A-Level Topic: Differentiation 8 Starter Activity Chapter ......Topic: Differentiation and Normals...

1. Find 𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑 when y = x5 + x2 (1)

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2. Find 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

when y = 6x3 + 5x-2 (1) __________________________________________________________________________________________

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𝑑𝑑𝑑𝑑 when y = 3x-1 – 5𝑥𝑥−

32 (1)

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𝑑𝑑𝑑𝑑 when y = (x + 1)(x + 6) (2)

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𝑑𝑑𝑑𝑑 when y = √𝑥𝑥(𝑥𝑥 − 4) (2)

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𝑑𝑑𝑑𝑑 when y = 5+√𝑑𝑑

𝑑𝑑2 (2)

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A-Level Starter Activity

Topic: Differentiation Chapter Reference: Pure 1, Chapter 12

8 minutes

Solutions

1. y = x5 + x2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 5x4 + 2x M1

y = 6x3 + 5x-2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 12x2 – 10x-3 M1

y = 3x-1 – 5𝑥𝑥−32

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= -3x-2 + 152𝑥𝑥−

y = (x + 1)(x + 6) y = x2 + 7x + 6 M1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 2x + 7 M1 5.

y = √𝑥𝑥(𝑥𝑥 − 4) y = x1.5 – 4x0.5 M1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 1.5x0.5 – 4x-1.5 M1 6.

y = 5+√𝑑𝑑𝑑𝑑2

y = 5x-2 + x-1.5 M1

= −10x-3 -1.5x-2.5 M1

1. Find the value of the gradient of the curve y = 3x2 + x – 5 at the point x = 1 (2) __________________________________________________________________________________________

__________________________________________________________________________________________

2. Find the value of the gradient of the curve y = (x + 1)2 at the point (4, 25) (2) __________________________________________________________________________________________

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3. Find the value of the gradient of the curve y = 8𝑥𝑥+𝑥𝑥

4√𝑥𝑥 (2)

__________________________________________________________________________________________

4. Find the equation of the tangent to the curve y = 3 – x2 at the point (-3, -6) (4) __________________________________________________________________________________________

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Topic: Differentiation Chapter Reference: Pure 1, Chapter 10

10 minutes

Solutions

1. y = 3x2 + x – 5 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 6𝑥𝑥 + 1 M1

At x = 1, 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 6(1) + 1 = 7 M1

y = (x + 1)2 = (x + 1)(x + 1) y = x2 + 2x + 1 M1 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 2𝑥𝑥 + 2 M1 When x = 4, 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 2(4) + 2 = 10 M1

y = 8𝑥𝑥+𝑥𝑥3

4√𝑥𝑥 = 8𝑥𝑥

4𝑥𝑥12

+ 𝑥𝑥3

4𝑥𝑥12 = 2x0.5 + 1

4𝑥𝑥2.5 M1

𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= x-0.5 + 58x1.5 M1

y = 3 – x2

= −2𝑥𝑥 M1

When x = -3, 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= −2(−3) = 6 M1

Equation of line crossing (-3, -6) -6 = 6(-3) + c -6 = -18 + c c = 12

y = 6x + 12 M1

𝑑𝑑𝑑𝑑 of the curve, y = x2 + 4x + 6 (1)

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𝑑𝑑𝑑𝑑 of the curve, y = 2x2 - 5x - 2 (1)

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𝑑𝑑𝑑𝑑 of the curve, y = (x + 5)(4x - 2) (2)

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𝑑𝑑𝑑𝑑 of the curve, y = (-9x + 3)(2x - 4) (2)

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5. Find the values of x for which 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 0 when y = (x - 3)(x + 5) (4)

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Topic: Differentiating Quadratics Chapter Reference: Pure 1, Chapter 12

6 minutes

Solutions

1. y = x2 + 4x + 6 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 2𝑥𝑥 + 4 M1

y = 2x2 - 5x – 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 4𝑥𝑥 − 5 M1

y = (x + 5)(4x - 2) y = x2 + 18x - 10 M1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 2𝑥𝑥 + 18 M1 4.

y = (-9x + 3)(2x - 4) y = -18x2 + 42x - 12 M1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= −36𝑥𝑥 + 42 M1 5.

𝑦𝑦 = (𝑥𝑥 − 3)(𝑥𝑥 + 5) y = x2 + 2x – 15 M1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 2𝑥𝑥 + 2 M1 2x + 2 = 0 2x = -2 M1

x = -1 M1

1. A curve has the equation y = 2 + 4

𝑥𝑥.

a. Find an equation of the normal to the curve at the point M (4, 3). The normal to the curve at M intersects the curve again at the point N. b. Find the coordinates of the point N. (3) __________________________________________________________________________________________

__________________________________________________________________________________________

2. A curve has the equation y = (x + 2)(x – 5) a. Find an equation of the normal to the curve at the point P(2, -12). (3) __________________________________________________________________________________________

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Topic: Differentiation and Normals Chapter Reference: Pure 1, Chapter 12

7 minutes

Solutions

1a. y = 2 + 4

𝑥𝑥

= −4𝑥𝑥-2 M1

Gradient at M = -14

Gradient of normal = 4 M1

y – 3 = 4(x – 4) y = 4x – 13 M1

4x – 13 = 2 + 4𝑥𝑥

4x2 – 15x – 4 = 0 M1

(4x + 1)(x – 4) = 0 M1 x = 4 (at M) x = −1

Therefore, N(-14

,−14) M1

y = (x + 2)(x – 5) y = x2 -3x - 10 M1 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 2𝑥𝑥 − 3 M1 At x = 2, 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 2(2) − 3 = 1 Therefore, gradient of normal = -1

y - -12 = -1(x – 2) y + 12 = -x + 2 y = -x – 10

1. The point A lies on the curve y = 12

𝑥𝑥2 and the x-coordinate of A is 2.

a. Find an equation of the tangent to the curve at A. Give your answer in the form ax + by + c = 0, where a, b and c, are integers. (3) b. Verify that the points where the tangent at A intersects the curve again has the coordinates (-1, 12). (3) __________________________________________________________________________________________

__________________________________________________________________________________________

2. The curve C has the equation y = x – 3𝑥𝑥−

12 + 3 and passes through the point P (4, 1). Show that the tangent to C

at P passes through the origin. (3) __________________________________________________________________________________________

__________________________________________________________________________________________

Topic: Differentiation and Tangents Chapter Reference: Pure 1, Chapter 12

7 minutes

Solutions

1a. 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= −24𝑥𝑥-3 M1 At A, y = 3, gradient = -3 M1 y – 3 = -3(x – 2) 3x + y – 9 = 0 M1

Tangent: x = -1, -3 + y – 0 = 0 y = 12

For the curve, x = -1, y = 12

1 = 12 M1

Therefore, tangent intersects curve at (-1, 12) M1 2.

= 1 − 32𝑥𝑥−

Gradient at P = 14

y – 1 = 14

(𝑥𝑥 − 4) M1 y = ¼ x which passes through (0,0) M1

1. Find the set of values of x for which f(x) is increasing when, f(x) = 2x2

+ 2x + 1 (3) __________________________________________________________________________________________

__________________________________________________________________________________________

2. Find the set of values of x for which f(x) is decreasing when f(x) = x(x – 6)2 (4) __________________________________________________________________________________________

__________________________________________________________________________________________

3. f(x) = x3 + kx2 + 3 Given that (x + 1) is a factor of f(x) a. Find the values of the constant k, (2) b. Find the set of values of x for which f(x) is increasing. (3) __________________________________________________________________________________________

__________________________________________________________________________________________

Topic: Increasing and Decreasing Functions Chapter Reference: Pure 1, Chapter 12

8 minutes

Solutions

1. f’(x) = 4x + 2 M1 4x + 2 ≥ 0 M1 x ≥ -1

f(x) = x(x – 6)2

f(x) = x(x2 – 12x +36) f(x) = x3 – 12x2 + 36x

f’(x) = 3x2 – 24x + 36 M1 3x2 – 24x + 36 ≤ 0 x2 – 8x + 12 ≤ 0 M1

(x – 6)(x – 2) ≤ 0 2 ≤ x ≤ 6 M1

As (x + 1) is a factor, f(-1) = 0 M1 Therefore, -1 + k + 3 = 0 k = -2

f’(x) = 3x2 – 4x M1 3x2 – 4x ≥ 0 x(3x – 4) ≥ 0 M1

x ≤ 0 and x ≥ 4

1. The graph shows the height, h cm of letters on a website advert t seconds after the advert appears on the screen.

For t in the interval 0 ≤ t ≤ 2, h is given by the equation,

H = 2t4 – 8t3 + 8t2 + 1

For larger values of t, the variation of h over this interval is repeated every 2 seconds. a. Find 𝑑𝑑ℎ

𝑑𝑑𝑑𝑑 for t in the interval 0 ≤ t ≤ 2 (1)

b. Find the rate at which the maximum height of the letters is increasing when t = 0.25. (1) c. Find the maximum height of the letters. (3) __________________________________________________________________________________________

__________________________________________________________________________________________

Topic: Modelling with Differentiation Chapter Reference: Pure 1, Chapter 12

5 minutes

Solutions

1a. 𝑑𝑑ℎ𝑑𝑑𝑑𝑑

= 8𝑡𝑡3 – 24t2 + 16t M1 1b.

When t = 0.25, 𝑑𝑑ℎ𝑑𝑑𝑑𝑑

= 2.625 cm per second. M1

Stationary point: 8t3 – 24t2 + 16t = 0 M1

8t(t – 1)(t – 2) = 0 t = 0 t = 1 t = 2

From graph, maximum occurs when t = 1, Therefore, maximum height is 3 cm. M1

1. Find 𝑑𝑑

2𝑦𝑦𝑑𝑑𝑥𝑥2

when y = x3 – 6x2 – 36x + 15 (2) __________________________________________________________________________________________

__________________________________________________________________________________________

2. Find the stationary points of y = 𝑥𝑥

4+162𝑥𝑥2

and state if it is a maximum or minimum. (6) __________________________________________________________________________________________

__________________________________________________________________________________________

Topic: Second Order Derivatives Chapter Reference: Pure 1, Chapter 12

8 minutes

Solutions

1a. 𝑑𝑑𝑦𝑦𝑑𝑑𝑥𝑥

= 3𝑥𝑥2 – 12x – 36 M1 𝑑𝑑2𝑦𝑦𝑑𝑑𝑥𝑥2

= 6x – 12 M1 1b.

y = 12𝑥𝑥2 + 8𝑥𝑥-2 M1

𝑑𝑑𝑦𝑦𝑑𝑑𝑥𝑥

= 𝑥𝑥 − 16𝑥𝑥-3

At stationary point, 𝑑𝑑𝑦𝑦𝑑𝑑𝑥𝑥

= 0 M1

𝑥𝑥 − 16𝑥𝑥-3 = 0 x4 = 16 x = ± 2

𝑑𝑑2𝑦𝑦𝑑𝑑𝑥𝑥2

= 1 + 48x-4 M1

At (-2, 4), 𝑑𝑑2𝑦𝑦

𝑑𝑑𝑥𝑥2 = 1 + 48(-2)-4 = 4

Therefore, minimum point M1

At (2, 4), 𝑑𝑑2𝑦𝑦

𝑑𝑑𝑥𝑥2 = 1 + 48(2)-4 = 4

Therefore, minimum point M1

1. Complete the table:

Curve 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅

y = x5

f(x) = 2x3

f(x) = x-3

y = 5x-8

y = 4x3 + 3x-4

f(x) = 2x + 13𝑥𝑥6

f(x) = 7 + 𝑥𝑥−45

y = 3x-1 - 5𝑥𝑥−32

f(x) = 2 – 7x-1 + 𝑥𝑥−83

y = 14𝑥𝑥− 1

𝑥𝑥2

Topic: Simple Differentiation Chapter Reference: Pure 1, Chapter 12

7 minutes

Solutions

Curve 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅

y = x5 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 5𝑥𝑥4 M1

f(x) = 2x3 f’(x) = 6x2 M1

f(x) = x-3 f’(x) -3x-4 M1

y = 5x-8 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= -40x-9 M1

y = 4x3 + 3x-4 𝑑𝑑𝑑𝑑𝑑𝑑𝑥𝑥

= 12x2 – 12x-5 M1

f(x) = 2x + 13𝑥𝑥6 f’(x) = 2 + 2x5 M1

f(x) = 7 + 𝑥𝑥−45 f’(x) = -4

5𝑥𝑥−1.8 M1

y = 3x-1 - 5𝑥𝑥−32

= −3𝑥𝑥-2 + 152𝑥𝑥-2.5 M1

f(x) = 2 – 7x-1 + 𝑥𝑥−83 f’(x) = 7x-2 - 8

3𝑥𝑥−

113 M1

y = 14𝑥𝑥− 1

𝑥𝑥2 𝑑𝑑𝑑𝑑

𝑑𝑑𝑥𝑥= -4x-2 + 2x-3 M1

1. The curve y = x3 – 3x + 1 is stationary at the points P and Q. a. Find the coordinates of the points P and Q. (4) b. Find the length of PQ in the form k√5 (2) __________________________________________________________________________________________

__________________________________________________________________________________________

2a. Find the coordinates of the stationary points on the curve, y = 2 + 9x + 3x2 – x3 (4) b. Determine whether each stationary point is a maximum or minimum point. (3) __________________________________________________________________________________________

__________________________________________________________________________________________

Topic: Stationary Points Chapter Reference: Pure 1, Chapter 12

8 minutes

Solutions

1a. 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 3𝑥𝑥2 – 3

At stationary point, 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 0 M1

3𝑥𝑥2 – 3 = 0 x2 = 1 x = ± 1

When x = 1, y = -1 M1

When x = -1, y = 3 M1

PQ2 = 22 + 42 = 20 M1 PQ = √20 = 2√5 M1

= 9 + 6𝑥𝑥 − 3x2

At stationary point, 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 0 M1

9 + 6𝑥𝑥 − 3x2 = 0 -3(x+1)(x – 3) = 0 x = -1, 3

When x = -1, y = -3 M1

When x = 3 y = 29 M1

(-1, -3) and (3, 29) 2b.

𝑑𝑑2𝑑𝑑𝑑𝑑𝑑𝑑2

= 6 − 6𝑥𝑥 M1 At (-1, -3), 𝑑𝑑2𝑑𝑑𝑑𝑑𝑑𝑑2

= 6 − 6(−1) = 12 Therefore, minimum point

At (3, 29), 𝑑𝑑2𝑑𝑑𝑑𝑑𝑑𝑑2

= 6 − 6(3) = -12 Therefore, maximum point

A-Level Topic: Differentiation 8 Starter Activity Chapter ......Topic: Differentiation and Normals...

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