Basic Skills in Higher Mathematics

1
Basic Skills in Higher Mathematics
Mathematics 1(H)Outcome 3
Robert GlenAdviser in Mathematics
2
Mathematics 1(Higher)
Outcome 2 Use basic differentiation
Differentiation
dydx
f? (x)
3
Mathematics 1(Higher)
Outcome 2 Use basic differentiation
PC Index
Click on the PC you want
PC(a) Basic differentiation
dydx
f? (x)
PC(b) Gradient of a tangent
PC(c) Stationary points
4
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basicdifferentiation
dydx
f? (x)
5
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
Click on the section you want
1 Simple functions
2 Simple functions multiplied by a constant
3 Negative indices
4 Fractional indices
5 Negative and fractional indices with
constant
6 Sums of functions (simple cases)
7 Sums of functions (negative indices)
8 Sums of functions ( algebraic fractions)
6
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
1 Simple functions
dydx
f? (x)
7
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Some examples
Every function f(x) has a related function called
the derived function.The derived function is
written f ?(x) f dash x
f ?(x)
f(x)
x3
3x2
x6
6x5
The derived function is also called the
derivative.
x10
10x9
x
( x1)
1
To find the derivative of a functionyou
differentiate the function.
3
0
( 3x0)
8
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Rule No. 1 for differentiation
If f(x) xn , then f ?(x) nxn -1
9
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x) x4
1 f ?(x) 4x3
2 f(x) x5
2 f ?(x) 5x4
3 g(x) x8
3 g?(x) 8x7
4 h(x) x2
4 h?(x) 2x
5 f ?(x) 12x11
5 f(x) x12
6 f ?(x) 1
6 f(x) x
7 f ?(x) 0
7 f(x) 5
10
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
2 Simple functions multiplied by a
constant
dydx
f? (x)
11
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
f(x)
f ?(x)
2x3
6x2
2 ? 3x2
3x6
18x5
3 ? 6x5
2x10
2 ? 10x9
20x9
5x
5 ? 1
5
12
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Rule No. 2 for differentiation
If f(x) axn , then f ?(x) anxn -1
13
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x) 3x4
1 f ?(x) 12x3
2 f(x) 2x5
2 f ?(x) 10x4
3 g(x) ½ x8
3 g?(x) 4x7
4 h(x) 5x2
4 h?(x) 10x
5 f ?(x) 3x11
5 f(x) ¼ x12
6 f ?(x) 8
6 f(x) 8x
6 f ?(x) 0
7 f(x) 10
14
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
3 Negative indices
dydx
f? (x)
15
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Rule No. 1 for differentiation
If f(x) xn , then f ?(x) nxn -1
16
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
1 f(x)
-3
-4
x -3
-1
x -4
f? (x) -3
x ?
-4
-3
-2
-5
-1
0

NoteThis is an example of using Rule No.1with
a negative index.
17
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
2 f(x)
-1
-2
x -1
-1
f? (x) -1
x -2
x ?
-2
-1
0

NoteThis is an example of using Rule No.1with
a negative index.
18
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x) x -2
1 f ?(x) -2x -3
2 f(x) x -4
2 f ?(x) -4x -5
3 g(x)
3 g?(x) -5x -6
4 h?(x) -1x -2
4 h(x)
5 f(x)
5 f ?(x) -10x -11
19
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
4 Fractional indices
dydx
f? (x)
20
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Rule No. 1 for differentiation
If f(x) xn , then f ?(x) nxn -1
21
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
3 f(x)
f? (x)
x ?
-1

-1
0
1
NoteThis is an example of using Rule No.1with
a fractional index.
22
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
4 f(x)
x ?
f? (x)
-1

-1
0
1
NoteThis is an example of using Rule No.1with
a fractional index.
23
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x)
1 f ?(x)
2 f(x)
2 f ?(x)
3 g?(x)
3 g(x)
4 h(x)
4 h?(x)
24
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
5 Negative and fractional indices
with constant
dydx
f? (x)
25
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Rule No. 2 for differentiation
If f(x) axn , then f ?(x) anxn -1
26
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
5 f(x)
-3
-4
2x -3
-1
x -4
f? (x) -6
x ?
-4
-3
-2
-5
-1
0

NoteThis is an example of using Rule No.2with
a negative index.
27
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
6 f(x)
-1
-2
5x -1
-1
f? (x)
x -2
x ?
-5
-2
-1
0

NoteThis is an example of using Rule No.2with
a negative index.
28
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x) 3x -2
1 f ?(x) -6x -3
2 f(x) 5x -4
2 f ?(x) -20x -5
3 g(x)
3 g?(x) -10x -6
4 h?(x) -2x -2
4 h(x)
5 f(x)
5 f ?(x) -30x -11
29
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
7 f(x)
f? (x)
x ?
6 ? ½ 3
3
-1

-1
0
1
NoteThis is an example of using Rule No.2with
a fractional index.
30
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
8 f(x)
f? (x)
x ?
9 ? 2/3 6
-1

-1
0
1
NoteThis is an example of using Rule No.2with
a fractional index.
31
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x)
1 f ?(x)
2 f(x)
2 f ?(x)
3 g?(x)
3 g(x)
4 h(x)
4 h?(x)
32
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
When a function is given in the form of an
equation, the derivative is written in the form

f ?(x)
Examples
2 f(x) 4x -3 f ?(x) -12x -4
1 f(x) 3x2 f ?(x) 6x
y 3x2
y 4x -3
-12x -4
6x
33
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
6 Sums of functions (simple cases)

dydx
f? (x)
34
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
g(x)
h(x)
k(x)
1
2
f(x) 5x2 - 3x 1
g(x) (x 3)(x - 2)
x2 x - 6
f? (x)
- 3
0
10x
10x - 3
g? (x)
2x
1
35
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Rule No. 3 for differentiation
If f(x) g(x) h(x) k(x) ... , then f
?(x) g ?(x) h ?(x) k ?(x) ..
36
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x) x2 7x - 3
1 f ?(x) 2x 7
2 f(x) 3x2 - 4x 10
2 f ?(x) 6x - 4
3 g(x) x(x2 x - 5)
3 g?(x) 3x2 2x - 5
4 h(x) 4x3 - 10x2
4 h?(x) 12x2 - 20x
5 f ?(x) 35x4 - 35x6
5 f(x) x5(7 - 5x2)
6 f ?(x) 3x2 2x 1
6 f(x) x3 x2 x 1
7 f ?(x) x3 x
7 f(x) ¼ x4 ½ x2
37
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
7 Sums of functions (negative
indices)
dydx
f? (x)
38
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
1
2
- 3x
-
f(x)
-
y
1

2x -2
5x -1
- 3x
-
-
x -2
x -1
1

f? (x)
-4x -3
- 3
5x -2
x -2
0

-2x -3


- 3
-

-


39
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x)
1 f ?(x)
-
- 2
- 2x
x2 - x
2 y
-
2
2x - 1
-
3x2
3 y
-
3


6x
4 g(x)
-
-
-

4 g?(x)

40
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
PC(a) - Basic differentiation
8 Sums of functions (algebraic
fractions)
dydx
f? (x)
41
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
More examples
1
2
f(x)
y
-




-
1
3x -1
- 2x -2
x
5
- 3x -1
4x -3
0
- 3x -2
f ? (x)
1
3x -2


0
-


1
42
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(a) Differentiate a function
reducible to a sum of powers of x
Differentiate each of these functions.
Here are the answers
1 f(x)
1 f ?(x) 2x - 2
2 y
2
1

3 g?(x) 1
3 g(x)
-


4 y
4
43
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
PC(b) -Gradientof a tangent
dydx
f? (x)
44
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
Rule No. 4 for differentiation
The gradient of a tangent to thecurve y f (x)
is f? (x) or
45
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
Example 1
y x2 - 5x
Find the gradient of the tangentto the curve y
x2 - 5x at each of the points A and B.
?
B (-1, 6)
2x
- 5
Gradient of tangent
x
m -7
?
A (3, -6)

2(3)
- 5
When x 3 ,
1
A
m 1

B
When x -1,
2(-1)
-7
- 5
So gradient of tangent at A is 1and gradient of
tangent at B is -7
46
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
Example 1 (continued)
y x2 - 5x
Find the equation of the tangentto the curve y
x2 - 5x at each of the points A and B.
?
B (-1, 6)
x
m 1
Tangent at A
m -7
?
A (3, -6)
Point on line is (3, -6)
Equation of tan is y - b m(x - a)
m 1
y
- (-6)

1
(x - 3)
y x - 9

6
x - 3
y
x
- 9
y
47
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
Example 1 (continued)
y x2 - 5x
Find the equation of the tangentto the curve y
x2 - 5x at each of the points A and B.
?
B (-1, 6)
x
m -7
Tangent at B
m -7
?
A (3, -6)
Point on line is (-1, 6)
y -7x - 1
Equation of tan is y - b m(x - a)
m 1
y -
6
-7
(x - (-1))

y x - 9
NB 3 negatives
- 7
y - 6
-7x
-7x
- 1
y
48
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
Example 2
y x3 - 5x 3
Find the gradient of the tangentto the curve y
x3 - 5x 3at each of the points P and Q.
Q (-2, 5)
?
3x2 - 5
Gradient of tangent
x
P (1, -1)
?
m 7

3(1)2 - 5
When x 1 ,
-2
P
m -2

Q
When x -2,
3(-2)2 - 5
7
So gradient of tangent at P is -2and gradient of
tangent at Q is 7
49
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
Example 2 (continued)
y x3 - 5x 3
Find the equation of the tangentto the curve y
x3 - 5x 3at each of the points P and Q.
Q (-2, 5)
?
x
m -2
Tangent at P
P (1, -1)
?
m 7
Point on line is (1, -1)
Equation of tan is y - b m(x - a)
m -2
y -
(-1)

-2
(x - 1)
y -2x 1

y
1
-2x
2
y
-2x
1
50
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
Example 2 (continued)
y x3 - 5x 3
Find the equation of the tangentto the curve y
x3 - 5x 3at each of the points P and Q.
Q (-2, 5)
?
x
m 7
Tangent at Q
P (1, -1)
?
m 7
y 7x 19
Point on line is (-2, 5)
Equation of tan is y - b m(x - a)
m -2
y -
y -2x 1
5

7
(x - (-2))
y - 5
7x
14
y
7x
19
51
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
1 Find the gradient of the tangent to the curve
y x2 5 at each of the points A and B.Find
the equation of each tangent.
y x2 5
B(-4, 21)
?
?
A (2, 9)
x
2x
Answers
Tangent at Am 4Equation is y 4x 1
Tangent at Bm -8Equation is y -8x - 11
52
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
2 Find the gradient of the tangent to the curve
y (3 x)(3 - x) at each of the points E and
F.Find the equation of each tangent.
y (3 x)(3 - x)
E (2, 5)
?
F
?
x
-2x
Answers
Tangent at Em -4Equation is y -4x 13
Tangent at F (-3, 0)m 6Equation is y 6x
18
53
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(b) Determine the gradient of a
tangent to a curve by differentiation
y
3 Find the gradient of the tangent to the curve
y x4 - 5x2 4 at each of the points A , B and
C.Find the equation of each tangent.
y x4 - 5x2 4
A (0, 4)
?
C(1, 0)
B(-2, 0)
?
?
x
4x3 - 10x
Answers
Tangent at Am 0Equation is y 4
Tangent at Bm -12Equation is y -12x - 24
Tangent at Cm -6Equation is y -6x 6
54
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
PC(c) -Stationary points
dydx
f? (x)
55
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
Rule No. 5 for differentiation
Stationary points occur when f? (x) 0 or
0
56
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
To determine the nature of a stationary point,
find out the gradient of the tangent before and
after the stationary point.
y
0
x
before
after
?
-

0
after
before
?
?
ve
- ve
slope
x
This is a maximum turning point
57
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
To determine the nature of a stationary point,
find out the gradient of the tangent before and
after the stationary point.
y
x
before
after
-
0

before
after
- ve
ve
slope
?
?
x
This is a minimumturning point
?
0
58
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
To determine the nature of a stationary point,
find out the gradient of the tangent before and
after the stationary point.
x
x
before
before
after
after
-

-
0
0

slope
slope
This is a minimumturning point
This is a maximum turning point
59
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
Example 1
Find the stationary point on the curve with
equation y x2 - 6x 5.
Using differentiation determine its nature.
y x2 - 6x 5
When x 3,

2x
- 6
(3)2
y
5
-6(3)
For stationary values,
0
9
-18
5
-4
ie 2x - 6 0
x 3
So a stationary point occurs at (3, -4).
60
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
Example 1 (continued)
Find the stationary point on the curve with
equation y x2 - 6x 5.
Using differentiation determine its nature.
x
3 -
3
3
A stationary point occurs at (3, -4).
-
0

2x - 6
slope
(3, -4) is a minimum turning point.
61
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
Example 2
Find the stationary points on the curve with
equation y 1/3x3 - x2 - 3x 10.
Using differentiation determine their nature.
When x 3,
x2
- 2x

- 3
y
-32
-3(3)
1/3(3)3
10

When x -1,
0
For stationary values,
1/3(-1)3
-(-1)2
-3(-1)
10
y

ie x2 - 2x - 3 0
So stationary points occur at (3, 1) and(-1,
)
x - 3
x 1
( )( ) 0
x
x
3,
-1
62
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
Example 2
Find the stationary points on the curve with
equation y 1/3x3 - x2 - 3x 10.
Using differentiation determine their nature.
x
3 -
3
3
A stationary point occurs at (3, 1).
-
0

slope
x2

- 2x
- 3
(3, 1) is a minimum turning point.
63
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
Example 2
Find the stationary points on the curve with
equation y 1/3x3 - x2 - 3x 10.
Using differentiation determine their nature.
x
-1 -
-1
-1
A stationary point occurs at (-1, ).

-
0
slope
x2

- 2x
- 3
(-1, ) is a maximumturning point.
64
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
Example 2
The graph of the curve y 1/3x3 - x2 - 3x
10looks something like this
Maximumturning point.
y
12
?
(-1, )
?
10
8
Minimum turning point.
y 1/3x3 - x2 - 3x 10
6
4
(3, 1)
2
?
x
0
1
-4
2
3
4
-3
5
-5
-2
-1
65
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
1 Find the stationary point on the curve y x2
3 .
Using differentiationdetermine its nature.
Solution
When x 0,
y
(0)2 3

2x
0 3
0
For stationary values,
3
ie 2x 0
So a stationary point occurs at (0, 3).
x 0
66
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
1 Find the stationary point on the curve y x2
3 .
Using differentiationdetermine its nature.
x
Solution (continued)
0 -
0
0
-
0

A stationary point occurs at (0, 3).
slope

2x
(0, 3) is a minimum turning point.
67
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
2 Find the stationary point on the curve y 16
- x2 .
Using differentiationdetermine its nature.
Solution
When x 0,
y
16 - (0)2

-2x
16 - 0
0
For stationary values,
16
ie -2x 0
So a stationary point occurs at (0, 16).
x 0
68
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
2 Find the stationary point on the curve y 16
- x2.
Using differentiationdetermine its nature.
x
Solution (continued)
0 -
0
0

0
-
A stationary point occurs at (0, 16).
slope

-2x
(0, 16) is a maximumturning point.
69
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
3 Find the stationary point on the curve y (x
5)(x - 3) .
Using differentiationdetermine its nature.
Solution
When x -1,
y x2
2x
- 15
y
(-1)2
- 15
2(-1)
1
- 2
- 15

2x
2
-16
For stationary values,
0
So a stationary point occurs at (-1, -16).
ie 2x 2 0
x -1
70
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
3 Find the stationary point on the curve y (x
5)( x - 3) .
Using differentiationdetermine its nature.
x
Solution (continued)
-1 -
-1
-1
-
0

A stationary point occurs at (-1, -16).
slope

2x 2
(-1, -16) is a minimum turning point.
71
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
4 Find the stationary points on the curve y
1/3 x3 - x2 - 15x.
Using differentiationdetermine their nature.
Solution
When x 5,
y

1/3(5)3
- (5)2
- 15(5)

x2
- 2x
- 15
When x -3,
For stationary values,
0
y
1/3(-3)3
-(-3)2

-15(-3)
27
ie
( )( ) 0
x - 5
x 3
So stationary points occur at (-3, 27) and(5,
)
x
x
5,
-3
72
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
4 Find the stationary points on the curve y
1/3 x3 - x2 - 15x.
Using differentiationdetermine their nature.
x
Solution (continued)
-3 -
-3
-3

-
0
A stationary point occurs at (-3, 27).
slope
- 2x
- 15

x2
(-3, 27) is a maximum turning point.
73
Mathematics 1(Higher)
Outcome 3 Use basic differentiation
PC(c) Determine the coordinates of
the stationary points on a curve.
4 Find the stationary points on the curve y
1/3 x3 - x2 - 15x.
Using differentiationdetermine their nature.
x
Solution (continued)
5 -
5
5
-
0

A stationary point occurs at (5, ).
slope
- 2x
- 15

x2
(-3, ) is a minimum turning point.