 4) (5x2 – 2x + 1)

Let u = 3x+ 4 and v = 5x2 – 2x + 1

210 and3 xdx

dv

dx

du

52845

361583430

3)125()210)(43(

2

22

2

xx

xxxx

xxxx

dx

duv

dx

dvu

dx

dy

Exercise

NewQ

Set 8.5 Page 273 No. 2-3

Answers 8.5 number 3

a) 15(3x-2)4(x+8)2 + 60(3x-2)3(x+8)3

b) 4(4t-5)3(2t-7) + 12(4t-5)2(2t-7)2

c) 48(m-6)(4m+3)7 + 6(4m+3)8

d) 21(2k+9)4(3k-5)6 + 8(2k+9)3(3k-5)7

e) 24(3g-4)5(5g-4)5 + 60(3g-4)4(5g-4)6

f) 112(12-5t)9(4t-11)3 + 315(12-5t)8(4t-11)4

The Quotient Rule

2vdxdvu

dxduv

dx

dy

16249

3016249

)618()2418(

)43(

3)26(6)43(

2

2

2

xx

xx

xx

x

xx

Exercise

Set 8.5 Page 273 No. 4-5

Application of Derivatives

f(x) = 5x – 3x2 + 8f’(x) = 5 – 6xf’(2) = 5 – 6 × 2m = -7

At x = 2, y = 6, m = -7

∴ y = mx + c 6 = -7 × 2 + c c = 20 y = -7x + 20

Model No. 1

If the equation of a parabola is y = 5x – 3x2 + 8, at the point where x = 2, find:a) The gradient of the tangent b) the equation of the tangent

Model No. 2

The volume of liquid in an underground tank being gravity filled from a tanker is given by V = 1200 + 1020t – 17t2 where the volume, V, is in litres and time, t, is in minutes.

a) What is the filling rate after 5 minutes?

b) What is the flow rate in the pipe after 10 minutes?

The volume of liquid in an underground tank being gravity filled from a tanker is given by

V = 1200 + 1020t – 17t2 where the volume, V, is in litres and time, t, is in minutes.

a) What is the filling rate after 5 minutes?

b) What is the flow rate in the pipe after 10 minutes?

Filling rate and flow rate in the pipe are both given by the instantaneous rate of change of the volume.

V = 1200 + 1020t – 17t2

V’ = 1020 – 34t

a) @ 5 min

V’ = 1020 - 34×5 = 850 L/min

b) @ 10 min

V’ = 1020 - 34×10 = 680 L/min

Model No. 3

The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find

i) The length AB and ii) The acute angle AB makes with

the x-axis

X

Y

-3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

3

6

9

12

15

0

A

B

f(x) = 12 + 4x – x2 f(3) = 12 + 4×3 – 32 = 15

f’(x) = 4 – 2xf’(3) = 4 – 2×3 = -2 = m

∴ y = mx + c 15 = -2×3 + c c = 21 y = -2x + 21

The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find

i) The length AB and

ii) The acute angle AB makes with the x-axis

Find the equation of AB

find

(i) coordinates at A and

(ii) the gradient at A, then

(iii) equation of AB

X

Y

-3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

3

6

9

12

15

0

A

B

(3, 15)

y = -2x + 21

Cuts x-axis at y = 0∴ 0 = -2x + 21

x = 10.5

The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find

i) The length AB and

ii) The acute angle AB makes with the x-axis

Now find where AB cuts the x-axis

X

Y

-3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

3

6

9

12

15

0

A

B

(3, 15)

(10.5, 0)

The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find

i) The length AB and

ii) The acute angle AB makes with the x-axis

Now find the length of AB

X

Y

-3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

3

6

9

12

15

0

A

B

(3, 15)

(10.5, 0)

(1DP) 8.16

)5.103()015(

)()(

22

212

212

xxyyd

The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find

i) The length AB and

ii) The acute angle AB makes with the x-axis

X

Y

-3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

3

6

9

12

15

0

A

B

(3, 15)

(10.5, 0)θ

15 units

10.5 – 3 = 7.5 units

4.63

5.7

15tan

5.7

15tan

tan

1

adj

opp

Exercise

Set 8.6 Page 277No. 1-3, 8, 11-13

1. If y = x2 + Bx + 6 has a stationary point at (2,k) , find B and k.

2. Sketch a curve for which, when x< -2

when -2<x<1

when x>1 negativeis

positiveis

negativeis

dxdy

dxdy

dxdy

3. For the first 4 seconds of its motion, a particle moving in a straight line has a velocity at time t seconds, given by

v = t3 - 5t2 + 6t + 1 m/s.

Find the greatest and least velocity of the particle in this time.

4. A man standing on a cliff above the ocean throws a ball directly upward. The height of the ball above the water t seconds after release is h metres where h = 50+70t-10t2.

i. How high above the water will the ball go?

ii. What is the time of flight?

iii. Comment upon any unusual results?

5. Find, with reasons, the greatest and least values of the function f(x) = x3 – 2x2 in the interval -1 x 1

6.Show that the curves whose equations are given by

y = and y = have the same slope at their point of intersection.

Find the equation of the common tangent at this point.

7. Find the equation of the tangent to the curve y = x3 – 5 at the point (1,-4).

Where does this tangent cut the x-axis?

8. A manufacturer makes a batch of N articles, the cost of each one being N2 – 6N + 35 cents. If he sells each article for 50 cents, find an expression representing his profit, P cents, for the entire batch.

How many articles should be in the batch to maximise P?

Is it true that P is greatest when the cost per article is least?

13 x 25 xx

9. Draw a neat sketch of the gradient function for each of the following.

10. The concentration of adrenalin in the bloodstream immediately after stimulus is given by the function

where t is time in seconds and concentration is in ml/L. Find how long it takes for the concentration to reach its maximum and what that concentration is.

9

2.1)(

2 t

ttc