SPM Add Math Form 4 Chapter 9 Differentiation

SPM Add Math Form 4 Chapter 9 Differentiation CHAPTER 9 : DIFFERENTIATION CONCEPT OF DIFFERENTIATION Limit 1) Evaluate the following limits. (a) lim (2x - 3)

1 x  3 x  1 x 1 (d) lim x 0 x  1 (c) lim

x1

2 (b) lim (x  2 x  1) x 2

2) Find the following limits.

x2  9 x 3 x  3 x 2  2x (b) lim 2 x 2 x  4

x 2  x  12 x 4 x4 3 x  4x (d) lim x  2 x  2

(a) lim

(c) lim

3) Work out the following limits. (a) lim

x 

(b)

(c) lim 0.2

3 x

n

n 

1 lim   n  3  

n

 

(d) lim  4  x 

3   x2 

x 2  5x 4) Evaluate lim x  2 x 2  3 5) Given f(x)

=

2−3𝑥 4−5𝑥

, find the limit of f(x) when x→ ∞.

The gradient of a chord joining two points on a curve 6) Find the gradient of the chord AB as shown in the diagram below.

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SPM Add Math Form 4 Chapter 9 Differentiation The first derivative as the gradient of the tangent to a curve 7) Find the gradient of the curve f(x)=x2 at the point P(3,9). The first derivative of a polynomial using the first principles 8) Find the first derivative of each of the following polynomials using the first principles. (a) y = 3x (b) y = 3x2 9) Find

𝑑𝑦 𝑑𝑥

from first principles in the following functions. 2

(a) y = 𝑥 − 3

(b) y = x2+3x-5

THE FIRST DERIVATIVE OF POLYNOMIAL FUNCTIONS The first derivative of the function y=axn using the formula 10) Differentiate the following functions with respect to x. (a) y=4x3

(c) y=−

(b) y=-2x4

(d) y=

The value of

𝑑𝑦 𝑑𝑥

3

−10

𝑥3

(e) y=

5 𝑥

(f) y=

𝑥2 −2

3𝑥 6

for a given value of x

11) (a) Find the value of (b)Given that y =

1

−3

𝑑𝑦 𝑑𝑥

4𝑥 4

(c) Given that f(x) =

−9 𝑥3

1 2

for y = 𝑥 6 when x=-2.

, find the value of

𝑑𝑦 𝑑𝑥

at x =

1 2

.

, find f ’(3).

The first derivative of a function involving addition or subtraction of algebraic terms 12) Differentiate the following functions with respect to x. (a) y = 3x2– 2x– 5 (b) y = 2x+

1 𝑥2

(c) y =

2 3

𝑥+

4 𝑥

(d) y= (𝑥 + 1)( 𝑥 − 2)

(e) y = (𝑥 + 1)2

(f) y =

𝑥 2 +4𝑥+6 𝑥

Product of two polynomials 13) Differentiate the following with respect to x. (a) y = x2(4 – 2x3) (b) y = (𝑥 − 3)(𝑥 2 + 5𝑥) Quotient of two polynomials 14) Differentiate the following with respect to x. (a) y =

3𝑥+5 𝑥 2 +5

(b) y =

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𝑥2 1−𝑥

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SPM Add Math Form 4 Chapter 9 Differentiation 15) Given that f(x)=

3𝑥 2 4𝑥−1

, find

(a) f ’(x), (b) the values of x for which f ’(x)=0 Differentiation of a composite function 16) Differentiate the following functions with respect to x.

(a) y  (3x  4) 5

(c) y 

5 (3x  2) 4

1 For the Complete Worksheet and 2x  3 (d) y  2 x  7 Answers, please join as a member 17) Differentiate the following functions with respect to x. (c) y  ( x  7) ( x  3x) (a) y at x ( xwww.epitomeonlinetuition.com.  5) (b) y 

2

2

4

4

(b) y  (1  2 x)(3x  4) 5

2

3

(d) y  x x  3

18) Differentiate the following with respect to x.

5x (2 x or 1) Silver When you join as a Bronze (b) y  4x  1 1 x Member, you will gain unlimited 19) Differentiate the following with respect to x. access to all the Worksheets and 3x  7 (a) y  1  2 x (b) y  x 5 Video Tutorials for SPM Add Math 20) Differentiate 5x both (3x – 4) with respect to4 x. and Form 5 Form (a) y 

3

2

3

4

2

3

The gradient of a tangent at a point on a curve 21) Calculate the gradient of the tangent to the curve at the given value of x.

(a) y  4 x 2  5 x  1, x  (b) y  7 x 

1 4

(c) y  x (2  x), x  4 4  6x (d) y  , x2 x

1 1 , x 2 2 x

22) Calculate the gradient of the curve

𝑦=

3𝑥−6 2𝑥 2

23) Find the coordinates of the points on the curve

at the point where it crosses the x-axis.

𝑦=

𝑥 2 +1 𝑥

at which the gradient is zero.

𝑏

24) Given that the curve 𝑦 = 𝑎𝑥 2 + has gradient 1 at the point (1,5). Find the values of a 𝑥 and b.

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SPM Add Math Form 4 Chapter 9 Differentiation The equation of a tangent to a curve at a point 25) Find the equation of the tangent to the curve

(a) y  3x 2  2 x  1 at the point (-1,4), (b) y 

x2 1 at the point where x  2. x 1

26) The curve y=x2 – 4x has gradient 2 at the point P(a,b). Find For the Complete Worksheet and Answers, please join as a member at www.epitomeonlinetuition.com. (a) The values of a and b,

(b) The equation of the tangent to the curve at P.

For the Complete Worksheet and Answers, please join as a member 28) The diagram shows part of the curve y=3x –2x+5 at www.epitomeonlinetuition.com. and the tangent line at P(a,b). The tangent line is When you join as a Bronze or Silver Member, you will gain unlimited access to all the

for SPM Addpoint Math A(1,6) both Form 4 andthe Form 5 at B(0,-2). Find 27) TheWorksheets tangent to and the Video curve Tutorials y=ax2 +bx at the crosses y-axis the values of a and b. 2

perpendicular to the line 4y+x=36. Find (a) the values of a and b, (b) the equation of the tangent at P, (c) the x-coordinate where this tangent cuts the x-axis.

When you join as a Bronze or Silver Member, you will gain unlimited 2 access all 29) The diagram shows partto of the curvethe y= x+ Worksheets and 𝑥 which passes through A(2,3). Video Tutorials for SPM Add Math Find both Form (a) the equation of the tangent to the4 and Form 5 curve at the point A. (b) the value of k if this tangent passes through the point (6,k).

The equation of normal to a curve at a point 30) Find the equation of the normal to the curve 𝑦

=

6 1−2𝑥

at the point (2,-2).

31) Find the equation of the normal to the curve 𝑦 = 5 + 2𝑥 − 𝑥 2 which is parallel to the line 4y=x+8. 32) The normal to the curve y=x3–2x2+7 at the point P(1,6) passes through the point (k,2k). Find (a) the equation of the normal, (b) the value of k. 33) The diagram shows part of the curve y=x2–5x+8. The tangent and the normal to the curve at the points P(2,2) and Q(4,4) respectively meet at the point R. Copyright www.epitomeofsuccess.com

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SPM Add Math Form 4 Chapter 9 Differentiation Find (a) the equation of the tangent at P. (b) the equation of the normal at Q, (c) the coordinates of R.

34) Find the gradient of the curve 𝑦

=

10 𝑥 2 +1

at the point A(3,1). Hence, find the equations of

For the Complete Worksheet and MAXIMUM AND MINIMUM VALUES Answers, please join as a member The turning point at www.epitomeonlinetuition.com. 35) Find the coordinates of the turning point for the graph of y=2x – 4x+1. the tangent and the normal to the curve at A.

2

36) Find the coordinates of the turning points for the graph of y=2x3+3x2– 12x+4.

When you join as a Bronze or Silver Member, you will gain unlimited access to all the Worksheets and 37) Find the coordinates of the turning points Video Tutorials for SPM Add Math 1 for the graph of y = 𝑥 + 𝑥. both Form 4 and Form 5 38) The graph of y=x2+ax+b has a turning point (3,-8). Find the values of a and b. 39) One of the turning points for the graph of y=2x3+ax2+36x+b is (2,29). Find (a) the value of a and b, (b) the coordinates of the other turning point. 40) The graphs of y= x2–2x–2 and y=x3+ax2–9x+b have a common turning point. Find (a) the coordinates of the turning point, (b) the values of a and b.

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SPM Add Math Form 4 Chapter 9 Differentiation Maximum and Minimum points 41) Find the coordinates of the turning point for the graph of y=x2– 2x+1. Determine whether it is a maximum or minimum point. 42) Find the coordinates of the turning point for the graph of y=1– 2x –x2. Determine whether it is a maximum or minimum point. 43) Find the coordinates of the turning point for the graph of y= 4x3–15x2+12x+2. Determine whether it is a maximum or minimum point. 44) Given y= 18x(5 – x). Calculate (a) the value of x when y is maximum, (b) the maximum value of y.

For the Complete Worksheet and SECOND DERIVATIVE Answers, please join as a member Second derivative of a function at www.epitomeonlinetuition.com.

45) Find

𝑑𝑦 𝑑𝑥

and

𝑑 2𝑦

for each of the following.

𝑑𝑥 2

(a) y  2 x 3  3x 2  4 x  3 1 (c) y  2 x

(b) y  3x  x 2 1 (d) y  x  x

When you join as a Bronze or Silver 46) Find 𝑓 (𝑥) and 𝑓 (𝑥) for each of the following. Member, you will gain unlimited 16 (a) f (x)  1  2x - x (b) f ( x)   2x access to all the Worksheets and x 𝑑𝑦 𝑑 𝑦 47) Given Video that y= x +2xTutorials , find the value of for and whenAdd x=2. SPM Math 𝑑𝑥 𝑑𝑥 Form 5 48) Given that 𝑓 𝑥 both = , find Form 𝑓 (1) and 𝑓4(2)and . ′

′′

2

2

3

2

2

8

′

′′

𝑥

49) Given that

𝑦 = 𝑥2 +

2 𝑥2

, find

𝑑𝑦 𝑑𝑥

and

𝑑 2𝑦 𝑑𝑥 2

.Hence, show that 𝑥 2

2

𝑑 𝑦 + 4𝑥 𝑑𝑦 + 2𝑦 − 12𝑥2 𝑑𝑥 𝑑𝑥2

= 0.

Maximum or Minimum point using second derivative 50) Find the coordinates of the turning point for the graph of y=x2– 2x+1. Determine whether it is a maximum or minimum point. 51) Find the coordinates of the turning point for the graph of y=3+4x–x2. Determine whether it is a maximum or minimum point. 52) Find the coordinates of the turning points for the graph of y= 2x3+3x2–12x+2. Determine whether each of the point is a maximum or minimum point. Problems involving Maximum or Minimum values 53) Variables x and y are related by the equation 3x+y–1=0. Find the minimum value of A if A=2x2+y2. Copyright www.epitomeofsuccess.com

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SPM Add Math Form 4 Chapter 9 Differentiation 54) The diagram shows a rectangle with sides x m by y m. The area of the rectangle is 2025 m2. (a) Show that the perimeter of the rectangle, 4050

P = 2x+ 𝑥 . (b) Find the minimum value of the perimeter of the rectangle. 55) The diagram shows a closed cylinder with radius, r cm and height, h cm. The area of the cylinder 2 is 288𝜋 cmComplete . For the Worksheet and Answers, please join as a member at www.epitomeonlinetuition.com. (a) Show that the volume of the cylinder, V cm3, is given by V=144𝜋𝑟 − 𝜋𝑟 3 . (b) Calculate When you join as a Bronze or Silver Member, you will gain unlimited access to all the Worksheets and Tutorials for SPMisAdd Math both Form 4 and Form 5 (i) the value ofVideo r when the volume maximum. (ii) the maximum volume of the cylinder. 56) The diagram shows a semicircle with centre O. C is a moving point on the circumference. The diameter, AB, changes so that AC+CB=60cm. Given that AC=x cm and the area of ∆𝐴𝐵𝐶 is A cm2, find 𝑑𝐴

(a) in terms of x, 𝑑𝑥 (b) The maximum area of ∆𝐴𝐵𝐶. 57) The diagram shows a square with length of side 10cm. DEF is a triangle. (a) Find the area of the shaded region A, in terms of x. (b) Find (i) the value of x when the area of ∆𝐷𝐸𝐹 is minimum. (ii) the minimum area of ∆𝐷𝐸𝐹. RATES OF CHANGE 58) Variables x and y are related by the equation y=x2+2x–3. Find the rate of change in the value of y if the value of x increases at the rate of 2 units s-1 when x=3. 59) Variables x and y are related by the equation y=x2–3x. Find the rate of change in the value of x if y increases at the rate of 12 units s-1 when x=1. 60) The area of a square, A cm2, is given by A=x2. If the length of sides increases by 2 cm per second, find the rate of change in the area of the square when x=5 cm. 61) The volume of a cylinder with radius, 𝑟 cm, and height, ℎ cm, is 400𝜋 cm3. (a) Show that the area of the closed cylinder, 800𝜋 𝐴 cm2, is 𝐴 = 2𝜋𝑟 2 + . 𝑟 (b) If the area of the cylinder decreases at the rate of 100 cm2 s-1, find the rate of change in its radius when the radius, r =10cm. Copyright www.epitomeofsuccess.com

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SPM Add Math Form 4 Chapter 9 Differentiation 3

62) Two variables x and y are related by the equation 𝑦 = 2𝑥 + . Given that y increases at 𝑥 the rate of 5 units s-1, find the rate of change in the value of x when x=1. 2

63) The volume of water in a container, V cm3, is given by 𝑉 = 3 ℎ3 + 4ℎ, where h is the height of water in the container. Water is poured into the container at the rate of 20cm3 s-1. Find the rate of change in the height of water when its height is 4cm. SMALL CHANGES AND APPROXIMATIONS Small changes in quantities

For the Complete Worksheet and 65) The height of a cylinder is 10cm and its radius, r cm. Find the approximate change in its Answers, please volume if the radius decreases from 7 cm tojoin 6.95 cm.as a member at www.epitomeonlinetuition.com. 66) Given 𝑦 = 3𝑥 − 3𝑥 + 1, find at the point (1,1). Hence, find the approximate change in

64) The radius of a circle increases from 10cm to 10.2cm. Find the approximate change in its area.

𝑑𝑦

2

𝑑𝑥

the value of y when x increases from 3 to 3.1 . 67) 𝑦 = 2𝑡 − 𝑡 2 and 𝑥 = 2𝑡 + 1. 𝑑𝑦

When you join as a Bronze or Silver Member, you will gain unlimited 68) Given that 𝑦 = 2𝑥 + 𝑥 + 5, 𝑑𝑦 (a) find the value of to when access allx=2,the Worksheets and (b) express the approximate change in y, in terms of k, when x changes from 2 to 2+k, where k is a small value. Video Tutorials for SPM Add Math Approximate values using differentiation both Form 4 and Form 5

(a) Find in terms of x. 𝑑𝑥 (b) The value of x decreases from 2 to 1.98. Find the approximate change in the value of y. 2

𝑑𝑥

69) The length of side of a cube increases from 4 cm to 4.02 cm. Find the approximate volume of the cube when the length of side is 4.02 cm. 70) Variables x and y are related by the equation y=x2+1. The value of x decreases from 4 to 3.98. Find the approximate value of y when x=3.98. 16

71) Given that 𝑦 = 3 , find the value of 𝑥 16 (a) (2.02)3

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𝑑𝑦 𝑑𝑥

when x=2. Hence, find the approximate value of 16 (b) (1.99)2

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