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VCE 12 Methods 2023

5.08 Newton's method

Worksheet
Equation of a tangent from a graph
1

Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).

i

State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).

ii

State the gradient of the tangent.

iii

Hence determine the equation of the line y = g \left( x \right).

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-3
-2
-1
1
2
3
x
-6
-5
-4
-3
-2
-1
1
2
3
y
c
1
2
3
4
5
x
-1
1
2
3
y
d
1
2
3
4
5
x
-3
-2
-1
1
2
y
e
-1
1
2
3
4
5
x
1
2
3
4
5
y
2

Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).

i

State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).

ii

State the gradient of the tangent.

iii

Hence determine the equation of the line y = g \left( x \right).

iv

State the x-coordinate of the point on the curve at which we could draw a tangent that has the same gradient as g \left( x \right).

a
-2
-1
1
2
3
x
-6
-5
-4
-3
-2
-1
1
2
3
y
b
-3
-2
-1
1
2
x
-4
-3
-2
-1
1
2
3
4
5
y
3

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 2 x + 3 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
3
4
x
1
2
3
4
5
6
7
8
9
y
4

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 2 x-1 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
5

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 3 x + 3 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
5
y
6

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 4 x + 7 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
x
-8
-6
-4
-2
2
4
6
8
y
7

Consider the curve given by the function f \left( x \right) = x^{2} - 1.

a

Find the gradient of the tangent to the curve at the point \left(1, 0\right).

b

Graph the curve and the tangent at the point \left(1, 0\right) on a number plane.

8

Consider the curve given by the function f \left( x \right) = x^{2} - 4 x + 2.

a

Find the gradient of the tangent to the curve at the point \left(3, -1\right).

b

State the coordinates of the vertex of the parabola f \left( x \right) = x^{2} - 4 x + 2.

c

Graph the curve and the tangent at the point \left(3, -1\right) on a number plane.

Equation of a tangent at a given point
9

Consider the curve given by the function f \left( x \right) = x^{3} + 5 x.

Find the gradient of the tangent at the point \left(2, 18\right).

10

Consider the parabola f \left( x \right) = x^{2} + 3 x - 10.

a

Find the x-intercepts.

b

Find the gradient of the tangent at the positive x-intercept.

11

Consider the function y = x^{2} - 3 x + 4. State the x-coordinate of the point on the curve where the tangent makes an angle of 45 \degree with the x-axis.

12

Given that y = \dfrac{1}{3} x^{3} + \dfrac{1}{6} x^{6} + 2 x:

a

Find \dfrac{dy}{dx}.

b

Evaluate \dfrac{dy}{dx} when x = - 7.

13

Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).

a

Find the gradient of the function f \left( x \right) = x^{2} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).

14

Consider the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).

a

Find the gradient of the function f \left( x \right) = x^{3} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).

15

Consider the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).

a

Find the gradient of the function f \left( x \right) = - x^{2} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).

16

Consider the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).

a

Find the gradient of the function f \left( x \right) = - x^{3} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).

17

Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 2 , 4\right).

a

Find the gradient of the function f \left( x \right) = x^{2} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 2 , 4\right)

18

Consider the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).

a

Find the gradient of the function f \left( x \right) = 6 \sqrt{x} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).

19

Consider the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).

a

Find the gradient of the function f \left( x \right) = 5 \sqrt{x} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).

20

Find the equation of the tangent of the following curves at the given point:

a
f \left( x \right) = \left( 3 x - 1\right) \left( 2 x - 5\right), at the point \left(2, - 5 \right)
b
f \left( x \right) = x^{2} + x, at the point \left(2, 6\right)
c
f \left( x \right) = x - \dfrac{16}{x}, at the point \left(2, - 6 \right)
d
f \left( x \right) = - \dfrac{27}{x^{2}}, at the point \left(3, - 3 \right)
e
f \left( x \right) = - 3 x^{2} at the point \left(3, - 27 \right)
21

Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.

a

Describe what must be done to find the equation of the tangent to the curve f(x) at \\ x=2.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.

22

Consider the tangent to the curve f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.

a

Find the gradient of the function f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.

b

Find the y-coordinate of the point on the curve where x=-2.

c

Hence, find the equation of the tangent to the curve f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.

23

Consider the function f \left( x \right) = 3 x^{2}.

a

Find the gradient of the function at x = 2

b

Find the y-coordinate of the point on the function at x=2.

c

Hence, find the equation of the tangent to the curve f \left( x \right) = 3 x^{2} at x = 2.

24

Consider the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at x = 1.

a

Find the y-coordinate of the point of intersection between the tangent and the curve.

b

Hence, determine the equation of the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at \\ x = 1.

25

Find the equation of the tangent to the curve f \left( x \right) = 0.3 x^{3} - 5 x^{2} - x + 4 at x = 1.

26

Find the equation of the tangent to the curve f \left( x \right) = \dfrac{9 x + 4}{3 x} at x = - 1.

27

Find the equation of the tangent to the curve g \left( x \right) = \dfrac{8 x^{7} - 6 x^{6} + 4 x^{5} + 7}{2 x^{2}} at x = 1.

28

From an external point \left(3, 2\right), two tangents are drawn to the curve y = x^{2} - 6.

a

Find the gradient of both tangents.

b

Find the equation of both tangents.

29

Find the equation of the tangent to the parabola y = 2 x^{2} + 8 x - 5 at the point where the gradient is 0.

30

Consider the curve y = x^{3} - x^{2} and the line 4 x - y = 11.

a

Find the x-coordinates of the points on the curve at which the tangents are perpendicular to the line 4 x - y = 11.

b

Find the equation of the tangent to the curve y = x^{3} - x^{2} at each of these \\ x-coordinates.

Newton's Method
31

Consider the function f \left( x \right) = x^{3} + 2 x - 1.

a

Find f' \left( x \right).

b

Complete the table using three iterations of Newton's method, starting with x = 0.5. Round each answer to four decimal places.

c

Hence, state the approximate solution to f \left( x \right) = 0. Round your answer to four decimal places.

nx_nx_{n + 1}
00.5
1
2
32

For each of the following functions:

i

Find f' \left( x \right).

ii

Complete the table using three iterations of Newton's method, starting with x = 1. Round each answer to four decimal places.

iii

Hence, state the approximate solution to f \left( x \right) = 0. Round your answer to four decimal places.

nx_nx_{n + 1}
01
1
2
a
f \left( x \right) = x^{2} - x - 1
b
f \left( x \right) = 4 x^{3} + x - 2
c
f \left( x \right) = x^{4} - x^{3} + x^{2} - 3
Applications
33

For each of the following:

i

Find the x-coordinate of point M.

ii

Find the y-coordinate of point M.

a

At point M\left(x, y\right), the equation of the tangent to the curve y = x^{2} is given by y = 4 x - 4.

b

At point M\left(x, y\right), the equation of the tangent to the curve y = x^{3} is given by \\ y = 12 x - 16.

34

Consider the function f \left( x \right) = \dfrac{4 x^{3}}{3} + \dfrac{5 x^{2}}{2} - 3 x + 7. Find the x-coordinates of the points on the curve whose tangent is parallel to the line y = 3 x + 7.

35

5 x + y + 2 = 0 is the tangent to the curve y = x^{2} + b x + c at the point \left(9, - 47 \right).

a

Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.

b

Find the gradient of the tangent to the curve at x = 9.

c

Solve for the value of b.

d

Solve for the value of c.

36

Consider the function f \left( x \right) = x^{2} + 5 x.

a

Find the x-coordinate of the point at which f \left( x \right) has a gradient of 13.

b

Hence, state the coordinates of the point on the curve where the gradient is 13.

37

Consider the function f \left( x \right) = x^{3} - 6 x^{2}.

a

Find the x-coordinates of the points at which f \left( x \right) has a gradient of 495.

b

Hence, state the coordinates of the points on the curve where the gradient is 495.

38

Consider the function f \left( x \right) = x^{3} + 3 x^{2} - 19 x + 2.

a

Find the x-coordinates of the points at which f \left( x \right) has a gradient of 5.

b

Hence, state the coordinates of the points on the curve where the gradient is 5.

39

Consider the function f \left( x \right) = x^{3} + 6 x^{2} - 14 x - 2.

a

Find the x-coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.

b

Hence, state the coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.

40

The curve f \left( x \right) = k \sqrt{x} - 5 x has a gradient of 0 at x = 16.Find the value of k.

41

Consider the function y = 4 x^{2} - 5 x + 2.

a

Find \dfrac{dy}{dx}.

b

Hence, find the value of x at which the tangent to the parabola is parallel to the x-axis.

42

Consider the function f \left( x \right) = 5 x^{2} + \dfrac{4}{x} - 1. The tangent to the curve at the point \left(2, 21\right) makes an angle of \theta with the x-axis. Find \theta, correct to the nearest degree.

43

The curve y = a x^{3} + b x^{2} + 2 x - 17 has a gradient of 58 at the point \left(2, 31\right). Find the values of a and b.

44

The graph of y = a x^{3} + b x^{2} + c x + d intersects the x-axis at \left(2, 0\right), where it has a gradient of 36. It also intersects the y-axis at y = - 28, where the tangent is parallel to the x-axis.

Find the values of a, b, c and d.

45

In the following graph, the line y = \dfrac{x}{10} + b is a tangent to the graph of f \left( x \right) = 6 \sqrt{x} at x = a.

a

Find the value of a.

b

Find the value of b.

x
y
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Outcomes

U34.AoS2.9

apply a range of analytical, graphical and numerical processes (including the algorithm for Newton’s method), as appropriate, to obtain general and specific solutions (exact or approximate) to equations (including literal equations) over a given domain and be able to verify solutions to a particular equation or equations over a given domain

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