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Higher Mathematics > Differentiation > Flashcards

Differentiation Flashcards

1
Q

What is the gradient of the tangent equal to

A

The gradient of the curve at a specified point

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2
Q

Find the gradient and equation of the tangent to y = 3x^2 at x = 1

A

y = 6x - 3

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3
Q

Method for finding equation of a tangent

A
  • differentiate the equation given
  • sub x into differentiated equation to find m
  • find y, state x and y in the form (x, y)
  • find equation of tangent by subbing known data into equation
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4
Q

A parabola has the equation y = x^2 - 4x + 1. Calculate the gradient and equation of the tangent at x = 3.

A

y = 2x - 8

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5
Q

The gradient of a tangent to the curve y = x^4 + 1 is 32.

a) find the point of contact of the tangent
b) find the equation of the tangent

A

y = 32x - 47

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6
Q

Find the interval for which the function y = 3x^2 + 2x - 5 is increasing

A

function is increasing when x > -1/3

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7
Q

Finding interval for when a function is increasing method

A
  • find derivative
  • set derivative > 0
  • solve for x
  • state interval
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8
Q

Finding interval when function is decreasing method

A
  • find derivative
  • set derivative < 0
  • solve for x
  • state interval
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9
Q

If a graph slopes down from left to right, the function is ________

A

Decreasing

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10
Q

If a graph slopes up from left to right, the function is ________

A

Increasing

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11
Q

The derivative of a function is equal to what

A

The gradient

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12
Q

If the derivative > 0, the curve is what

A

Increasing as m is positive

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13
Q

If the curve is increasing, what is the value of the derivative

A

> 0

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14
Q

If the derivative < 0, the curve is what

A

The curve is decreasing as m is negative

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15
Q

Value of derivative when curve is decreasing

A

< 0

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16
Q

Method for determining if function is increasing or decreasing

A
  • find derivative
  • sub x into derivative
  • stare if function is increasing or decreasing
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17
Q

Is y = x^2 - 5x increasing or decreasing when x=4?

A

Increasing at x = 4 since dy/dx > 0.

(= 3)

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18
Q
A
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19
Q
A
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20
Q

Stationary point

A

Graph is neither increasing or decreasing

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21
Q

Gradient at stationary points

A

0

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22
Q

Nature of stationary point

A
  • Maximum turning point
  • Minimum turning point
  • Point of inflection
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23
Q
  • Maximum turning point
  • Minimum turning point
  • Point of inflection
A

Nature of turning point Minimum

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24
Q

Method for finding stationary point

A
  • differentiate and set equal to 0
  • solve to find x
  • nature table
  • sub x into original equation to find y
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25
Q
A

Minimum stationary point at (3, -11)

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26
Q
A

Maximum stationary point at (-1, 4)

Minimum stationary point at (1, -4)

B) increasing

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27
Q
A

Minimum turning point (-1, -2)

POI (0, 0)

Maximum turning points (1, 2)

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28
Q

The gradient of the tangent is equal to what

A

The gradient of the curve at a specific point

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29
Q

The gradient of the curve at a specific point is equal to what

A

The gradient of the tangent

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30
Q

Method for finding the equation of a tangent when given equation and x?

A
  • differentiate equation
  • sub x into derivative to find m
  • find y, state x and y as (x, y)
  • find equation by subbing point and gradient
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31
Q

Find the gradient and equation of the tangent to y = 3x^2 at x = 1

A

y = 6x - 3

point = (1, 3)
m = 6

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32
Q
A

y = 2x - 8

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33
Q

The gradient of a tangent to the curve y = x^4 + 1 is 32.

a) find point of contact
b) find equation

A

(2, 17)

y = 32x - 47

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34
Q

Stationary point

A

Graph that is neither increasing or decreasing

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35
Q

Graph that is neither increasing or decreasing

A

Stationary point

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36
Q

Gradient at stationary points

A

0

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37
Q

Nature of stationary point can be…

A
  • Maximum turning point
  • Minimum turning point
  • Point of inflection
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38
Q
  • Maximum turning point
  • Minimum turning point
  • Point of inflection
A

Nature of stationary point

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39
Q

Method for finding stationary point

A
  • Differentiate and set equation = 0
  • Solve to find x
  • Nature table
  • Sub x into original equation to find y
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40
Q
  • Differentiate and set equation = 0
  • Solve to find x
  • Nature table
  • Sub x into original equation to find y
A

Method for finding stationary point

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41
Q
A

Minimum stationary point at (3, -11)

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42
Q

When do the max and min values of a function occur in a closed interval?

A

The maximum or minimum turning points or at the ends of the interval

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43
Q
A

Minimum TP (3, -18)

Maximum value = 32
Minimum value = -18

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44
Q
A

a) 17 and 189

b) POI (0, 0), minimum TP (2, -64)

c) max value = 189, min value = -64

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45
Q

method for finding max and min values of a function in a closed interval

A
  • sub max value into function
  • sub min value into function
  • find stationary points using derivative
  • state largest and smallest values
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46
Q

Find the roots of 3x^2 - 12 = 0

A

2 and -2

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47
Q

What is differentiating finding

A

The rate of change

48
Q

What finds the rate of change

A

Differentiate

49
Q

y when differentiated…

50
Q

How to differentiate

A

Multiply the coefficient by the power and decrease the power by 1

56
Q
59
Q

a0 = ?

60
Q

a^-m = ?

65
Q

Write with positive indices:

66
Q

Write with positive indices:

67
Q

Write in the form ax^n:

68
Q

Write in the form ax^n:

69
Q

Write in the form ax^n:

70
Q

Write in the form ax^n:

71
Q

Simplify:

72
Q

Simplify:

73
Q

Simplify:

74
Q

What to ensure before differentiating

A
  • no x term on the bottom of a fraction
  • change square root to fractional indices
75
Q

How to subtract ‘1’ from a fraction

A

Top - bottom

76
Q

3/2 - 1

77
Q

6/5 - 1

78
Q

4/7 - 1

79
Q

-4/5 - 1

80
Q

-7/3 - 1

81
Q

-1/2 - 1

82
Q

Differentiate 1/x^3

83
Q

Differentiate:

84
Q

Differentiate:

85
Q

Find f ’(9) for the following:

86
Q

Differentiate:

87
Q

Displacement

A
  • Distance from the origin at time
  • A distance with direction
  • Units: metres
88
Q
  • Distance from the origin at time
  • A distance with direction
  • Units: metres
A

Displacement

89
Q

Velocity

A
  • Rate of change of displacement with respect to time
  • A speed with direction
  • Units: m/s
90
Q
  • Rate of change of displacement with respect to time
  • A speed with direction
  • Units: m/s
91
Q

Acceleration

A
  • The rate of change of velocity with respect to time
  • The rate at which an object changes its speed
  • Units: m/s^2
92
Q
  • The rate of change of velocity with respect to time
  • The rate at which an object changes its speed
  • Units: m/s^2
A

Acceleration

93
Q

Relationship between displacement, velocity, and acceleration

A

Displacement
V
Velocity
V
Acceleration

By differentiation

94
Q

A car is travelling along a straight road. The distance, x metres, travelled in t seconds is x = 10t - 5t^2

Find the velocity when t = 0.5

95
Q

A car travelling has a velocity of v = 10 + 6t^2 - t^3

Find the acceleration when t = 3

96
Q
A

a) velocity = -4 + 3t^2
Acceleration = 6t

b) displacement = -2m
velocity = -1m/s
acceleration = 6 m/s^2

c) 2 seconds

97
Q

Name, equation and degree of the following graph:

A

Constant

Y = a

Degree = 0

98
Q

Name, equation and degree of the following graph:

A

Linear graph

Y = mx + c

Degree = 1

99
Q

Name, equation and degree of the following graph:

100
Q

Name, equation and degree of the following graph:

101
Q

Name, equation and degree of the following graph:

102
Q

What happens to the graph when you differentiate a function

A

It’s degree is reduced by one. This corresponds to a graph

103
Q

f (x) gradient -> f’(x) graph relationships

A

+ve -> above x axis
zero -> on x axis
-ve -> below x axis

107
Q

Express with +ive indices:

108
Q

Write with +ive indices:

109
Q

Write with +ive indices:

110
Q

To sketch a curve, what do we need to know?

A
  • it’s stationary points and their nature
  • where it crosses the x axis (set y=0)
  • where it crosses the y axis (set x = 0)
111
Q
A

Min TP (-1, -2)
Max TP (1, 2)
Graph passes y axis at (0, 0)
Graph crosses x axis at (/3, 0) and (-/3, 0)

112
Q
A

POI at (0, 0)

Max TP at (2, 16)

y intercept at (0, 0)

x intercept at (0, 0) and (8/3, 0)

Higher Mathematics (11 decks)

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